Volume 3 Number 11 Electronic Journal of Theoretical Physics ISSN 1729-5254 EJTP Editors Ammar Sakaji http://www.ejtp.com Ignazio Licata June, 2006 E-mail:info@ejtp.com Volume 3 Number 11 Electronic Journal of Theoretical Physics ISSN 1729-5254 EJTP Editors Ammar Sakaji http://www.ejtp.com Ignazio Licata June, 2006 E-mail:info@ejtp.com Editor in Chief A. J. Sakaji EJTP Publisher P. O. Box 48210 Abu Dhabi, UAE info@ejtp.com Info@ejtp.info Editorial Board Co-Editor Ignazio Licata, Foundations of Quantum Mechanics Complex System & Computation in Physics and Biology IxtuCyber for Complex Systems Sicily – Italy editor@ejtp.info ignazio.licata@ejtp.info ignazio.licata@ixtucyber.org Wai-ning Mei Condensed matter Theory Physics Department University of Nebraska at Omaha, Omaha, Nebraska, USA e-mail: wmei@mail.unomaha.edu physmei@unomaha.edu Tepper L. Gill F.K. 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Cardano" Via Alfredo Casella 3 00013 Mentana RM - ITALY Applied Mathematics Department of Mathematics Naval Postgraduate School 1141 Cunningham Road Monterey, CA 93943, USA e-mail: solitone@yahoo.it e-mail: byneta@gmail.com Haret C. Rosu A. Abdelkader Advanced Materials Division Institute for Scientific and Technological Research (IPICyT) Camino a la Presa San José 2055 Col. Lomas 4a. sección, C.P. 78216 San Luis Potosí, San Luis Potosí, México Experimental Physics Physics Department, AjmanUniversity Ajman-UAE e-mail: atef28@gmail.com atef@ejtp.info e-mail: hcr@titan.ipicyt.edu.mx Leonardo Chiatti Medical Physics Laboratory ASL VT Via S. Lorenzo 101, 01100 Viterbo (Italy) Tel : (0039) 0761 236903 Fax (0039) 0761 237904 e-mail: fisica1.san@asl.vt.it Zdenek Stuchlik Relativistic Astrophysics Department of Physics, Faculty of Philosophy and Science, Silesian University, Bezru covo n´am. 13, 746 01 Opava, Czech Republic e-mail: Zdenek.Stuchlik@fpf.slu.cz chiatti@ejtp.info Copyright © 2003-2006 Electronic Journal of Theoretical Physics (EJTP) All rights reserved Table of Contents No 1 Articles Non-Minimal Coupling Effects of the Ultra-Light Particles on Photons Velocities in the Radiation Dominated Era of the Universe. Page 1 El-Nabulsi Ahmad Rami 2 A Toy Model of Financial Markets 11 J. P. Singh and S. Prabakaran 3 Rayleigh process and matrix elements for the onedimensional harmonic oscillator 29 J.H. Caltenco, J.L. López-Bonilla, and J. Morales 4 Identical synchronization in chaotic jerk dynamical systems 33 Vinod Patidar and K. K. Sud 5 Second Order Perturbation of Heisenberg Hamiltonian for Non-Oriented Ultra-Thin Ferromagnetic Films 71 P. Samarasekara 6 Frameable Processes with Stochastic Dynamics 85 Enrico Capobianco 7 Ab-initio Calculations for Forbidden M1/E2 Decay Rates in Ti XIX ion 111 A. Farrag 8 Some Properties of Generalized Hypergeometric Thermal Coherent States 123 Dusan Popov 9 Space-Filling Curves for Quantum Control Parameters 133 Fariel Shafee 10 The Spectrum of the Lagrange Velocity Autocorrelation Function in Confined Anisotropic Liquids 143 Sakhnenko Elena I and Zatovsky Alexander V. 11 On the Quantum Correction of Black Hole Thermodynamics 151 Kourosh Nozari and S. Hamid Mehdipour 12 A Graphic Representation of States for Quantum Copying Sara Felloni and Giuliano Strini 159 EJTP 10 (2006) 1–10 Electronic Journal of Theoretical Physics Non-Minimal Coupling Effects of the Ultra-Light Particles on Photons Velocities in the Radiation Dominated Era of the Universe El-Nabulsi Ahmad Rami ∗ Plasma Application Laboratory, Department of Nuclear and Energy Engineering and Faculty of Mechanical, Energy and Production Engineering, Cheju National University, Ara-dong 1, Jeju 690-756, Korea Received 6 September 2005 , Accepted 27 November 2005, Published 25 May 2006 Abstract: The effect of the ultra-light masses of the order of the Hubble constant, implemented in Einstein’s field equations from non-minimal coupling and supergravities arguments, on photons velocities in the radiation dominated epoch of the Universe within the framework of non-minimal interaction of electromagnetic fields with gravity is developed and discussed in details. c Electronic Journal of Theoretical Physics. All rights reserved.  Keywords: Non-minimal coupling, ultra-light masses, effective cosmological constant, photons velocity. PACS (2006): 04.40 Nr, 98.80 -k, 95.30 SI 1. Introduction A standard result of Einstein’s gravity is that massless particles, in particular photons, move at light celerity ‘c’. A question worth examining is whether the velocity of massless photons is shifted when these later propagate in exotic background filled of ultra-light particles (ULP) of tiny masses in the order of the Hubble constant (m ≈ H). This could have important cosmological and astrophysical implications. In fact, the possibility of shifting photon propagation (SPP) in gravitational fields (or non-trivial topologies) is an interesting prediction of quantum field theory in curved space-time. It appears that photon propagation may depend on their direction and polarisation, travel with speeds exceeding the normal speed of light ‘c’ [1]. It is a quantum effect induced by vacuum ∗ atomicnuclearengineering@yahoo.com 2 Electronic Journal of Theoretical Physics 10 (2006) 1–10 polarisation (allowing the photons to exist as a virtual e+ e− pair so that at the quantum level it is characterized by the Compton wavelength of the electron) and implies that the Principle of Equivalence does not hold for interacting quantum field theories such as QED. The propagation of photon in Schwarzschild, Robertson-Walker, gravitational wave, de Sitter backgrounds, charged black hole were done and remarkable results were discovered [2]. In each case (except the totally isotropic de Sitter space-time) it was possible to find directions and polarisations for which the photon velocity exceeds ‘c’. Generalization to neutrino propagation in a Robertson-Walker metric using the WeinbergSalam model was done in [3]. In a gravitational field, the photon propagation is sensitive to an anisotropic space-time curvature and may depend on this later [4,5]. Recently, a series of papers has appeared in which the light velocity varies in the early Universe and this solves the horizon, monopole and the flatness problems in standard cosmology [6,7,8,9]. In this work, we will investigate further the consequences of non-minimal coupling on light velocity in the presence of ultra-light particles. 2. Non-minimal Coupling, Supergravities Arguments and Einstein Fields Equations We start with the non-minimal interaction of electromagnetic fields with gravity in the √ following form L̃ = gξRμν Fαμ F νμ , ξ being the coupling constant, Rμν the Riemann tensor, g the metric scalar and Fμν = ∂μ Aν − ∂ν Aμ is the electromagnetic strength (Aμ is the vector potential ). Terms like RF 2 are neglected here. In this way, the field equations read [10,11,12]: Dν [F μν − 2ξ (Rαμ F αν − Rαν F αμ )] = 0 where Dν is the covariant derivative. Let us now restrict our attention to the form of the Riemann tensor we will choose for this work. In a recent paper [13], we introduced, for some scalar field , a non-minimal coupling between the scalar curvature and the density of √ the scalar field in the following form L = −ξ gRφ∗ φ, ξ = 1/6 . R is the scalar curvature and is the complex conjugate of . From a view point of quantum field theory in curved space-time, it is natural to consider such a non-minimal coupling. In fact, the conformal case results in an extension of the property of conformal invariance for massless fields, which is attractive from physical point of view. This parameter describes the strength of the coupling between the curvature of spacetime and the inflation. Minimal coupling corresponds to ξ = 0. It was shown that in this case and for a particular scalar negative complex potential field V (φφ∗ ) = 3/4m2 (ωφ2 φ∗2 − 1),ω being a tiny parameter inspired from supergravity inflation theories, ultra-light masses ‘m’ are implemented naturally in Einstein field equations (EFE), leading to a cosmological constant ‘Λ’ in accord with observations2 . In matter-free background, the scalar curvature was found to be =4Λ̄ 2 It has been argued that a non-minimal coupling term-generated by quantum corrections-is to be expected whenever the space-time curvature is large; in most theories that describe inflationary scenarios, it turns out that a value of ξ different from zero is unavoidable. As a matter of fact, it seems sensible to consider an explicit non-minimal coupling in the supergravities inflationary paradigm. Electronic Journal of Theoretical Physics 10 (2006) 1–10 3 where Λ̄ = Λ−3/4m2 is the effective cosmological constant 3 (in natural units, m ≈ H/c2 where ‘’ is the Planck constant and ‘c’ being the celerity of light). As a result, one candidate Einstein field equations is( = c = 1): 1 Gμν ≡ Rμν − gμν R = −Λ̄gμν 2 (1) Λ̄ = Λ − 3m2 /4 ≡ Λ1 + Λ2 is the effective cosmological constant (Λ2 = −3λ̄−2 C where λC = /mc is the Compton wavelength in natural units). Remark that for Λ1 = 0, the scalar curvature is negative and the space-time is not Minkowskian. Other field equations exist also but correspond only for p = ρ/3 (radiation era):    1 3m2 Rμν − gμν R + Λgμν = −8πG p + ρ + (2a) uμ uν + pgμν 2 8πG Rμν 1 − gμν R = −8πG 2 = ≡ ≡ ≡  3m2 p+ρ+ 8πG   Λ uμ uν + p + 8πG    Λ gμν −8πG (p + ρ̄) uμ uν + p + 8πG   Λ gμν + tμν (m) −8πG Tμν (p, ρ) + 8πG −8πG [Tμν (p, ρ) + Tμν (Λ, m)]  −8πG Tμν   gμν  (2b) (2c) (2d) (2e) (2f) p and ρ are the pressure and density of matter,ρ̄ = ρ + 3m2 /8πG, tμν = 3m2 /8πGuμ uν and: (3) Tμν (p, ρ) = (p + ρ) uμ uν + pgμν  1  2 Λ gμν = (4) 3m uμ uν + Λgμν 8πG 8πG Contracting equations (2) with g μν using g μν uμ uν = −1 yields of course. In this way, the ultra-light masses and the cosmological constant are parts of the matter contents of the Universe rather than geometrical entities. The radiative field equations (RFE) (2a,b,c,d,e,f) are identical to that of Einstein standard ones but with an additional energy density ρm = 3m2 /8πG(m ≤ H). One can also refer to equation (4) as the stress-energy tensor of vacuum and light particles, which is a ”microscopic stress-energy tensor ”. In fact, the conservation law holds and we have:  Tμν ≡ ∇ν Tμν (p, ρ) + ∇ν Tμν (Λ, m) = 0 (5) ∇ν Tμν (Λ, m) = tμν + When Tμν (p, ρ) = 0, the microscopic stress-energy tensor will behave as the macroscopic one if we assume that: P (Λ, m) ≡ PΛ = Λ/8πG (6) 3 In [13], 8πG ≡ κ was set equal to unity. 4 Electronic Journal of Theoretical Physics 10 (2006) 1–10 3m2 − Λ (7) 8πG That is, in the microscopic version, if Λ = 0, P = 0 but the density is positive. While for Λ > 3m2 , the pressure is positive and the density is negative. If 0 < Λ < 3m2 , then both the pressure and the density are positive. Before treating the non-minimal coupling scenario, we will discuss briefly the implications of equations (2) in standard cosmology. For this, we consider a homogenous and isotropic Universe in the radiation dominated epoch described by Friedman-RobertsonWalker line element with scale factor a (t)[14]. The radiative field equations read: ρ (Λ, m) = k 8πGρ Λ ȧ2 + + m2 + 2 = 2 a a 3 3 (8) ä 8πGρ Λ m2 =− + − (9) a 3 3 2 k = −1, 0, +1 is the curvature constant for open, flat or closed space-time and dim (Λ) = dim (m2 ) = length−2 . If the cosmological constant and the ultra-light masses are assumed to be constant with time, then from the energy conservation law: ρ ∝ a−4 . For zero density, 2Λ > (<) 3m2 and the acceleration of the Universe accelerate (decelerate) with time(ä > (<) 0). Combining equation (8) and (9), we can eliminate the density factor and obtains:   2 3m2 k 2Λ m2 ä ȧ2 2 + 2+ 2 = + = Λ 1+ (10) ≡ Λ̂ a a a 3 2 3 4Λ 3 where Λ̂ = Λ (1 + 3m2 /4Λ). If we restrict ourselves to spatially flat universes (k = 0) and we use the definition da H ≡ ȧ/a = H0 adT where time is assumed to be measured in units of Hubble times T ≡ H0 t, then: H2 dH 2Λ̂ +2 = (11) dT H0 3H0 Assuming that Λ̂ ∝ T −α , that is the cosmological constant and the ultra-light masses decrease with time, this model was found to be singular but can significantly be older than models with constant Λ and m2 [15,16,17,18,19]. For m2 << Λ, Λ̂ → Λ, while for m2 >> Λ, Λ̂ → 3/4m2 . If for instance, we assume that m2 = β/a2 and Λ = δ ȧ2 /a2 + ηä/a, β, δ, η are constants [20,21], than from equation (10)4 :       ä 2δ 2η ȧ2 β 1 =0 (12) 1− + 2 1− + k− a 3 a 3 2 a2 which gives: ȧ2 = 4 2(3−2δ) 3 (β − 2k) + Da− 3−2η , D = const. 2 (3 − 2δ) (13) The fact that the two terms Λ and m2 play the role of two cosmological constant in the theory, we have the freedom to choose Λ = δ1 ȧ2 a2 + η1 ä/a + β1 a2 and m2 = δ2 ȧ2 a2 + η2 ä/a + β2 a2 where δ1,2 , η1,2 , β1,2 are constants. In this work, we simplified our assumptions just to have at the beginning a simple idea about the effects of the ultra-light masses in the theory. Electronic Journal of Theoretical Physics 10 (2006) 1–10 5 For D = 0 which corresponds to singular solutions, one finds for flat space-time(k = 0): a= 3β t 2 (3 − 2δ) (14) 2 (3 − 2δ) 3t2 (15) where δ < 3/2, β > 0. In this way : m2 = Λ= δ t2 (16) From equation (8), we find: 3 (δ − 1) (17) 8πGt2 with 1 < δ < 3/2. In this way, we don’t have an inflationary phase and no horizon problem appears. From the above equations, we see that the ultra-light masses, the cosmological constant and the density are independent of the value of β and η. The Hubble parameter is H = ȧ/a = 1/t and the density matter of the Universe is given by Ωr = ρ/ρc = δ − 1 < 1/2 where ρc = 3H 2 /8πG is the critical density. The deceleration parameter is q ≡ −äa/ȧ2 = 0. The density parameter due to vacuum contribution is ΩΛ = Λ/3H 2 = δ/3 and that due to ultra-light particles contributions is Ωm = m2 /H 2 = 2 (3 − 2δ)/3. In this way ΩT otal = Ωr + Ωm + ΩΛ = 1 as required by inflation [22]. The ultra-light particles than contribute to the total energy density and their masses decrease as inverse to time. Note from equations (15) and (16) that Λ = 3δm2 /2 (3 − 2δ) < 9m2 /4. Finally, note that when the ‘Λ’ and ‘m2 ’ terms dominate the dynamics of equation (8) with the assumption that the Universe undergoes a long period of evolution during which the celerity of light changes as c = c0 an , c0 , n =constants [8]:   Λc2 (t) ȧ2 Λ 2 2 2 2n = + m c (t) → + m c2n (18) 0 a a2 3 3 ρ= So at large times, we have a ∝ t−1/n and it was found in [8] that for negative ”n”, there is a solution to the quasi-lambda problem. In order to have a very simple idea about the role of the ultra-light masses in the theory, we suppose that the space-time is flat, that is k = 0 with the following behavior of the ultra-light masses m2 = β/a2 and the cosmological constant Λ = δ/a2 , β, η=constants (see footnote 4 ) [23,24,25]. In this case, when ‘Λ’ and ‘m2 ’ terms dominate at large times the dynamics of equation (8):   Λc2 (t) δ 2 2n−2 ȧ2 2 2 = (19) + m c (t) = β + ca a2 3 3 0 That is a ∝ t−1/n−1 and from [8,26], it is required that n < 0 and c = c0 t−n/n−1 . In summary, m2 ∝ t2/n−1 and as a result mc ∝ 1/t. Another way to study shifting and time-varying photons velocities is by using the non-minimal coupling of electromagnetic fields and gravity. 6 3. Electronic Journal of Theoretical Physics 10 (2006) 1–10 Varying Photons Velocities From Non-Minimal Coupling Following [10,11], we admit the existence of a surface S represented by φ (x) = 0. The wavenumber of the photon trajectories is given by the gradient of its phase kλ = ∇λ φ where the Faraday tensor vanishes at its hypersurface, that is (Fμν )S = 0. Its derivative defines a function φμν such that: (∂λ Fμν )S = (Dλ Fμν )S = kλ φμν (20) As a consequence, equation (1) takes the form: [φμν − 2ξ (Rαμ φαν − Rαν φαμ )] kν = 0 (21) In the radiation dominated era, it follows that:     4ρ ρ ρm Λ μν αν μ + ρ m u uν + + − φ kν − 2ξ φ . (−χ) δνμ 3 3 2 χ     ρ ρm Λ 4ρ ν αμ + ρm u uα + + − kν = 0 + −φ . (−χ) δαν 3 3 2 χ (22) where χ = 8πG ( = c = 1). For simplicity, we let k0 = kμ uμ and we use the antisymmetric fact of (φμν − φνμ = 2φμν ) as well as Maxwell equations: φμν kλ + ϕνλ kμ + φλμ kν = 0 (23) By contracting by k λ the last equation, equation (23) reduces to: φμν kν = −2ξχ 4ρ + ρm  3  k0 φμν uν ≡ (N ) k0 φμν uν ρ ρm Λ 1 + 4χ 3 + 2 − χ ξ (24) Replacing (24) into (23), then: φμν k 2 + N (−φμν kλ − ϕλμ kν − kν φμλ ) k0 uλ = 0 (25) The antisymmetric of φμν eliminates all the terms in the parentheses of equation (25) and we are left with: (26) φμν k 2 − N k02 = 0 ⇒ k 2 − N k02 = 0, ∀φμν The effective photons velocity, in case allρ, ρm , Λ = 0, ∀ξ is then given by [6]:     ρ i   1 − 4ξΛ − 4ξχ 3 |ki k | 2     v = = |1 + N | =   k02  1 + 4ξχ ρ + ρm − Λ  3 3 (27) χ and the light velocities is not equal to ‘c’. Adopting equations (15), (16) and (17), equation (27) takes the form in normal units:    1 − 4 ξ (2δ − 1)    t2 (28) v2 =    1 + 4 tξ2 (5 − 4δ)  Electronic Journal of Theoretical Physics 10 (2006) 1–10 7 with 1 < δ < 3/2. As a result, for ξ > (<) 0, vphotons < (>) c (light celerity). Adopting the fact c = c0 an with n < 0, than the photons velocities decreases with time whatever is the sign of the coupling constant. An interesting case is when the background is ‘free from matter ’ . From (27) we get:     1 − 4ξΛ 2  (29) v =  1 + 4ξ (m2 − Λ)  If m2 = 0, than v 2 = 1 which is light celerity in units ( = c = 1). Assuming m2 = β/t2 , 4Λ = 3m2 or Λ̄ = 0 and as a result R = 0. In this case, equation (28) gives:       3β  2  2    1 − ξ 1 − 3m ξ t − 3ξβ 2   t =   v 2 =  = (30)  1 + m2 ξ   1 + ξ tβ2   t2 + ξβ  Again, if ξ > (<) 0, vphotons < (>) c (light celerity). As a result, the velocity of photons is affected and shifted by the presence of the ultra-light tiny masses and depends on the sign of the coupling constant. It doesn’t correspond in fact to null geodesics as in the standard case. Positive coupling constant corresponds to friction and negative one corresponds to superluminal case [27,28]. If we adopt the fact c = c0 an , then the photons velocities not only is shifted but also decrease with time if n < 0 and increase if n > 0. The constancy of the speed of light is not preserved in this analysis. It depends on how is filled the background space and how is used a coupling constant different of zero that modifies presentation of the Einstein’s Field Equations (EFT ), with an additional term. It is important to notice that the environment where speed of light reaches its maximum value is the lightest one: the empty space, all because of the constancy of the speed of light law, which in time, originates the fourth time-coordinate. In our case, the red-shift coefficient ‘z’ varies with time according to cz = Hr combined to equations (29) or (30) for a matter free background. ‘r’ is supposed to be the distance form the galaxy to the earth [14]. If the coupling constant is assume to be positive, one can than have a cosmological model based on interpretation of the red shift by decrease of the light speed with time everywhere in the universe beginning with a certain moment of time in the past. Of course, the agreement with the fundamental physics laws will be completed by introducing in a future work the evolution of other fundamental constants synchronously with the variation of the light speed [29]. Finally, we note that recently, growing amount of astrophysical data show important evidence for statistical and apparent physical association between low-redshift galaxies and high-redshift quasi-stellar objects suggesting noncosmological origin of their redshift and as a result failure of classical quasar explanation [30]. The author found analytical solution of Einstein equations describing bubbles made from axions with periodic interaction potential considered as one of the leading dark matter candidate. Remember that in our model [13], the ultra-light masses implemented in Einstein field equations enabled us to solve the ‘missing mass problem’ and as result considered as dark matter candidate. In Minkowski space, objects at constant proper distance with respect to an observer have zero redshift. However, in an expanding universe special relativistic concepts do not generally apply. In fact, a galaxy with zero total velocity does not have zero redshift even 8 Electronic Journal of Theoretical Physics 10 (2006) 1–10 in the empty universe case. This demonstrates that cosmological redshifts are not special relativity Doppler shifts [31,32]. It was also proved that Minkowski coordinate and the Robertson-Walker coordinates (FRW universe) are interchangeable descriptions for an empty universe. However, velocities in the Minkowski universe are not equivalent to velocities in the FRW universe because of the different definitions of time and distance in these two models. A coordinate transform relates velocities in the Minkowski universe to velocities in the FRW universe. Superluminal recession velocities in the FRW universe do not violate special relativity because they are not in the observer’s inertial frame. 4. Conclusions: In this work, we used the Einstein’s field equations with effective cosmological constant inspired from non-minimal coupling and supergravities arguments to study the consequences of non-minimal coupling between electromagnetic fields and gravity on light velocity in the presence of ultra-light particles at the radiation dominated epoch of the Universe. We showed that if the cosmological constant and the ultra-light square masses varies as Λ = δ ȧ2 /a2 +ηä/a and m2 = β/a2 , then at singular solutions and for a flat spacetime, Λ, m2 and ρ decreases with time as 1/t2 , the Hubble parameter vary as H = 1/t, the deceleration parameter is zero and ΩT otal = Ωr +Ωm +ΩΛ = 1 as required by inflation. As a result, the ultra-light particles than contribute to the total energy density and their masses decrease as inverse to time. When the ‘Λ’ and ‘m2 ’ terms dominate the dynamics of our field equations with the assumption that the Universe undergoes a long period of evolution during which the celerity of light changes as c = c0 an , it was found that at large times a ∝ t−1/n and that for negative ‘n’, there is a solution to the quasi-lambda problem. Finally, we studied varying light velocities from non-minimal coupling. We found that photons velocities depends on the coupling constant and only on δ ∈ (1, 3/2)in a way that ξ > (<) 0, vphotons < (>) c with m2 > 0. The model described in this paper could have important implications in various systems, in particular cosmological scenarios, black hole physics and quantum interactions [33,34,35,36,37]. 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Phys. 7, 27-46 (2005). [36] R. A. El-Nabulsi, Elec. J. Theor. Phys. 9, 1-5 (2006). [37] R. A. El-Nabulsi, Supergravity, Non-mimimal Coupling and Eternal Accelerated Expansion: Some important Cosmological Features of Ultra-Light Particles and Induced Cosmological Constant, Submitted to Elec. J. Theor. Phys. EJTP 3, No. 11 (2006) 11–27 Electronic Journal of Theoretical Physics A Toy Model of Financial Markets J. P. Singh and S. Prabakaran ∗ † Department of Management Studies Indian Institute of Technology Roorkee Roorkee 247667, India Received 27 October 2005 , Accepted 9 January 2006, Published 25 June 2006 Abstract: Several techniques of fundamental physics like quantum mechanics, field theory and related tools of non-commutative probability, gauge theory, path integral etc. are being applied for pricing of contemporary financial products and for explaining various phenomena of financial markets like stock price patterns, critical crashes etc.. In this paper, we apply the well entrenched methods of quantum mechanics and quantum field theory to the modeling of the financial markets and the behaviour of stock prices. After defining the various constituents of the model including creation & annihilation operators and buying & selling operators for securities, we examine the time evolution of the financial markets and obtain the Hamiltonian for the trading activities of the market. We finally obtain the probability distribution of stock prices in terms of the propagators of the evolution equations. c Electronic Journal of Theoretical Physics. All rights reserved.  Keywords: Economic Physics , Financial Markets , Stock Prices, Quantum Models PACS (2006): 89.65.Gh , 03.65.-w, 02.50.-r, 05.30.-d, 02.30.Tb 1. Introductionn The specialty of “physics” is the study of interactions between the various manifestations of matter and its constituents. The development of this subject over the last several centuries has led to a gradual refining of our understanding of natural phenomena. Accompanying this has been a spectacular evolution of sophisticated mathematical tools for the modeling of complex systems. These analytical tools are versatile enough to find application not only in point processes involving particles but also aggregates thereof leading to field theoretic generalizations and condensed matter physics. Furthermore, with the rapid advancements in the evolution and study of disordered ∗ † jatinfdm@iitr.ernet.in Jatinder pal2000@yahoo.com 12 Electronic Journal of Theoretical Physics 3, No. 11 (2006) 11–27 systems and the associated phenomena of nonlinearity, chaos, self organized criticality etc., the importance of generalizations of the extant mathematical apparatus to enhance its domain of applicability to such disordered systems is cardinal to the further development of science. A considerable amount of work has already been done and success achieved in the broad areas of q-deformed harmonic oscillators [1], representations of q-deformed rotation and Lorentz groups [2-3]. q-deformed quantum stochastic processes have also been studied with realization of q-white noise on bialgebras [4]. Deformations of the Fokker Planck’s equation [5], Langevin equation [6] and Levy processes [7-8] have also been analysed and results reported. Though at a nascent stage, the winds of convergence of physics and finance are unmistakably perceptible with several concepts of fundamental physics like quantum mechanics, field theory and related tools of non-commutative probability, gauge theory, path integral etc. being applied for pricing of contemporary financial products and for explaining various phenomena of financial markets like stock price patterns, critical crashes etc. [8-19]. The origin of the association between physics and finance, though, can be traced way back to the seminal works of Pareto [20] and Batchlier [21], the former being instrumental in establishing empirically that the distribution of wealth in several nations follows a power law with an exponent of 1.5, while the latter pioneered the modeling of speculative prices by the random walk and Brownian motion. The cardinal contribution of physicists to the world of finance came from Fischer Black & Myron Scholes through the option pricing formula [22] which bears their epitaph and which won them the Nobel Prize for economics in 1997 together with Robert Merton [23]. They obtained closed form expressions for the pricing of financial derivatives by converting the problem to a heat equation and then solving it for specific boundary conditions. The theory of stochastic processes constitutes the “golden thread” that unites the disciplines of physics and finance. Modeling of non relativistic quantum mechanics as energy conserving diffusion processes is, by now, well known [24]. Unification of the general theory of relativity and quantum mechanics to enable a consistent theory of quantum gravity has also been attempted on “stochastic spaces” [25]. Time evolution of stock prices has been, by suitable algebraic manipulations, shown to be equivalent to a diffusion process [26]. Contemporary empirical research into the behavior of stock market price /return patterns has found significant evidence that financial markets exhibit the phenomenon of anomalous diffusion, primarily superdiffusion, wherein the variance evolves with time according to a power law tα with α > 1.0. The standard technique for the study of superdiffusive processes is through a stochastic process that evolves according to a Langevin equation and whose probability distribution function satisfies a nonlinear Fokker Planck equation [27]. There is an intricate yet natural relationship between the power law tails observed in stock market data and probability distributions that emanate as the solution of the Fokker Planck equation. The Fokker Planck equation is known to describe anomalous diffusion Electronic Journal of Theoretical Physics 3, No. 11 (2006) 11–27 13 under time evolution. Empirical results [28-31] establish that temporal changes of several financial market indices have variances that that are shown to undergo anomalous super diffusion under time evolution. One of the most exhaustive set of studies on stock market data in varying dimensions has been reported in [32-36]. In [36], a phenomenological study was conducted of stock price fluctuations of individual companies using data from two different databases covering three major US stock markets. The probability distributions of returns over varying timescales ranging from 5 min. to 4 years were examined. It was observed that for timescales from 5 minutes upto 16 days the tails of the distributions were well described by a power law decay. For larger timescales results consistent with a gradual convergence to Gaussian behaviour was observed. In another study [32] the probability distributions of the returns on the S & P 500 were computed over varying timescales. It was, again, seen that the distributions were consistent with an asymptotic power law behaviour with a slow convergence to Gaussian behaviour. Similar findings were obtained on the analysis of the NIKKEI and the Hang –Sang indices [32]. Stock market phenomena are assumed to result from complicated interactions among many degrees of freedom, and thus they were analyzed as random processes and one could go to the extent of saying that the Efficient Market Hypothesis [37-38] was formulated with one primary objective – to create a scenario which would justify the use of stochastic calculus [39] for the modeling of capital markets. The Efficient Market Hypothesis contemplates a market where all assets are fairly priced according to the information available and neither buyers nor sellers enjoy any advantage. Market prices are believed to reflect all public information, both fundamental and price history and prices move only as sequel to new information entering the market. Further, the presence of large number of investors is believed to ensure that all prices are fair. Memory effects, if any at all, are assumed to be extremely short ranging and dissipate rapidly. Feedback effects on prices are, thus, assumed to be marginal. The investor community is assumed rational as benchmarked by the traditional concepts of risk and return. An immediate corollary to the Efficient Market Hypothesis is the independence of single period returns, so that they can be modeled as a random walk and the defining probability distribution, in the limit of the number of observations being large, would be Gaussian. Anomalous diffusion is a hallmark of several intensively studied physical systems. It is observed, for example, in the chaotic dynamics of fluid in rapidly rotating annulus [40], conservative motion in a periodic potential [41], transport of fluid in a porous media [42], percolation of gases in porous media [43], crystal growth spreading of thin films under gravity [44], radiative heat transfer [45], systems exhibiting surface to surface growth [46] and so on. Several analogies between physical systems and financial processes have been explored in the last decade, some of which have already been mentioned above. Perhaps, the most striking one is that between financial crashes witnessed in stock markets and critical 14 Electronic Journal of Theoretical Physics 3, No. 11 (2006) 11–27 phenomena like phase transitions is discussed here to place the main theme of this paper in its proper perspective. Stock market crashes are believed to exhibit log periodic oscillations which are characteristic of systems exhibiting discrete scale invariance i.e. invariance through rescaling by integral powers of some length scale like the Serpinski triangle and other similar fractal shapes. In the years preceding the infamous crash of October 19, 1987, the S & P market index was seen to fit the following expression exceedingly precisely [47-48], (S & P )t = Ω + Γ (tc − t)γ {1 + Ξ cos [θ ln (tc − t) + φ]} Physicists working in solid state and condensed matter physics would immediately recognize the analogy of the above expression with the one obtained for critical phenomenon in spin model of ferromagnetism [49]. We briefly elucidate the salient features of this model. Crystalline solids comprise of atoms arranged in a lattice. Each such atom generates a magnetic field parallel to the direction of the atom’s spin. In the case of substances that do not exhibit ferromagnetic character, these spin directions are randomly oriented so that the aggregate magnetic field vanishes. However, in ferromagnetic substances these spins are polarized in a particular direction resulting in a nonzero aggregate field. Ferromagnetic substances usually exhibit two distinct phases. one in which the spins orient themselves in a particular direction resulting in an aggregate magnetic moment at temperatures below a well defined critical temperature tc and the other where the spins are disoriented with a zero aggregate moment above the critical temperature. At temperatures below tc , the coupling force between neighboring atoms predominates resulting in an alignment of spins whereas above tc the additional energy manifests itself in disorienting (randomizing) the spins. Renormalization group theory enables us to group these atoms in blocks of spins whose composite spins are equal to the algebraic sum of the spins of the atoms constituting the block. It then provides that a model involving interactions between these composite spins of a block can be constructed that replicates the macroscopic properties of the block and yet cannot depend on the size of the block. That is, the system would exhibit a scaling symmetry, which is discrete, if we allow for the finite size of the atom and continuous otherwise. The magnetic susceptibility of such a mag , where the symbols have their usual meannetic substance defined by χ (T ) = ∂M ∂B B=0   ing, obeys a power law of the form χ (T ) = Re (T − Tc )α+iβ or equivalently χ (T ) = (T − Tc )α {1 + β cos [ln (T − Tc )] + O (β 2 )}which is reminiscent of op cited expression for log periodic oscillations in financial crashes. Furthermore, the access to enhanced computing power during the last decade has enabled analysts to try refined methods like the phase space reconstruction methods for determining the Lyapunov Exponents [50] of stock market price data, besides doing Rescaled Range Analysis [51] etc. A set of several studies has indicated the existence of strong evidence that the stock market shows chaotic behavior with fractal return structures and positive Lyapunov exponents. Results of these studies have unambiguously established the existence of significant nonlinearities and chaotic behavior in these time Electronic Journal of Theoretical Physics 3, No. 11 (2006) 11–27 15 series [52-55]. In this paper, we attempt one such model. The objective is to apply the well entrenched methods of quantum mechanics and quantum field theory to the study of the financial markets and the behaviour of stock prices. Section 2, which forms the essence of this paper, arrives at various results for financial markets by modeling them as quantum Hamiltonian systems. The probability distribution for stock prices in efficient markets is also obtained. Section 3 concludes. 1.1 Quantum Model of Financial Markets We consider an “isolated” financial market comprising of n investors and m type of securities. The market is “isolated” in the sense that new types of securities are neither created nor are existing ones destroyed. Further, the number of investors is also constant. The investor i, i = 1, 2, 3.......n is assumed to possess a cash balance of xi , i = 1, 2, 3.......n (which may be negative, representing borrowings) and yij (z) , i = 1, 2, 3.....n; j = 1, 2, 3....m units of security j at a unit price of z . Obviously, yij ≥ 0, ∀i, j. Towards constructing a basis for our Hilbert space representing the financial market, we define a pure state of the system as |Ψi  = |{xi , {yij (z) , j = 1, 2, ...m} , i = 1, 2, ...n} (1) Thus, a pure state represents a state of the market where the entire holdings of cash and securities of every investor are known with certainty. This represents a complete measurement of the market and hence, is in conformity with the standard definition of “pure state” of a system. A basis for our Hilbert space may then be constituted by the set of all the pure states of the type (1) i.e. Ψ = {|{xi , {yij (z) , j = 1, 2, ...m} , i = 1, 2, ...n}} (2) The elements of this basis set Ψ satisfy the orthogonality condition Ψi | Ψj  = δij with respect to the scalar product defined in the sequel. The orthogonality condition makes sense in the financial world – it implies that if a market is in a pure state |Ψi  then it cannot be in any other pure state. However, a complete measurement of the market is, obviously, not practicable in real life. At any point in time, we are likely to have certain information only about a fraction of the market constituents. Hence, the instantaneous state of the market |ψ (t) may be represented by a linear combination of the pure states |Ψl (t) i.e.  |ψ (t) = Cl |Ψl (t) (3) l We endow our Hilbert space H with the scalar product    ψ (t) | ξ (t) = Cl∗ Dm Ψl (t) | Ψm (t) = Cl∗ Dm δlm = Cl∗ Dl l,m l,m l (4) 16 Electronic Journal of Theoretical Physics 3, No. 11 (2006) 11–27 where we have assumed the orthogonality of the pure states. The components of the state space vector |ψ (t)are given byCl = Ψl (t) | ψ (t) and are related to the probability of finding the market in the pure state |Ψl (t). Since our basis comprises of all possible measurable pure states, the completeness of the basis is ensured so that  I= |Ψl (t) Ψl (t)| (5) l In analogy with the no particle state or ground state in quantum mechanics, we can define a ground state of our financial market as |0 = |xi = 0, yij (z) = 0∀i, j, z (6) i.e. the ground state is the market state in which no investor has any cash balances nor any securities. This state is, obviously, a pure state being fully measurable and would also not evolve in time since no trade can take place in this market. We define the cash and security coordinate operators x̂i & ŷij (z) by their action on the basis state (1) to provide respectively the balances of cash and the j th security (at price z) with the ith investor as the eigenvalues i.e. x̂i |{xi , {yij (z) , j = 1, 2, ...m} , i = 1, 2, ...n} = xi |{xi , {yij (z) , j = 1, 2, ...m} , i = 1, 2, ...n} (7) ŷij (z) |{xi , {yij (z) , j = 1, 2, ...m} , i = 1, 2, ...n} = yij (z) |{xi , {yij (z) , j = 1, 2, ...m} , i = 1, 2, ...n} (8) A cash translation operator T̂i (z) is also defined by the following T̂i (z ′ ) |{xi , {yij (z) , j = 1, 2, ...m} , i = 1, 2, ...n} = |{xi + z ′ , {yij (z) , j = 1, 2, ...m} , i = 1, 2, ...n} (9) i.e. it transfers an amount of cash z to the ith investor. The operator T̂i (z) obviously satisfies the following properties  T̂i (z1 ) T̂i (z2 ) = T̂i (z1 + z2 ) (10) T̂i (0) = Iˆ (11) T̂i (z) , x̂j = T̂i (z) x̂j − x̂j T̂i (z) = −zδij T̂i (z) (12) T̂i↑ (z) = T̂i (−z) (13)  Towards obtaining an explicit representation of the cash translation operator, we assume  dT̂i (z)  p̂i = dz  as the generator of infinitesimal cash translations dz to the investor i. z=0 Expanding T̂i (z) as a Taylor’s series and using eqs. (10), (11) we have  T̂i (0) + T̂i (dz) − 1 T̂i (z) dT̂i (z) T̂i (z + dz) − T̂i (z) = lim = lim = lim dz→0 dz→0 dz→0 dz dz dz  dT̂i (z)  dz z=0 dz  dz... − 1 T̂i (z) = p̂i T̂i (z) (14) Electronic Journal of Theoretical Physics 3, No. 11 (2006) 11–27 17 with the solution T̂i (z) = ezp̂i . Furthermore, we have (suppressing the yij indices for the sake of brevity)   |{xi + dz, i = 1, 2, ...n} = T̂i (dz) |{xi , i = 1, 2, ...n} = T̂i (0) + dT̂i (z)  dz  = [I + p̂i dz...] |{xi , i = 1, 2, ...n} z=0  dz... |{xi , i = 1, 2, ...n} (15) Hence, ∂{xi ,i=1,2,...n}|ψ ∂xi = p̂↑i = lim dz→0 {xi +dz,i=1,2,...n}|ψ−{xi ,i=1,2,...n}|ψ dz {xi , i = 1, 2, ...n} | ψ = −p̂i {xi , i = 1, 2, ...n} | ψ ⇒ p̂i = ∂ −z ∂x so that T̂i (z) = e i (16) − ∂x∂ i . The following commutation rule holds between x̂i and p̂i [x̂i , p̂j ] = δij (17) The condition of an isolated market ensures that the basis and hence the Hilbert space does not depend on time. This implies that the temporal evolution of the system is unitary. Creation & Annihilation Operators for Securities We define âij (z)as the annihilation operator of the security j from the portfolio of investor i for a price z i.e. when operator âij (z) acts on a state, the number of units of security j is reduced by one from the portfolio of investor i for a price z.Similarly, we define creation operators â↑ij (z) as the adjoint of the annihilation operators that increase the number of units of security j in the portfolio of investor i for a price z.The precise action of these operators on a state vector is defined by the following âij (z) |{xi , {yij (z) , j = 1, 2, ...m} , i = 1, 2, ...n} =  y ij (z).z |{xi , {yij (z) − 1, j = 1, 2, ...m} , i = 1, 2, ...n} (18) and â↑ij (z) |{xi , {yij (z) , j = 1, 2, ...m} , i = 1, 2, ...n} =  (yij (z) + 1)z |{xi , {yij (z) + 1, j = 1, 2, ...m} , i = 1, 2, ...n} (19) where the factor ‘z’ has been introduced in the eigenvalues to ensure “scale invariance” of the theory. These operators satisfy the following commutation relations:  ↑ ′ âij (z), âij (z ) = zδzz′ δik δjl (20) and ′   â↑ij (z), â↑ij (z ′ ) [âij (z), âkl (z )] = =0     T̂i (z) , âjk (z ′ ) = T̂i (z) , â↑jk (z ′ ) = 0     T̂i↑ (z) , âjk (z ′ ) = T̂i↑ (z) , â↑jk (z ′ ) = 0 (21) (22) (23) 18 Electronic Journal of Theoretical Physics 3, No. 11 (2006) 11–27 Further more â↑ij (z) âij (z) |{xi , {yij (z) , j = 1, 2, ...m} , i = 1, 2, ...n} = zyij (z) |{xi , {yij (z) , j = 1, 2, ...m} , i = 1, 2, ...n} (24) which implies that the number operator would be â↑ij (z) âij (z) ŷij (z) = z (25) Using the aforesaid operators we can construct an arbitrary basis state from the ground state as follows |{xi , {yij (z) , j = 1, 2, ...m} , i = 1, 2, ...n} = n  T̂i (xi ) i=1 m   j=1 {z,yij (z)∈N}  yij (z) â↑ij (z) |0 (26) Buying and selling operators The buying (selling) operation of a security is, in each case, a composite operation consisting of the following:i. the creation (annihilation) of a security at the relevant price z; and ii. the decrease (increase) in the cash balance by z of the investor undertaking the trade. Hence we can define the buying (selling) operator as composite of the cash translation operator and the creation (annihilation) operators for securities as follows:b̂↑ij (z) = â↑ij (z) T̂i↑ (z) = â↑ij (z) T̂i (−z) (27) for the “buying” operation and b̂ij (z) = âij (z) T̂i (z) (28) for the “selling” operation. These operators satisfy the following commutation rules    b̂ij (z) , b̂↑kl (z ′ ) = zδzz′ δik δjl    ↑ ↑ ′ b̂ij (z) , b̂kl (z ) = b̂ij (z) , bkl (z ) = 0 ′     b̂ij (z) , T̂k (z ′ ) = b̂↑ij (z) , T̂k (z ′ ) = 0   ↑ b̂ij (z) , x̂k = zδik b̂↑ij (z)   b̂ij (z) , x̂k = −zδik b̂ij (z) (29) (30) (31) (32) (33) Electronic Journal of Theoretical Physics 3, No. 11 (2006) 11–27 19 1.2 Temporal Evolution of Financial Markets In analogy with quantum mechanics, we mandate that the state of the market at a given instant of time ‘t’, is represented by a vector in the Hilbert space H whose components determine the statistical nature of the market. Hence the temporal evolution of the market in essentially determined by the evolution of this vector with the flow of time. In the Schrödinger picture, the time evolution of a system can be characterized through a ∧ unitary evolution operator U (t, t0 ) in H, that acts on the initial state |ψ (t0 ) to transform it to |ψ (t) i.e |ψ (t) = Û (t, t0 ) |ψ (t0 ) (34) The assumption of the market being isolated and hence Ψ = {|{xi , {yij (z) , j = 1, 2, ...m} , i = 1, 2, ...n}} being a complete basis at all times, and  |Cl (t)|2 = 1, ∀t together with the group property the conservation of probability i.e. ∧ l of U (t, t0 ) implies that the temporal evolution is unitary i.e. U (t, t0 ) U ↑ (t, t0 ) = U ↑ (t, t0 ) U (t, t0 ) = 1 (35)   ∂ Furthermore Û (t0 , t0 ) = 1. Defining the Hamiltonian Ĥ (t) = i ∂t Û (t, t0 ) t=t0 as the in- finitesimal generator of time translations (evolution) we obtain, through a Taylor’s ex Û (t, t0 ) δt + ...or pansion up to first order Û (t + δt, t0 ) = Û (t, t0 ) + ∂∂tÛ (t + δt, t) δt=0 ∂ Û (t, t0 ) Û (t + δt, t0 ) − Û (t, t0 ) = lim = −iĤ (t) Û (t, t0 ) δt→0 ∂t δt − 4t (36) Ĥ(t)dt where time ordering of the operators with the immediate solution Û (t, t0 ) = e t0 constituting the Hamiltonian is assumed. Before progressing further with the development of the model, some observations are in order about the theory developed thus far. (1) In standard quantum mechanics, Ĥ (t) is usually a bounded operator and hence the 4 − tt Ĥ(t)dt exponential series in Û (t, t0 ) = e 0 converges so that its approximation to first ∂|ψ(t) order is acceptable giving i ∂t = Ĥ (t) |ψ (t) which is the Schrödinger equation of wave mechanics. This may not always be the case in financial markets. (2) Since time evolution of financial market, essentially, occurs through trades in securities, it is appropriate to infer that the Hamiltonian represents the trading activities of the market. (3) In order that the evolution operator Û (t, t0 ) is well defined, we mandate that the Hilbert space His so constructed that the kernel of Û (t, t0 ) is empty. 1.3 Modeling Time Value of Money Time value of money and interest rate instruments are classically modeled through the 4 dB(t) r(t)dt first order differential equation dt = r (t) B (t)with the solution B (t) = B (0) e . 20 Electronic Journal of Theoretical Physics 3, No. 11 (2006) 11–27 A possible candidate for the Hamiltonian function H(in the classical picture) that would generate this temporal development as the equations of motion is H (x, p; t) = n  Hi (xi , pi ; t) = i=1 n  ri (t) xi (t) pi (t) (37) i=1 This Hamiltonian leads to the following equations of motion dpi (t) ∂Hi (xi , pi ; t) = −ri (t) pi (t) =− dt ∂xi dxi (t) ∂Hi (xi , pi ; t) = ri (t) xi (t) , = dt ∂pi (38) While the interpretation of first of these equations is straightforward being the growth of cash reserves of the ith investor with the instantaneous rate ri (t), the implications of second equation are more subtle. To provide a financial logic to this equation, we note that pi is the infinitesimal generator of cash translations in the classical picture and hence dpi (t) i ,pi ;t) = −ri (t) pi (t) represents the rate of change of the cash translations = − ∂Hi (x dt ∂xi generator which, given a fixed rate of growth of cash, would decrease with the amount of cash translations. Using the Weyl formalism for transformation from the classical to the quantum picture, we require that the quantum mechanical analog of H (x, p; t) be Hermitian and symmetric in its component operators. Hence, we postulate the ansatz   n n n    iri (t) 1ˆ Ĥ (x̂ (t) , p̂ (t) ; t) = (x̂i p̂i + p̂i x̂i ) = Ĥ (x̂i (t) , p̂i (t) ; t) = iri (t) x̂i p̂i + I 2 2 i=1 i=1 i=1 (39) for the quantum mechanical Hamiltonian representing the time value of money, so that the time development operator is  Û (t, t0 ) = e −i 4t  n 4t 2 Ĥ(t)dt =e t0 i=1 t0 ri (t)xi (p̂i + 12 Iˆ)dt (40) which may be evaluated using standard methods like Green’s functions and Feynmann propagator theory. 1.4 Representation of Trading Activity Let us consider a deal in which an investor ‘i’ buys a security ‘j’ at a price of ‘z’ units and immediately thereafter sells the same security to another investor ‘k’ at a price of ‘z ′ ’ units and credits/debits the difference z ′ − z to his cash account. The composite transaction will, in our operator formalism, take the form b̂ij (z ′ ) b̂↑ij (z). In analogy with this argument, we can represent the Hamiltonian for trading activity of the market as ∞ ∞   dz  dz ′ HT r (t) = hijkl (ξ, t) b̂↑ij (z) b̂kl (z ′ ) ′ z z i,j,k,l 0 ′ 0 where ξ = ln zz ensures that the amplitudes are scale invariant. (41) Electronic Journal of Theoretical Physics 3, No. 11 (2006) 11–27 21 1.5 Probability Distribution of Stock Prices We now derive the probability distribution of stock prices in market equilibrium and show that the prices follow a lognormal distribution, thereby vindicating the efficacy of this model. For this purpose, we assume that an investor i = α buys one unit of a security j = β at time t = ti for a price z. We need to ascertain the probability PT (z ′ |z )i.e. the probability of the security j = β having a price z ′ at time tf = ti + T .We assume that during the period tf − ti , investor α holds exactly one unit of β and that before ti and after tf ,α  holdsno unit of β.  z (ti ) be the state that represents investor α holding exactly one unit of β at a Let ψαβ      z  ↑ price z at time ti in the Hilbert space H. Hence, we have ψαβ (ti ) = b̂αβ (z) ψαβ (ti ) where    ψ (t )  αβ i is the state that represents investor α not holding any unit of β. This also im   plies that b̂αβ (z) ψαβ (ti ) = 0.Let us assume that the final state corresponding to the      z  z initial state ψαβ (ti ) is represented by ψαβ (tf ) so that tf 4   −i Ĥdt  z  z    ↑ ψαβ (tf ) = Û (ti , tf ) ψαβ (ti ) = e ti b̂αβ (z) ψαβ (ti ) (42)  z′  The amplitudes of ψαβ (tf ) are determined in the usual way by taking scalar product     z z′ (tf ) and we have, for the matrix elements of the propagator (tf )  ψαβ ψαβ tf 4     −i Ĥdt   ′ ′ ti G (z , tf ; z, ti ) = ψαβ (tf ) b̂αβ (z ) e b̂↑αβ (z) ψαβ (ti ) (43) In this case, the trading Hamiltonian will contain creation and annihilation operators relating to the investor α and those relating to the securityβ i.e., it will be of the form ∞ ∞   dz  dz ′ ĤT r (t) = h (ξ, t) b̂↑αβ (z) b̂kl (z ′ ) ′ αβkl z z k,l 0 (44) 0 We further make the assumption that the amplitudes can be approximated by their first two moments about ξ = 0, being sharply peaked about z ′ = zsince, in the timescales being considered, most trades would occur around z. Hence, we have   hαβkl ∼ Ωαβkl (t) − iξ −1 Ξαβkl (t) δ (ξ) (45) ′ ′ −1 ′ −1 = zz − 1 = z′z−z to first order and Noting that ξ = ln zz , we have ξ −1 = ln zz   −1 ′ d ln zz ′ δ (ξ) = δ ln zz = δ (z ′ − z) = z ′ δ (z ′ − z). Using these results and eqs. (44) dz ′ & (45), we obtain ∞ ∞     dz  dz ′ z ′ ĤT r (t) ∼ Ξαβkl (t) b̂↑αβ (z) b̂kl (z ′ ) zδ (z − z) Ωαβkl (t) − i ′ ′ z z z − z k,l 0 0 22 Electronic Journal of Theoretical Physics 3, No. 11 (2006) 11–27     dz ↑ ∂ b̂αβ (z) Ωαβkl (t) + izΞαβkl (t) b̂kl (z ′ ) = z ∂z k,l ∞ (46) 0 ∂ and hence it can be We note that this expression for the Hamiltonian is linear in ∂z diagonalized in the “momentum space” through a Fourier transformation and we have ∞ ∞ ′ ∞ 1  dz dz ĤT r (t) = dpb̂↑αβ (z) [Ωαβkl (t) + iΞαβkl (t) p] b̂kl (z ′ ) eipξ (47) ′ 2π k,l z z −∞ 0 0 The assumption of market equilibrium implies that the Hamiltonian should be independent of time over the relevant timescales that would be much smaller than those determining aggregate market behaviour so that we may write eq. (43) as       (48) G (z ′ , tf ; z, ti ) = ψαβ (ti ) b̂αβ (z ′ ) e−iĤ(ti )T b̂↑αβ (z) ψαβ (ti ) Because of the Hamiltonian being diagonal in momentum space, it is more convenient to work in momentum space for evaluating the propagators and we have, for the equivalent of eq. (48) in momentum space as ⎡ ⎤ ⎤ ⎡∞    ∞ dz ′ ′   ′ dz −ipln(z/κ) ↑   e G̃ (p′ , p; T, ti ) = ψαβ (ti ) ⎣ eip ln(z /κ) b̂αβ (z ′ )⎦ e−iĤ(ti )T ⎣ b̂αβ (z)⎦ b̂↑αβ (z) ψαβ (ti ) ′ z z 0 0 (49) To solve the problem further, we make use of second order perturbation theory. The first step is to split the Hamiltonian into components as follows ∞ ∞   dz ↑   dz ↑ Ĥ (t) = b̂ (z) [Ωαβαl (t) + Ξαβαl (t) p] b̂αβ (z) + b̂ (z) [Ωαβkl (t) + iΞαβkl (t) p] b̂kl (z ′ ) z αβ z αβ l k,l,k=α 0 0 (50) Let Ei be the energy eigenstate of the unperturbed Hamiltonian i.e. of the state of the market before the purchase of security β by the investor α , then the energy eigenstate of the Hamiltonian Ĥ (ti )i.e. after the purchase of security β by the investor α will be of  the form Ep = Ei + [Ωαβαl (ti ) + iΞαβαl (ti ) p] − ip2 σ 2 where the second term represents l the impact on the energy eigenstates of the transactions involving investor α or security β and the last term is the second order perturbation term due to the overall fluctuations of the market. Substituting this value of Ep in eq. (49) and noting that the Hamiltonian and hence the propagator G̃ (p′ , p; T, ti )is also diagonal in “momentum space”, we have G̃ (p′ , p; T, ti ) ∼ 2πδ (p′ − p) e−iT Ep = 2πδ (p′ − p) e−iT [Ei +Ω(ti )+iΞ(ti )p−ip 2 σ2 where  Ωαβαl (ti ) = Ω (ti ) , l  ] (51) Ξαβαl (ti ) = Ξ (ti ) . l Inverting back to “coordinate space”, we obtain   ′ z +ΞT 2 ∞ / ) ) 1 eiT (Ei +Ω) (ln(z 4σ 2T −iT [Ei +Ω(ti )+iΞ(ti )p−ip2 σ 2 ]−ip ln z ′ /z ′ √ G (z , tf ; z, ti ) = e ∼ dpe 2π 2σ πT −∞ (52) Electronic Journal of Theoretical Physics 3, No. 11 (2006) 11–27 23 The probability PT (z ′ |z )i.e. the probability of the security j = β having a price z ′ at time tf = ti + T will then be proportional to the square of the above amplitude and hence, we finally obtain  2 PT (z ′ |z ) α |G (z ′ , tf ; z, ti )| = 4πσ 2 T −1 e (In(z′ /z)+ΞT )2  2σ 2 T (53) which agrees perfectly with the standard stochastic theory of finance wherein stock returns are modeled extensively through lognormal distributions. 2. Conclusions The following interesting observations emanate from the above analysis:- (1) Eq. (53), on comparison with the standard expression for probability distribution of stock price in the conventional stochastic calculus based approach to the Black Scholes formula, identifies Ξ with the expected return on stock. This return is independent of the eigenvalue Ei and hence, the state of the market. A financial interpretation of this could be that the stock returns are dependent on the performance of the company and independent of market dynamics. (2) Independence of stock returns of the market dynamics would, however, mandate that the stock volatility measured by the standard deviation σ is related to the stock market dynamics which seems justified since higher trading volumes would imply greater volatility and vice versa. (3) If we define the uncertainty of measurement of a random variate by its standard deviation, then, from eqs. (52) & (53), we have the uncertainty for the stock price process √ zand its Fourier conjugate p, after a time T , as σz = σ T and σp = 2σ1√T so that σz σp = 21 as it should be, since the distribution of z is assumed Gaussian in the aforesaid analysis. (4) Furthermore, σσpz = 2σ 2 T which enables the identification of σ 2 as the reciprocal of the mass and hence, the inertia of the stock price process. It is intriguing to note that the same analogy follows through another completely independent analysis i.e. the Black = −rSt ∂C − Scholes equation for the option price in its standard form is given by ∂C ∂t ∂S 2 1 2 2∂ C σ St ∂S 2 + rC, where C (St , t) denotes the instantaneous price of a call option with 2 exercise price E at any time t before maturity when the price per unit of the underlying 2 = −r ∂C − 21 σ 2 ∂∂xC2 + 12 σ 2 ∂C + rC is St . Making the substitution St = ex we obtain ∂C ∂t ∂x ∂x which, when compared with the standard quantum mechanical Hamiltonian in one degree of freedom identifies σ 2 as the reciprocal of the mass of the underlying system. Contemporary quantitative finance is dominated by stochastic modeling of market behaviour. These models are essentially in the nature of tools of data analysis that aim to predict future events by applying probabilistic methods to historical data. Empirical evidence testifies that probability distributions of stock returns are negatively skewed, have fat tails and show leptokurtosis [56]. Some of the features of empirical distributions of stock prices are modeled through Levy distributions [57-60], stochastic volatility [61] or cumulant expansions around the lognormal case. Each of these models, however, 24 Electronic Journal of Theoretical Physics 3, No. 11 (2006) 11–27 attempts to empirically attune the model parameters to fit observed data and hence, is equivalent to interpolating or extrapolating observed data in one form or the other. Hence, stochastic models fail to take cognizance of causal factors that get submerged in the superficial patterns exhibited by the avalanche of data being analysed. In actual fact, every new price determination of a security and hence, the fluctuation of prices is attributable to a new trade in the relevant security at that price. The “trading process” therefore manifests itself as a price history of a security. The fundamental limitation of stochastic tools in simulating extended memory effects is circumvented by this approach. An attempt has also been made through this “toy model” to establish that a quantum mechanical version of financial markets results in a temporal evolution of the probability distribution analogous to that of simple stochastic systems. Stochastic models also lack ability to accommodate collective effects like phase coherency in lasers that could, possibly, be built into this quantized description. It need be emphasized here that the above is purely a phenomenological model for modeling stock behavior. It is fair to say that the current stage of research in financial processes is dominated by the postulation of phenomenological models that attempt to explain a limited set of market behavior. There is a strong reason for this. A financial market consists of a huge number of market players. Each of them is endowed with his own set of beliefs about rational behavior and it is this set of beliefs that govern his actions. The market, therefore, invariably generates a heterogeneous response to any stimulus. Furthermore, “rationality” mandates that every market player should have knowledge and understanding about the “rationality” of all other players and should take full cognizance in modeling his response to the market. This logic would extend to each and every market player so that we have a situation where every market player should have knowledge about the beliefs of every other player who should have knowledge of beliefs of every other player and so on. We, thus, end up with an infinitely complicated problem that would defy a solution even with the most sophisticated mathematical procedures. Additionally, unlike as there is in physics, financial economics does not possess a basic set of postulates like General Relativity and Quantum Mechanics that find homogeneous applicability to all systems in their domain of validity. Electronic Journal of Theoretical Physics 3, No. 11 (2006) 11–27 25 References [1] A. J. Macfarlane, J.Phys., A 22, (1989), 4581; [2] L.C. Biedenharn, J.Phys., A 22, (1989), L873; [3] S. Zakrzewski, J.Phys., A 31, (1998), 2929 and references therein; Shahn Majid, J. Math. Phy., 34, (1993), 2045; [4] Michael Schurmann, Comm. Math. Phy., 140, (1991), 589; [5] C. Blecken and K.A. Muttalib, J.Phys., A 31, (1998), 2123; [6] J. P. Singh, Ind. J. Phys., 76, (2002), 285; [7] U. Franz and R. Schott, J. Phys., A 31, (1998), 1395; [8] V.I. Man’ko et al, Phy. Lett., A 176, (1993), 173; V.I. Man’ko and R.Vileta Mendes, J.Phys., A 31, (1998), 6037; [9] W. Paul & J. Nagel, Stochastic Processes, Springer, (1999); [10] J. 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Yakovenko, Statistical Mechanics of money, income and wealth: A Short Survey, arXiv:cond-mat/0211175 v1, 9 Nov 2002; [19] J. Doyne Farmer, Physics Attempt to Scale the Ivory Tower of Finance, adaporg/9912002 10 Dec 1999; [20] V. Pareto, Cours d’Economie Politique (Lausannes and Paris),(1897); [21] L. Batchlier, Annelas Scientifiques de l’Normal Superieure III-17 21-86,(1900); P. Cootner, The Random Character of Stock Market Prices (Cambridge, MA: MIT Press ) Reprint ,(1964); [22] F. Black & M. Scholes, Journal of Political Economy, 81, (1973), 637; [23] R. C. Merton, Journal of Financial Economics, (1976), 125; [24] M. Nagasawa, Stochastic Processes in Quantum Physics, Birkhauser, (2000); 26 Electronic Journal of Theoretical Physics 3, No. 11 (2006) 11–27 [25] E. Purgovecki, Stochastic Quantum Mechanics and Quantum Spacetime: A consistent unification of relativity and quantum theory based on stochastic spaces Reidel, Dordrecht, 1st printing 1984, revised printing, 1986 [26] M. Baxter & E. Rennie, Financial Calculus, Cambridge University Press, (1992). [27] H. Risken, The Fokker Planck Equation, Springer, (1996); [28] R.N. Mantegna & H.E. Stanley, An Introduction to Econophysics, Cambridge, (2000); [29] M.M. Dacrrogna et al, J. Int’l Money & Finance, 12, (1993), 413; [30] R.N. Mantegna & H.E. Stanley, Nature, 383, (1996), 587; [31] R.N. Mantegna, Physica A, 179, (1991), 232; [32] P. Gopikrishnan et al, Phys. Rev. E 60, (1999), 5305; [33] P. Gopikrishnan et al, Phys. Rev. E 62, (2000) R4493; [34] P. Gopikrishnan et al, Physica A, 299, (2001), 137; [35] P. Gopikrishnan et al, Phys. Rev. E 60, (1999) 5305; [36] V. Plerou et al, Phys. Rev. E 60, (1999) 6519; [37] W.F. Sharpe, Portfolio Theory & Capital Markets, McGraw Hill, (1970); [38] E.J. Elton & M.J. Gruber, Modern Portfolio Theory, & Investment Analysis, Wiley, (1981); [39] S.M. Ross, Stochastic Processes, John Wiley, (1999); [40] A. Ott el al., Phys. 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Vastano, Determining Lyapunov Exponents From a Time Series, Physica 16D, (1985), 285; [51] B.B. Mandlebrot, The Fractal Geometry of Nature, Freeman Press, (1977); [52] G. DeBoek, Ed., Trading on the Edge, Wiley, (1994); Electronic Journal of Theoretical Physics 3, No. 11 (2006) 11–27 27 [53] J.G. DeGooijer, Testing Nonlinearities in World Stock Market Prices, Economics Letters, 31, (1989); [54] E. Peters, A Chaotic Attractor for the S&P 500, Financial Analysts Journal, March/April, (1991); [55] E. Peters, Fractal Structure in the Capital Markets, Financial Analysts Journal, July/August, 31, (1989); P. Cootner, Ed., The Random Character of Stock Market Prices, Cambridge MIT Press, (1964); [56] E. Peters, Chaos & Order in the Capital Markets, Wiley, (1996) and references therein; [57] L. Andersen L & J. Andreasen, Review of Derivatives Research, 4, (2000), 231; [58] J.P. Bouchaud et al, Risk 93, (1996), 61; [59] J.P. Bouchaud & M.Potters, Theory of Financial Risks, Cambridge, (2000); [60] E. Eberlein et al, Journal of Business 71(3), (1998), 371; [61] B. Dupire, RISK Magazine, 8, January 1994; EJTP 3, No. 11 (2006) 29–32 Electronic Journal of Theoretical Physics Rayleigh process and matrix elements for the one-dimensional harmonic oscillator J.H. Caltenco1∗ , J.L. López-Bonilla1† , J. Morales2‡ 1 Escuela Superior de Ingenierı́a Mecánica y Eléctrica Instituto Politécnico Nacional Edif. Z, Acc. 3-3er Piso Col. Lindavista C.P. 07738 México D.F. 2 Area de Fı́sica AMA, CBI Universidad Autónoma Metropolitana-Azc. Apdo. Postal 16-306, CP 02200 México DF Received 21 December 2005 , Accepted 2 February 2005, Published 25 June 2006 Abstract: We show that, the matrix elements m |e−γ x |n  for the one-dimensional harmonic oscillator have application in Markov process theory, permitting thus to resolve the FokkerPlanck equation for the two-dimensional probability density corresponding to Rayleigh case. c Electronic Journal of Theoretical Physics. All rights reserved.  Keywords: Matrix elements, One-dimensional harmonic oscillator, Markov process theory, Fokker-Planck equation PACS (2006): 03.65.Ta , 03.65.Ge ,02.50.Ga, 46.25.Cc, 02.70.Ns 1. Introduction In [1-4] were calculated the matrix elements:  ∝ ∗ −γx f (γ) = m| e ψm (x) e−γ x ψn (x)dx |n = (1) −∝ for the harmonic oscillator in one dimension, where γ ≥ 0 is an arbitrary parameter. Then, it was deduced the following result for m ≥ n:   2 m−n 2  γ / m−n n! γ γ f (γ) = e 4 Ln −√ (2) − m! 2 2 ∗ † ‡ hcalte@maya.esimez.ipn.mx jlopezb@ipn,mx, lopezbjl@hotmail.com jmr@correo.azc.uam.mx 30 Electronic Journal of Theoretical Physics 3, No. 11 (2006) 29–32 in terms of the associated Laguerre polynomials Lqn . It is interesting to observe that the 2nd order differential equation [5-8] defining to p Lq permits to show, via (2), that f (γ)satisfies the equation: 1 1 df d2 f + − 2 γ 4 + 4Aγ 2 + 4Q f = 0 2 dγ γ dγ 4γ (3) where A = m + n + 1 and Q = (m − n)2 . That is, (2) is solution of (3), with which it is possible to obtain [9] the Morse’s radial wave function [10]. In Sec. 2, f (γ)is employed to resolve the nonstationary stochastic Fokker-Planck equation (FPE) [11] for the Rayleigh distribution. 2. Two-Dimensional Probability Density Associated to Rayleigh Process. The equation (3) has the structure: D1 (γ)f =  Q γ2 +A+ 2 4 γ  f (4) where it appears the important Bessel’s operator [12]: DC (γ) = d2 C d + 2 dγ γ dγ (5) for the case C = 1. The operator (5) has interesting applications in hydrodynamics, the theory of subharmonic functions, electrostatics, the Euler-Poisson-Darboux equation, elasticity, the generalized radiation problem, quantum mechanics and generalized axially symmetric potential. Here we shall show that, the Bessel’s operator D1 is useful for to determine the probability density ω associated to Rayleigh process in two dimensions, because the FPE can adopt a form similar to (4). In fact, the nonstationary stochastic FPE for the Rayleigh distribution is given by [11] p. 73:    ∂ k k ∂ 2ω ω̇ = (6) βx− ω + ∂x 2x 2 ∂ x2 being k and β positive parameters, then the corresponding eigenfunctions X (x) are solutions of:     2 σ2 λ σ2 d X 2d X + 1+ 2 + + x− σ X=0 (7) d x2 x dx x β k . The operator D1 participates when in (7) we make the following change with σ 2 = 2β of functions: x2 (8) F (x) = x−1 e 4σ2 X obtaining thus the relation: D1 (x)F =  1 λ x2 − 2− 2 4 4σ σ σ β  F (9) Electronic Journal of Theoretical Physics 3, No. 11 (2006) 29–32 31 On the other hand, in (4) it is possible to realize the formal change of variable: γ= i x , σ i= √ −1 (10) then it results the equation: D1 (x)f =  Q A x2 + 2− 2 4 4σ x σ  f , (11) with the same structure as (9); therefore Q = 0, that is m = n, and A = 2n + 1 = 1 + βλ , then λ = 2nβ , n= 0,1,2,. . . Besides, F is proportional to f given by (2) with the change (10):  2  2 x − x2 F ∝ e 4σ Ln (12) 2σ 2 then (8), (12) and factors of normalization lead us to the eigenfunctions:  2  x x − x22 2σ L e Xn (x) = n n!σ 2 2σ 2 (13) which are solutions of (7) for λ = 2nβ. The result (13) is our principal aim because it permits to write immediately the two-dimensional probability density associated to Rayleigh case, to see [11]. 32 Electronic Journal of Theoretical Physics 3, No. 11 (2006) 29–32 References [1] A. E. Glassgold and D. Holliday, Phys. Rev. A139 (1965) 1717. [2] J. Morales, J. López-Bonilla and A. Palma, J. Math. Phys. 28 (1987) 1032. [3] J. Morales and A. Flores-Riveros, J. Math. Phys. 30 (1989) 393. [4] J. López-Bonilla and G. Ovando, Bull. Irish Math. Soc. N.44 (2000) 61. [5] C. Lanczos, Linear differential operators, D. Van Nostrand Co., London (1961). [6] J. D. Talman, Special functions, W. A. Benjamin Inc. New York (1968). [7] H. Hochstadt, The functions of Mathematical Physics, Dover, New York (1971). [8] M. Abramowitz and I. A. Stegun, Handbook of mathematical functions, John Wiley and Sons, New York (1972). [9] J. H. Caltenco, J. López-Bonilla and R. Peña-Rivero, J. Sci. Res. 50 (2002) 125. [10] Ch. S. Johnson Jr. and L. G. Pedersen, Problems and solutions in Quantum Chemistry and Physics, Dover, New York (1986). [11] R. L. Stratonovich, Topics in the theory of random noise. Vol. I, Gordon and Breach, New York (1963). [12] A. Weinstein, Ann. Mat. Pura Appl. 49 (1960) 359. EJTP 3, No. 11 (2006) 33–70 Electronic Journal of Theoretical Physics Identical synchronization in chaotic jerk dynamical systems Vinod Patidar1∗ and K. K. Sud2 1 Department of Physics Banasthali Vidyapith Deemed University Banasthali - 304022, Rajasthan, INDIA 2 Department of Physics College of Science Campus M. L. S. University, Udaipur – 313002, INDIA Received 23 December 2005 , Accepted 24 February 2006, Published 25 June 2006 Abstract: It has been recently investigated that the jerk dynamical systems are the simplest ever systems, which possess variety of dynamical behaviours including chaotic motion. Interestingly, the jerk dynamical systems also describe various phenomena in physics and engineering such as electrical circuits, mechanical oscillators, laser physics, solar wind driven magnetosphere ionosphere (WINDMI) model, damped harmonic oscillator driven by nonlinear memory term, biological systems etc. In many practical situations chaos is undesirable phenomenon, which may lead to irregular operations in physical systems. Thus from a practical point of view, one would like to convert chaotic solutions into periodic limit cycle or fixed point solutions. On the other hand, there has been growing interest to use chaos profitably by synchronizing chaotic systems due to its potential applications in secure communication. In this paper, we have made a thorough investigation of synchronization of identical chaotic jerk dynamical systems by implementing three well-known techniques: (i) PecoraCarroll (PC) technique, (ii) Feedback (FB) technique and (iii) Active Passive decomposition (APD). We have given a detailed review of these techniques followed by the results of our investigations of identical synchronization of chaos in jerk dynamical systems. The stability of identical synchronization in all the aforesaid methods has also been discussed through the transversal stability analysis. Our extensive numerical calculation results reveal that in PC and FB techniques the x-drive configuration is able to produce the stable identical synchronization in all the chaotic jerk dynamical systems considered by us (except for a few cases), however y-drive and z-drive configurations do not lead to the stable identical synchronization. For the APD approach, we have suggested a generalized active passive decomposition, which leads to the stable identical synchronization without being bothered about the specific form of the jerk dynamical system. Several other active passive decompositions have also been listed with their corresponding conditional Lyapunov exponents to achieve the stable identical synchronization in various chaotic jerk dynamical systems. c Electronic Journal of Theoretical Physics. All rights reserved.  Keywords: Chaos, Jerk dynamical systems, Identical synchronization of chaos, synchronized chaos PACS (2006): 05.45.+b; 47.52.+j; 05.45.-a ∗ vinod r patidar@yahoo.co.in 34 1. Electronic Journal of Theoretical Physics 3, No. 11 (2006) 33–70 Introduction Various studies of nonlinear dynamical systems in the last four decades have significantly extended the notion of oscillations in these systems. It has been shown that the post-transient oscillations in dynamical systems can be associated not only with the regular behavior such as periodic or quasiperiodic oscillations, but also with chaotic behavior [1-5]. Chaos has long-term unpredictable behavior, which is usually couched mathematically as sensitivity to initial conditions i.e., where the system’s dynamics takes it, is hard to predict from the starting point. One way to demonstrate this is to run two identical chaotic systems side by side, starting both at very close, but not exactly equal initial conditions. The systems soon diverge from each other, but both retain the same attractor pattern. An interesting question to ask is: Can we force the two chaotic systems to follow the same path on the attractor? i.e., Can chaotic systems be made synchronized? The affirmative answer is possible to this question. It has been shown that some of the ideas of synchronization can also be extended for the description of particular type of behaviour in coupled systems possessing chaotic dynamics. For example, Fujisaka and Yamada [6-8] have demonstrated that two identical systems with chaotic individual dynamics can change their behaviour from uncorrelated chaotic oscillations to identical chaotic oscillations as the strength of the coupling between the systems is increased. Synchronization of chaotic systems is important as the noise-like behavior of chaotic systems suggests us that such behavior might be useful in some type of private communications. One glance at the Fourier spectrum from a chaotic system suggests the same. There are typically no dominant peaks, no special frequencies i.e., the spectrum is broadband. To use a chaotic signal in communications we are immediately led to the requirement that somehow the receiver must have a duplicate of the transmitter’s chaotic signal or, better yet, synchronize with the transmitter. In fact, synchronization is a requirement of many types of communication systems, not only chaotic ones. Unfortunately, if we look at how other signals are synchronized, we will get very little insight as to how to do it with chaos. New methods are therefore required. The organization of rest of the paper is as follows: In Section 2, we introduce the jerk dynamical systems, which are emerged as a sub class of dynamical systems with rich variety of dynamical behaviour including the chaotic motion. We also summarize in a tabular form the various classes of jerk dynamical systems, which we have used for studying the identical synchronization of chaos in these systems. In Section 3, we investigate the identical synchronization of chaos in jerk dynamical system using various algorithms along with their brief descriptions. Particularly in subsections 3.1, 3.2 and 3.3, we investigate the identical synchronization of various jerk dynamical systems using Pecora-Carroll (PC) technique, feedback (FB) technique and active-passive decomposition (APD) respectively. We also investigate the stability of identical synchronization in the aforesaid methods using extensive transversal stability analysis (i.e., extensive conditional Lyapunov exponents (CLE’s) calculation) as well as Lyapunov function con- Electronic Journal of Theoretical Physics 3, No. 11 (2006) 33–70 35 struction criterion. Finally in Section 4, we summarize the major conclusions drawn from our study of identical synchronization of chaotic jerk dynamical systems. 2. Chaotic Jerk Dynamical Systems In this section, we briefly discuss how the jerk dynamical systems came into existence in the studies of chaotic dynamics. In recent years, there has been a growing interest to find three dimensional dynamical systems which are functionally as simple as possible but nevertheless chaotic [10-22]. Using extensive computer search, Sprott [9] found 19 distinct chaotic models (one of them is conservative and remaining are dissipative, referred as Model A to S) with three-dimensional vector fields that consist of five terms including two nonlinearities or of six terms with one quadratic nonlinearity. Subsequently, Hoover [10] pointed out that the only conservative system (Model A) found by Sprott is an already known special case of Nose-Hoover thermostat dynamical system, which exhibits time reversible Hamiltonian chaos [11]. Except this the other models B to S were apparently unknown. On the other hand, in 1996 Gottlieb [12] reported that Sprott’s Model A can be recast into an explicit third order form x = J(x, ẋ, ẍ), which he called ‘jerk function’. Because it involves a third derivative of x, which in a mechanical system is a rate of change of acceleration and is called jerk [13]. Since it is known that any explicit ordinary differential equation can be recast in the form of a system of coupled first order differential equations but the converse does not hold in general. Even if one may reduce the dynamical system to a jerk form for each of the phase space variables, the resulting differential equation may look quite different i.e. there may be different possible jerk forms of a single dynamical system. With this, Gottlieb [12] asked a provocative question “What is the simplest jerk function that gives chaos?” By following the study of Gottlieb, Linz [14] reported that the original Rössler model, Lorenz model and the Sprott’s model R can be reduced to jerk form and further showed that the jerk form of Rössler and Lorenz models are rather complicated and are not suitable candidates for the Gottlieb’s simplest jerk function. However, the jerk form of Sprott’s model R demonstrates the existence of a much simpler form of jerk function that exhibits chaos. By following the Linz’s study [14], in 1998 Eichhorn et. al. [15] used the method of Görbner bases and showed that the fifteen of the Sprott’s chaotic flows [9] can be recast into jerk form. They also showed that these fifteen models , the Rössler toroidal model [16] and Sprott’s minimal chaotic flow [17] can be arranged into seven classes (referred as JD1 to JD7) of jerky dynamics as a hierarchy of quadratic jerk equations with increasingly many terms. Table 1 summarizes the classification simple polynomial chaotic flows suggested by Eichhorn et al. [15]. The alphabets in the first column are the labels assigned by Sprott [9], except for ‘TR’ and ‘SJ’ which are the abbereviations for Toroidal Rössler and Simplest Jerk systems respectively. Such a classification provides a simple mean to compare the functional complexity of different systems and also demonstrate the equivalence of cases not otherwise apparent. These seven different classes of chaotic jerk dynamical systems, we have used to study the identical synchronization in section 3. 36 Electronic Journal of Theoretical Physics 3, No. 11 (2006) 33–70 In a subsequent study Eichhorn et. al. [18] examined the simple cases of JD1 and JD2 in more detail and identified the regions of parameter space over which they exhibit chaos. Patidar and Sud [19] have studied in detail, the dynamical behaviour of a special family of jerk dynamical system x = −Aẍ − ẋ + G(x), where A is a system parameter, G(x)is nonlinear function containing one nonlinearity, one system parameter and a constant term. Table 1: Basic classes of chaotic dissipative jerky dynamics Model Basic classes and parameter values for which there is chaotic behavior JD1 I J L N R SJ JD2 k1 k1 k1 k1 k1 k1 M k1 = −1 Q S TR JD3 F G H = = = = = = −1 −β = −2 −1 −γ = −2 −1 −a = −2.017 k1 = β − 1 = −0.5 k1 = −1 k1 = −β = −0.2 k1 k4 k1 k4 k1 k4 = α − 1 = −0.5 1 = 2α = 1 = α − 1 = −0.6 1 = 4α = 0.625 = α − 1 = −0.5 1 = 1 = 2α JD4 O k1 = k4 = − 21 1 4 JD5 D JD6 k1 = −1 P k1 = −1 k4 = 1 JD7 K k1 = α − k4 = 1 1 2α ... x = k1 ẍ + k2 x + xẋ + k3 k2 = −2α = − 0.4 k2 = −αβ = − 4 k2 = −α = − 3.9 k2 = −αγ = − 4 k2 = −α = − 0.9 k2 = − 1 ... x = k1 ẍ + k2 ẋ + x2 + k3 k2 = −β = − 1.7 k2 = β − α = − 2.6 k2 = −α = − 4 k2 = − 1 ... x = k1 ẍ + k2 ẋ + k3 x2 + xẋ + k4 1 − 1 = −2.5 k2 = α − α k2 = α − 1 2α k2 = α − 1 α − 1 = −1.85 − 1 = −2.5 ... x = k1 ẍ + k2 ẋ + k3 x2 + xẍ + k4 k2 = 1 − α = − 1.7 k3 k3 k3 k3 k3 k3 = = = = = = −2α2 = −0.08 αβ (1 − α ) = −4 α (2βγ − α ) = −8.19 α (2β − α γ) = −4 −β = −0.4 0 k3 = −α − 2 β2 4 = −2.4225 k3 = − α4 = −2.4025 k3 = −α2 β = −16 k3 = − 14 (α + β )2 ≈ −0.0858 = −0.25 k3 = − α 2 k3 = −α = −0.4 k3 = − α = −0.25 2 k3 = −1 ... x = k1 ẍ + k2 x2 + k3 ẋ2 + xẍ k2 = 1 k3 = −α = −3 ... x = k1 ẍ + k2 ẋ + k3 x2 + k4 ẋ2 + xẍ + k5 k2 = 1 − α = − 1.7 k3 = − 12 k5 = 1/2 ... x = k1 ẍ + k2 ẋ + k3 x2 + k4 ẋ2 + k5 xẋ + xẍ + k6 1 k2 = α − α k3 = −α = −0.3 − 1 ≈ −2.37 − 12 ≈ −3.53 1 ≈ 0.83 k5 = 2 − α = 1.7 k6 = 4α They have identified the regions of parameter space, where different type of long time dynamical behaviour dominates, using some analytical methods as well as extensive Lyapunov spectra calculation in complete parameter space by considering five different forms of nonlinear function G(x) comprising of absolute, quadratic, cubic, quartic and quintic nonlinearities. They observed that only systems having absolute and quadratic nonlinearities in G(x) exibit chaos. As a result they made an important conclusion for these jerk dynamical systems that a certain amount of nonlinearity is sufficient for exhibiting chaotic behaviour but more is not necessarily better. Apart from the above studies on the dynamics of jerk dynamical system, Patidar et. al. [20] and Patidar and Sud [21] respectively, have attempted the problems of controlling chaos and synchronization of Electronic Journal of Theoretical Physics 3, No. 11 (2006) 33–70 37 chaos in jerk dynamical systems. In the next section, we investigate the identical synchronization of chaos in jerk dynamical system using various algorithms along with their brief descriptions. 3. Identical Synchronization In Chaotic Jerk Dynamical Systems Early work on synchronization of coupled chaotic systems was done by Fujisaka and Yamada [6-8]. In that work, some sense of how the dynamics might change was brought out by a study of the Lyapunov exponents of synchronized, coupled systems. Although Fujisaka and Yamada [6,-8] were the first to exploit local analysis for the study of synchronized chaos, their papers went relatively unnoticed. Later, a now-famous paper by Afraimovich et. al. [22] exposed many of the concepts necessary for analyzing synchronous chaos, although it was not until many years later that wide-spread study of synchronized, chaotic systems took hold. As already mentioned, chaotic systems are dynamical systems that defy synchronization due to their essential feature of displaying high sensitivity to initial conditions. As a result, two identical chaotic systems starting at nearly the same initial points in phase space develop onto trajectories which become uncorrelated in the course of time. However, it has been shown that it is possible to synchronize these kind of systems, to make them evolving on the same chaotic trajectory [6, 22, 23, 24, 25]. When we deal with coupled identical systems, synchronization appears as the equality of the state variables while evolving in time. We refer to this type of synchronization as identical synchronization (IS). Other names in literature for this notion are: complete synchronization or conventional synchronization [26]. In this section we will discuss main properties of this kind of synchronization. The appearance of this kind of synchronization have been estabilished by means of several different coupling mechanisms, such as Pecora and Carroll method [24, 25, 27], the negative feedback [28], the sporadic driving [29], the active-passive decomposition [30, 31], the diffusive coupling and some hybrid method of Guemez and Matias [32] etc. In this section, we will concentrate our attention to explain the properties and stability of synchronized motion in three particular coupling schemes namely Pecora and Carroll (PC) method, feedback technique and active passive decomposition (APD). 3.1 Pecora Carroll (PC) Technique Pecora and Carroll [24, 25] discovered a way to achieve identical synchronization. They take a complete chaotic system and choose a subsystem within it. Then they make a replica of this subsystem. The original system is called drive (master) system and the duplicate subsystem is called response (slave). The response is just like the drive except it is missing one or more variables. The missing variables are sent from the drive to the response, inputting the variable wherever it is needed in the response subsystem. If a stable response subsystem has been chosen, then the response’s dynamic variables will 38 Electronic Journal of Theoretical Physics 3, No. 11 (2006) 33–70 converge to their counterparts in the drive and will remain synchronized with them as long as the drive continues to supply the missing variables to the response subsystem i.e. the key idea for identical synchronization is to choose a stable subsystem. Consider an autonomous n-dimensional dynamical system Ẋ = F (X), (1) here X = (x1 , x2 , x3 , ..., xn )T and F (X) = (f1 (X), f2 (X), ..., fn (X))T . Divide the system into two parts in an arbitrary way as: X = (v, w), v̇ = g(v, w), ẇ = h(v, w), (2) where v = (x1 , x2 , ..., xm ), g = (f1 (X), f2 (X), ..., fm (X)), w = (xm+1 , xm+2 , ..., xn ) and h = (fm+1 (X), fm+2 (X), ..., fn (X)). Now create a new subsystem w′ identical to w system and, substitute the set of variables v for the corresponding v ′ in the function h i.e., ẇ′ = h(v ′ , w′ )and v ′ = v i.e. the new subsystem will be drived by the original system ẇ′ = h(v, w′ ). (3) In such a way we have the following compound system: v̇ = g(v, w), ẇ = h(v, w), ẇ′ = h(v, w′ ) (4) Now we examine the difference Δw = w − w′ , Under the right condition (i.e. if the chosen subsystem is stable) both the system will synchronize as time grows i.e. Δw → 0 as t → ∞. In order to study the dynamics of difference Δw, subtract Eq. (3) from Eq. (2), we get an equation for the dynamics of Δw as: ẇ − ẇ′ = d (Δw) = Dw h(v, w) Δw, dt (5) where Dw his the matrix of derivatives of hwith respect to w i.e. the Jacobian of the w subsystem vector field with respect to w only. The technique described above is also known as complete replacement technique because in this technique we completely replace all the v ′ variables of response subsystem by their counterparts (v) in the original system. It is clear that in the phase space of the compound system described by Eq. (4), there exists a manifold w = w′ such that if the initial conditions lie in this manifold, the consequent evolution of the system will take place in this manifold as well. In other words, the manifold w = w′ is an invariant manifold and is called synchronization manifold. In the above analysis, we have said that both systems will show identical chaotic oscillations if a stable subsystem is chosen. In order to define which subsystem is stable, Electronic Journal of Theoretical Physics 3, No. 11 (2006) 33–70 39 one needs to study the stability of synchronized oscillations. Since in our case the synchronized oscillations are chaotic, these trajectories are always unstable. However in the analysis of synchronized chaotic motions one has to distinguish between the instability for perturbation tangent to synchronization manifold and transverse to it. The regime of identical chaotic oscillations is stable, when the synchronized trajectories are stable for the perturbations in the transverse direction to the synchronization manifold. This analysis is called transversal stability analysis. The Eq. (5) is a basic equation for much of the discussion on synchronized chaotic systems. Three most frequent criteria for the stability of synchronized chaotic motions are: (1) The criterion based on eigenvalues of the Jacobian matrix corresponding to the flow evaluated on synchronization manifold (i.e. Jacobian matrix defined on difference system) has been introduced by Fujisaka and Yamada [6, 7], which requires that for the stable synchronization the largest eigenvalue of the aforesaid Jacobian matrix should be negative. For the case of Pecora Carroll technique discussed above, this criterion states that the largest eigenvalue of the Jacobian matrix Dw hmust be negative for onset of synchronization. If the Dw h is constant over the attractor i.e. response is linear then the calculation of eigenvalues (λ1 , λ2 , ..., λn−m ) is straightforward and hence one can easily conclude whether the synchronization is possible or not. But the complications arise; when the Jacobian is not constant over the attractor i.e. response system is nonlinear in nature (In case of PC method the response means the subsystem, but for the other coupling mechanisms, where subsystem is not required, we will consider the difference system instead of response system i.e. the basic equation (Eq. (5)), which describe the dynamics of difference of drive and response systems). (2) He and Vaidya [27] developed a criterion for chaos synchronization based on the notion of asymptotic stability of dynamical systems, which refers to the condition for a given chaotic system with drive-response (master-slave) configuration to reach the same eventual state at a fixed time irrespective of the choice of initial conditions. One of the practical way to establish the asymptotic stability of the response subsystem is to find a suitable Lyapunov function L(Δw)(where Δw represents the difference) that satisfy the following properties: a) L(Δw) ≥ 0in a certain region containing the chaotic set of synchronized motion. b) L(0) = 0, In other words, L(•)is zero everywhere on the synchronization manifold (i.e.w = w′ ). c) dL ≤ 0 everywhere in a certain region containing the chaotic set of synchronized dt motion and dL = 0 on the synchronization manifold. dt If one succeeds in construction of such Lyapunov function for a particular choice of response subsystem then one can conclude that the subsystem is stable and the chaotic set of synchronized oscillations is transversally stable therefore the composite system evolves towards the regime of identical synchronized chaotic oscillations. The criterion based on construction of Lyapunov function for the vector field 40 Electronic Journal of Theoretical Physics 3, No. 11 (2006) 33–70 of perturbation transverse to the synchronization manifold enables one to prove that all trajectories in the phase space of the coupled systems are attracted by the synchronization manifold. Despite the fact that it is a very reliable criterion, which guarantees the onset of chaotic synchronization, it has two serious disadvantages. First of all, there is no generic method for construction of Lyapunov function for an arbitrary system. As a result the search for it is time consuming or practically impossible even though these systems are known to be synchronized. Thus this method is not general. The second disadvantage is that the estimate for the minimum coupling strength necessary to synchronize two systems provided by this method is usually not accurate. It is found that the systems can be synchronized by much weaker coupling than predicted by the Lyapunov function method. (The second disadvantage is not applicable for the PC method as there is no coupling parameter involved in PC technique. However it is applicable for feedback methods, where we introduce the coupling between the drive and response systems rather than creating a new subsystem, as we did in PC method. These methods will be clear in the subsequent sections). (3) In contrast with the Lyapunov function criterion, the analysis of transversal Lyapunov exponents is quite straight forward and can be easily employed, even for rather complicated systems. This can be done by calculating the Lyapunov exponents of Eq. (5) by using methods similar to ones used for the computation of conventional Lyapunov exponents for any dynamical systems [33-36]. When all the (n − m) Lyapunov exponents are negative, then the compound system will move towards the synchronization manifold. Note that the full system (1) has a set of n Lyapunov exponents (at least one will be positive because we are considering the chaotic system), but the Lyapunov exponents for Eq. (5) do not form a subset of the Lyapunov exponents of full system because they depend upon the drive variables (v) so called conditional Lyapunov exponents (CLE’s). Using Pecora and Carroll approach described above, identical synchronization of chaos has been demonstrated numerically as well as experimentally on several chaotic systems such as Lorenz system [24, 25, 27, 37-32], Rossler system [24], the hysteretic circuit [24] Chua’s circuit [43], driven Chua’s circuit [44], DVP oscillator [45-47], phase-locked loops [48-50] etc. Now, we demonstrate Pecora and Carroll approach for identical synchronization by taking chaotic jerk dynamical systems as prototypical examples. First of all we consider the two chaotic jerk dynamical systems studied by Patidar and Sud [19], which are members of x = −Aẍ − ẋ + G(x)(for G(x) = B (x 2 + C) and G(x) = B ( |x| + C)), then we will extend the results to the jerky representations of the Sprott’s simple chaotic flows by considering their classification according to the hierarchy of quadratic jerk functions with increasingly many more terms (see Table 1 for detailed classification). We consider the jerk equation of the form: x + A ẍ + ẋ = G(x) (6) Electronic Journal of Theoretical Physics 3, No. 11 (2006) 33–70 41 The dynamical system representation of Eq. (6) in terms of ODE’s by using the transformation ẋ = yandẏ = zwill be: ẋ = y ẏ = z ż = − Az − y + G(x) (7) There are three possibilities to implement the PC method in this system. (i) x-drive configuration: If we consider the x as drive variable then the drive and response systems for jerk dynamical system (Eq. (7)) will be as ẋ = y ẏ = z ż = − Az − y + G(x) and ẏ ′ = z ′ ż ′ = −A z ′ − y ′ + G(x) and the difference system for Δy = y − y ′ and Δz = z − z ′ in matrix form is     d (Δy) Δy 0 1 dt = d Δz −1 − A (Δz) dt (8) (9) Eq. (9) is the basic equation for describing the stability of the perturbation transverse to the synchronization manifold under x-drive configuration for jerk dynamical system (Eq. (7)). Since in this equation the coefficient matrix (Jacobian matrix) is a constant matrix and it is very√easy to calculate the eigenvalues of this matrix. The eigenvalues for 2 this matrix are: −A ± 2 A −4 , for A = 0.6(chaotic case) the eigenvalues are −0.3 ± i 0.95, which give the following solution for Eq. (9) Δy = Δz = e−0.3t (K1 cos (0.95t) + K2 sin (0.95t)) (10) here K1 and K2 are constants of integration. It is clear that as t → ∞, Δy = Δz = 0, hence drive and response systems synchronize. If we numerically calculate the Lyapunov exponents of the subsystem (subjected to x-drive condition) i.e. conditional Lyapunov exponents (CLE’s), these come out to -2.9986E-1, -3.0013E-1 (for both the cases (i) G(x) = B(x 2 + C) with B = 0.58 and C = −1 (ii) G(x) = B( |x| + C) with B = 1.0 and C = −2.0). As all the CLE’s are negative and hence synchronization is possible. In the above paragraph, we have analyzed the possibility of identical synchronization in chaotic jerk dynamical system (Eq. (7)) for x-drive configuration using eigenvalues of the Jacobian matrix and CLE’s. Now we analyze the possibility of synchronization using the Lyapunov function construction method. If we consider the Lyapunov function    1 1+A 2 2 2 2 L(Δy, Δz) = (11) (Δy) + (Δz) + (Δy + Δz) + A (Δy) , 2 A here Δy = y − y ′ and Δz = z − z ′ . Lyapunov function L (Δy, Δz)is positive definite and L(0, 0) = 0 i.e., it is zero everywhere on the synchronization manifold (y = y ′ and z = z ′ ). Now the time rate of change of Lyapunov function is given by dL dt (Δy, Δz) = 1+A A (Δy) dtd (Δy) + (Δz) dtd (Δz) + (Δy + Δz) dtd (Δy) + d (Δz) dt + A (Δy) dtd (Δy) 42 Electronic Journal of Theoretical Physics 3, No. 11 (2006) 33–70 = − 2A (Δz)2 + (Δy)2 (12) ≤ 0and also dL = 0 on the synchronization manifold, hence the for A = 0.6 > 0, dL dt dt synchronization is possible i.e. Δy = Δz = 0 as t → ∞. From the analysis given above, we found that all the three criteria guarantee the identical synchronization of chaotic jerk dynamical systems described by Eq. (7) using the x-drive configuration. Fig. 1 Identical synchronization of chaos in jerk dynamical system having quadratic nonlinearity using Pecora-Carroll technique (i.e., Eq. (8) with G(x) = B(x2 + C)) for A = 0.6, B = 0.58 and C = 1.0: (a) time history of y-variables of drive and response systems (b) the difference between y-variables of drive and response as a function of time (c) time history of z-variables of drive and response systems (d) the difference between z-variables of drive and response as a function of time. We have also numerically solved the drive-response system of jerk dynamical system for x-drive configuration i.e., Eq. (8) by using fourth-order Runge-kutta integrator with the step size 0.01. The results of the numerical calculations are shown in Figures 1 and 2 for G(x) = B(x2 + C)with A=0.6, B=0.58 and C = −1. Particularly in Figures 1(a) and 1(b), we have shown the y−variables of both drive and response systems and difference between y−variables of both drive and response systems respectively. It is clear that ast increases the y-variables of drive and response systems synchronize i.e., difference becomes zero. Similar behaviour is discussed in Figures 1(c) and 1(d) for z-variables. There are more ways to visualize identical synchronization between the drive and response systems. Some of them we have shown in Figure 2. Particularly in Figure 2(a), we have plotted the y-variable of drive versus y-variable of response while in Figure 2(b) z-variable of drive versus z-variable of response (after some transient time die out). Both plots are straight lines inclined with an angle of 45o and passing through the origin (0,0) which also show the synchronized behaviour of drive and response systems. In Figure 2(c) we have shown the trajectories of the drive and response systems in the xy-plane. We see that the drive and response start with different initial conditions but after some time both converge to the same trajectory. A better view has been shown in Figure 2(d) in which we have plotted the trajectory in the ΔyΔz − plane (i.e. difference Electronic Journal of Theoretical Physics 3, No. 11 (2006) 33–70 43 Fig. 2 Identical synchronization of chaos in jerk dynamical system having quadratic nonlinearity using Pecora-Carroll technique (i.e., Eq. (8) with G(x) = B(x2 + C)) for A = 0.6, B = 0.58 and C = −1.0: (a) y-variable of drive versus y-variable of response system (b) z-variable of drive versus z-variable of response system (c) the trajectory of drive and response systems in yz-plane (d) the trajectory of the difference of y-variables and z-variables of drive and response systems in difference plane. Fig. 3 Identical synchronization of chaos in jerk dynamical system having absolute nonlinearity using Pecora-Carroll technique (i.e., Eq. (8) with G(x) = B(|x| +C)) for A = 0.6, B = 1.0 and C = −2.0: (a) time history of y-variables of drive and response systems (b) the difference between y-variables of drive and response as a function of time (c) time history of z-variables of drive and response systems (d) the difference between z-variables of drive and response as a function of time. plane), it is a spiral ending at a fixed point (0,0). Results of the similar numerical calculations for Eq. (8) with G(x) = B(|x | + C) for A = 0.6, B = 1.0 and C = -2.0 are shown in Figures 3 and 4. (ii) y-drive configuration: If we consider y as a drive variable then the drive and response systems for jerk dynamical system (Eq. (7)) will be as 44 Electronic Journal of Theoretical Physics 3, No. 11 (2006) 33–70 Fig. 4 Identical synchronization of chaos in jerk dynamical system having absolute nonlinearity using Pecora-Carroll technique (i.e., Eq. (8) with G(x) = B(|x| +C)) for A = 0.6, B = 1.0 and C = −2.0: (a) y-variable of drive versus y-variable of response system (b) z-variable of drive versus z-variable of response system (c) the trajectory of drive and response systems in yz-plane (d) the trajectory of the difference of y-variables and z-variables of drive and response systems in difference plane. ẋ = y ẏ = z ż = − Az − y + G(x) and ẋ′ = y ż ′ = −A z ′ − y + G(x′ ) (13) and the difference system for Δx = x − x′ and Δz = z − z ′ will be as: d (Δx) dt d (Δz) dt = y −y = 0 = −A(Δz) + G(x) − G(x′ ) (14) Eq. (14) is the basic equation for describing the stability of the perturbation transverse to the synchronization manifold under y-drive configuration for jerk dynamical system (Eq. (7)). Since in this equation the coefficient matrix (Jacobian matrix) is not a constant matrix as in case of x-drive configuration. So it is not easy to calculate the eigenvalues of this matrix because these will depend on the time evolution of x and z variables of both drive and response systems. Instead of finding the eigenvalues, we numerically calculate the conditional Lyapunov exponents and which are 2.9899E-4 (≈ 0) & -6.0029E-1 (for G(x) = B(x 2 + C);A = 0.6, B = 0.58 and C = −1.0) and 5.4307E-4 (≈ 0) & -6.0054E-1 (for G(x) = B(|x|+C);A = 0.6, B = 1.0 and C = −2.0). Since one of them (in both cases) is approximately zero or positive definite and hence identical synchronization is not possible. We have also numerically solved the drive-response system of jerk dynamical system for y-drive configuration i.e., Eq. (13) by using fourth-order Runge-kutta integrator with the step size 0.01. The results of the numerical calculations are shown in Figure 5 (for Electronic Journal of Theoretical Physics 3, No. 11 (2006) 33–70 Fig.5 Identical synchronization of chaos in Fig.6 Identical synchronization of chaos in jerk jerk dynamical system having quadratic nonlinearity using Pecora-Carroll technique (Eq. (13) with G(x) = B(x2 + C)) for A = 0.6, B = 0.58 and C = −1.0: (a) the difference between x-variables of drive and response as a function of time (b) the difference between zvariables of drive and response as a function of time (c) ) x-variable of drive versus x-variable of response system (d) z-variable of drive versus z-variable of response system. dynamical system having absolute nonlinearity using Pecora-Carroll technique (Eq. (13) with G(x) = B(|x| +C)) for A = 0.6, B = 1.0 and C = −2.0: (a) the difference between xvariables of drive and response as a function of time (b) the difference between z-variables of drive and response as a function of time (c) ) x-variable of drive versus x-variable of response system (d) z-variable of drive versus z-variable of response system. 45 G(x) = B(x 2 + C); A = 0.6, B = 0.58 and C = −1.0) and in Figure 6 (for G(x) = B(|x | + C);A = 0.6, B = 1.0 and C = −2.0). Particularly in Frames (a) of both figures, we have shown the difference between x-variable of the drive and response systems while in Frames (b) the difference between the z-variables of the drive and response systems. We observe that the difference between the x-variables of drive and response systems is constant and equal to the initial difference. It is so because one of the CLE’s is equal to zero for y-drive configuration, which suggests that neither the initially nearby trajectories diverge nor converge. However the difference between the z-variables of the drive and response systems is large and oscillatory and hence the drive and response systems do not synchronize for y-drive configuration. In Frames (c) of both the figures, we have plotted the x-variables of drive versus the x-variable of response system. It is a straight line passing through the point (0, 16) not through origin, which suggests that x-variables of drive and response systems do not synchronize (x = x′ ) but have a linear relationship 46 Electronic Journal of Theoretical Physics 3, No. 11 (2006) 33–70 (x′ = x + 16). However the plot of z versus z ′ (see Frames (d) of both the Figures) is a complicated geometrical structure, which shows the non-synchronized behaviour of z−variables of drive and response systems and also suggests that zand z ′ have a nonlinear relationship as evident from a complicated attractor type structure in Frames (d). (iii) z-drive configuration: If we consider z as the drive variable then the drive and response systems for jerk dynamical system (Eq. (7)) will be as: ẋ = y ẏ = z ż = − Az − y + G(x) and ẋ′ = y ′ ẏ ′ = z (15) and the difference system for Δx = x − x′ and Δy = y − y ′ in matrix form is     d (Δx) 0 1 Δx dt = (16) d 0 0 Δy (Δy) dt Eq. (16) is the basic equation for describing the stability of the perturbation transverse to the synchronization manifold under z-drive configuration for jerk dynamical system (Eq. (7)). Since in this equation the coefficient matrix (Jacobian matrix) is a constant matrix and it is very easy to calculate the eigenvalues of this matrix. The eigenvalues for this matrix are: 0, 0 for any value ofA and hence the trajectories for x(t)&x′ (t)and y(t) & y ′ (t)respectively, will remain apart by a constant distances |x(0) − x′ (0)|and |z(0) − z ′ (0)| , in this case also synchronization is not possible. If we numerically calculate the conditional Lyapunov exponents, these turn out to be (0.0, 0.0), which are consistent with the above results. From the analysis of identical synchronization using PC technique for the jerk dynamical system (Eq. (7)), we can summarize the results in the Table 2. Table 2: Conditional Lyapunov exponents (CLE’s) for different combination of drive and response systems for jerk dynamical system having quadratic and absolute nonlinearities. Drive Variable x y z x y z ... x + Aẍ + ẋ = B(x2 + C ); A = 0.6; B = 0.58; C = −1 Response Subsystem Conditional Lyapunov Exponents Whether IS is possible or not? (y, z) -2.9986E-1, -3.0013E-1 Yes (x, z) 2.9899E-4, -6.0029E-1 No (x, y) 0.0000E-0, 0.0000E-0 No ... x + Aẍ + ẋ = B(|x| + C); A = 0.6; B = 1; C = −2 (y, z) -2.9986E-1, -3.0013E-1 Yes (x, z) 5.4307E-4, -6.0054E-1 No (x, y) 0.0000E-0, 0.0000E-0 No In the above analysis, we have discussed in detail the possibilities of identical synchronization of chaos in jerk dynamical systems having quadratic and absolute nonlinearities using the Pecora-Carroll approach. Now we extend our calculations for the identical synchronization of chaos using PC approach to the jerk dynamical representations of the Sprott’s simple chaotic flows [9] by considering their classification [15] according to the hierarchy of quadratic jerk functions with increasing many more terms. In Table 3, we have summarized all these results. We observe from the Table 3 that the x-drive configuration for all the jerk dynamical systems guarantees the identical synchronization except Electronic Journal of Theoretical Physics 3, No. 11 (2006) 33–70 47 for the models L, R and D, where no subsystem is stable. However the y-drive and z-drive configurations are unstable for all the jerk dynamical systems. 48 Electronic Journal of Theoretical Physics 3, No. 11 (2006) 33–70 3.2 Feedback (FB) Technique In Section 3.1, we observed that Pecora and Carroll method of chaos synchronization works fairly good for the jerk dynamical systems. However it requires dividing the original Electronic Journal of Theoretical Physics 3, No. 11 (2006) 33–70 49 system into two stable subsystems. In this section, we address the same problem of synchronization but in different way by considering the following question: Can one make a chaotic trajectory of one system to synchronize with a chaotic trajectory of other identical chaotic system, starting with different initial conditions without dividing the original system into two stable subsystems? It is possible by means of feedback method or one-way coupling method [45, 47, 51-53]. By one-way coupling we mean that the behaviour of second chaotic system (response) is dependent on the behaviour of the first identical system (drive) but the first one (drive) is not influenced by the behaviour of second (response) system. The one way coupling method is also called as feedback method as in this we choose a drive variable from the drive system and feedback control is applied to the response system. The feedback control is proportional to the difference of the chosen variable from drive and its counterpart from response system. Under suitable conditions, as time elapses, the amount of feedback decreases and soon both the drive and response systems achieve complete synchronization by following the same trajectory and afterward the feedback amount becomes zero and the identical synchronization persists. The similar feedback method has been used by Singer et. al. [54], Pyragas [55], Chen and Dong [56], Pyragas [57], Kapitaniak and Chua [58], Kapitaniak et. al. [59] for the controlling of chaos, where the drive system exhibits the periodic motion of desired periodicity while the response system is chaotic and the aim of the feedback control is to bring the response system to the periodicity of drive system. In the following, we discuss the formulation and stability of the feedback method for identical synchronization of chaos. Consider an n-dimensional dynamical system, which is chaotic, as Ẋ = F (X), (17) where X = (x1 , x2 , x3 , ..., xn )T and F (X) = (f1 (X), f2 (X), ..., fn (X))T . Now choose a dynamical variable as drive variable from it e.g. xi (1 ≤ i ≤ n). Consider another chaotic system identical to Eq. (17) but starting from different initial conditions (i.e. with different variables), as Ẋ ′ = F (X ′ ), (18) where X ′ = (x1 , x2 , ..., xn )T and F (X ′ ) = (f1 (X ′ ), f2 (X ′ ), ..., fn (X ′ ))T . Now the feedback control, which is proportional to the difference of the drive variable xi and its counterpart xi in the response, is applied to response system. Hence the response system looks as ′ ′ ′ ′ ẋ1 = f1 (x1 , x2 , ...., xn ) . ′ ′ ′ ′ ′ ẋi = fi (x1 , x2 , ...., xn ) − c(xi − xi ) , (19) . . ′ ′ ′ ′ ẋn = fn (x1 , x2 , ...., xn ) where c is a constant and termed as feedback constant or coupling strength. The pair of drive and response dynamical systems (i.e. Eqs. (17) and (19)) synchronize if the 50 Electronic Journal of Theoretical Physics 3, No. 11 (2006) 33–70 dynamical system describing the evolution of the difference, ⎡ ⎤ ⎡ ′ ⎤ ⎡ ⎤ ′ ′ ′ ẋ1 − ẋ1 f1 (x1 , x2 , ..., xn ) − f1 (x1 , x2 , ..., xn ) ė1 ⎢. ⎥ ⎢. ⎥ ⎢. ⎥ ⎢ ⎥ ⎢ ′ ⎥ ⎢ ⎥ ′ ′ ′ ′ ⎢ ėi ⎥ ⎢ ẋi − ẋi ⎥ ⎢ fi (x1 , x2 , ..., xn ) − fi (x1 , x2 , ..., xn ) − c(xi − xi ) ⎥ = = ⎢. ⎥ ⎢ ⎥ ⎢ ⎥, ⎢ ⎥ ⎢. ⎥ ⎢. ⎥ ⎣. ⎦ ⎣. ⎦ ⎣. ⎦ ′ ′ ′ ′ ėn ẋn − ẋn fn (x1 , x2 , ..., xn ) − fn (x1 , x2 , ..., xn ) (20) possesses a stable fixed point at the origin E = 0,where E = (e1 , e2 , ..., en )T and ′ ei = xi − xi for i = 1 to n. The dynamics of the error system can be understood by studying the following linearized system for small E: Ė = DF · E, (21) where DF is the Jacobian (matrix of derivatives ) of response system with respect to Ei.e. ⎤ ⎡ ∂f1 ∂f1 1 . . ∂e . ∂f ∂e1 ∂ei n ⎢ . . . . . . ⎥ ⎥ ⎢ ∂f ∂f ∂fi ⎥ i i ⎢ . − c . . ⎢ ∂e ∂e ∂e n ⎥, i DF = ⎢ 1 (22) ⎥ . . . . . . ⎥ ⎢ ⎣ . . . . . . ⎦ ∂fn ∂fn n . ∂ei . . ∂f ∂e1 ∂en All the derivatives in the Jacobian matrix are evaluated at E = 0i.e., on the synchronization manifold. It is clear that the coupling strength c also appears in equation of error system and hence it also affects the stability of synchronization. Now the question arises: for which selection of drive variable and coupling strength the identical synchronization will be stable. To study the stability of identical synchronization in this case, one can use the three basic criteria introduced in the Section 3.1: (i) If the largest eigenvalue of the Jacobian matrix DF appeared in Eq. (21) is negative then drive and response systems will possess identical synchronization [6-8]. (ii) If a suitable Lyapunov function [27] L(E)exists, which satisfies: (a)L(E) ≥ 0, (b) L(0) = 0, (c) dL(E)/dt ≤ 0 and (d) dL(0)/ = 0. (iii) If all the Lyapunov exponent of the system described by Eq. (21) (condt ditional Lyapunov exponents) are negative, then the identical synchronization between drive and response systems will be stable [24, 25]. Now we use the feedback technique described above to demonstrate the identical synchronization of chaos in jerk dynamical systems. First of all we consider the two jerk dynamical systems, which have been studied in detail by Patidar and Sud [19], then we will extend the results to the jerky representations of the Sprott’s simple chaotic flows by considering their classification (as suggested by Eichhorn et al. [15]) according to the hierarchy of quadratic jerk functions with increasing many more terms [see Table 1 for detailed classification] First, we consider the following family of jerk dynamical system: x = −A ẍ − ẋ + G(x), (23) Electronic Journal of Theoretical Physics 3, No. 11 (2006) 33–70 51 where A is a system parameter and G(x) is a nonlinear function. The dynamical system representation of Eq. (23) in terms of first order ODE’s by using the transformations ẋ = yand ẏ = zwill be as: ẋ = y ẏ = z . (24) ż = −Az − y + G(x) Now choose a dynamical variable x as drive variable and applied the feedback control as explained in the previous subsection. The response system after inclusion of feedback term will look like as: ẋ′ = y ′ − c(x′ − x) ẏ ′ = z ′ , (25) ż ′ = −Az ′ − y ′ + G(x′ ) where c is the coupling strength between the drive and response system. The dynamical system for the evolution of difference between drive and response systems described by Eqs. (24) and (25) will be as: Ė = DF · E, (26) where E = (x′ − x, y ′ − y, z ′ − z)T and DF = " −c 0 ∂G/∂x 1 0 0 1 −1 −A # . (27) The derivative in the Jacobian matrix DF is evaluated atE = 0. It is clear that for a nonlinear functionG(x), the Jacobian matrix given above is not a constant matrix and it is not easy to calculate its eigenvalues. For analyzing the stability of identical synchronization in this case, we have performed the numerical calculation of the Lyapunov exponents of Eq. (26) (i.e. conditional Lyapunov exponents). First of all we have numerically integrated the coupled systems described by Eqs. (24) and (25) with A = 0.6, G(x) = 0.58 (x 2 − 1) and for different values of coupling strength (c). In Figure 7, We have shown the solutions for two specific values of coupling strength c = 0and c = 0.8. Particularly in Frames (a), (b) and (c) respectively, we have shown the solution of coupled systems in xx′ -plane, yy ′ -plane and zz ′ -plane for c = 0 i.e when both the systems are evolving independently from different initial conditions. It is clear that these figures correspond to an unsynchronized motion. While in Frames (d), (e) and (f) respectively, we have shown the solution of coupled systems in xx′ -plane, yy ′ -plane and zz ′ -plane for c = 0.8. In all these frames ((d), (e) and (f)), we observe a line inclined at an angle of 45o and passing through the origin, which suggests the equality of all three dynamical variable of drive with its counterparts in response systems i.e. identical synchronization. For both cases, we have also calculated the Lyapunov exponents of the response system. For c = 0, the drive and response systems are chaotic and evolve independently and hence both have the same set of Lyapunov exponents i.e.(0.6832E01, 0.0, -0.6683). For c = 0.8, the response system is dependent on the behaviour of drive system however the drive system is not influenced by the behaviour of response system. Hence the drive system possesses the same Lyapunov exponents as for c = 0, 52 Electronic Journal of Theoretical Physics 3, No. 11 (2006) 33–70 Fig. 5 Identical synchronization of chaos in jerk dynamical system using feedback technique (Eqs. (24) and (25)) for A = 0.6and G(x)=0.58(x 2 − 1): Frames (a), (b) and (c) respectively, show the unsynchronized motion of x, y and z variables of drive and response systems, when both systems are uncoupled i.e. c = 0.0 while Frames (d), (e) and (f) respectively, show the synchronized motion of x, y and z variables of drive and response systems, when both systems are coupled through coupling strength c = 0.8. while the values of Lyapunov exponents of response system i.e. conditional Lyapunov exponents (CLE’s) are (-0.4653E-1, -0.2705E-0, -0.1085E+1). From the values of CLE’s it is clear that all the CLE’s are negative or the largest CLE is negative and hence identical synchronization is stable for c = 0.8. For getting the complete information for the value of coupling strength, at which the identical synchronization is stable or unstable, we have calculated the conditional Lyapunov exponents for a certain range of coupling strength (c) in step of 0.01. In Figure 8, we have shown the plot of largest CLE as a function of coupling strength. We observe from this figure that when c is greater than 0.65 then the largest CLE is negative and hence identical synchronization is stable while for c less than 0.65, the largest CLE is positive and identical synchronization is unstable. We have also numerically integrated the coupled systems described by Eqs. (24) and (25) with A = 0.6, G(x) = ( |x| − 2) and for different values of coupling strength (c). In Figure 9, we have shown the solutions for two specific values of coupling strength c = 0 (unsynchronized) and c = 0.7(synchronized). For both cases, we have also calculated the Lyapunov exponents of the response system. For c = 0, when both the drive and response systems are chaotic and evolve independently hence have the same set of Lyapunov exponents i.e. (+0.3148E-01, 0.0, -0.6314E-0) while for c = 0.7 the response system is dependent on the behaviour of drive system however the drive system is not influenced by the behaviour of response system hence the drive system possess the same Lyapunov exponents as for c = 0, while the values of Lyapunov exponents of response system i.e. conditional Lyapunov exponents (CLE’s) are (-0.2269E-1, -0.3072E-0, -0.9716E-0). All Electronic Journal of Theoretical Physics 3, No. 11 (2006) 33–70 53 Fig. 6 Largest conditional Lyapunov exponent (CLE of Eq. (26)) as a function of coupling strength (c) for A = 0.6and G(x)=0.58(x 2 − 1). the CLE’s are negative or the largest CLE is negative and hence identical synchronization is stable for c = 0.7. For getting the complete information for the value of coupling strength at which identical synchronization is stable or unstable, we have calculated the conditional Lyapunov exponents for a certain range of coupling strength (c) in step of 0.01. In Figure 10, we have shown the plot of largest CLE as a function of coupling strength. We observe from this figure that when c is greater than 0.55 then the largest CLE is negative and hence identical synchronization is stable while for c less than 0.55 largest CLE is positive and identical synchronization is unstable. Finally, we extend the feedback technique for identical synchronization of chaos in jerk dynamical representations of the Sprott’s simple chaotic flows [9] by considering their classification [15] according to the hierarchy of quadratic jerk functions with increasing many more terms. In Figures 11 to 14, we have shown the largest CLE as a function of coupling strength for the jerk dynamical representations of Sprott’s simple chaotic flows, when x-variable is used as a drive for calculating the feedback control term. In the respective frames we have also depicted the threshold value of coupling strength beyond which the largest CLE is negative i.e. identical synchronization is possible. 3.3 Active Passive Decomposition (APD) In Sections 3.1 and 3.2, we have discussed two different techniques for synchronizing two identical chaotic systems as well as their application to various jerk dynamical systems. In Section 3.1, we observed that the drive and response technique introduced by Pecora 54 Electronic Journal of Theoretical Physics 3, No. 11 (2006) 33–70 Fig. 7 Identical synchronization of chaos in jerk dynamical system using feedback technique (Eqs. (24) and (25)) for A = 0.6and G(x)=(|x| -2): Frames (a), (b) and (c) respectively, show the unsynchronized motion of x, y and z variables of drive and response systems, when both systems are uncoupled i.e. c = 0.0 while Frames (d), (e) and (f) respectively, show the synchronized motion of x, y and z variables of drive and response systems, when both systems are coupled through coupling strength c = 0.7. Fig. 8 Largest conditional Lyapunov exponent (CLE of Eq. (26)) as a function of coupling strength (c) for A = 0.6and G(x)=(|x| -2). and Carroll [24, 25] works fairly for synchronizing various identical jerk dynamical systems. The x-drive configuration for all jerk dynamical systems guarantees the identical synchronization except for the models L, R and D, where no subsystem is found to be Electronic Journal of Theoretical Physics 3, No. 11 (2006) 33–70 55 Fig. 9 Largest conditional Lyapunov exponent (CLE of Eq. (26)) as a function of coupling strength (c) for JD1. Fig. 10 Largest conditional Lyapunov exponent (CLE of Eq. (26)) as a function of coupling strength (c) for JD2. stable. However the y-drive and z-drive configurations for all the jerk dynamical systems lead to unstable synchronization. One important point we note that in case of Pecora and Carroll technique of identical synchronization only a finite number of possible decompositions exist, which are bounded 56 Electronic Journal of Theoretical Physics 3, No. 11 (2006) 33–70 Fig. 11 Largest conditional Lyapunov exponent (CLE of Eq. (26)) as a function of coupling strength (c) for JD3. by the number of different subsystems. For a n-dimensional dynamical system, number of possible decomposition in terms of drive and response subsystem is n(n-1)/2. In general only a few of the possible response subsystems possess negative conditional Lyapunov exponents and may be used to implement synchronizing systems. Similarly, from Section 3.2, we observed that the feedback technique for synchronization of two identical chaotic systems also works for various jerk dynamical systems. However in this case also the identical synchronization of chaos in jerk systems is possible only when we use the xvariable as a drive i.e. the only x-drive configuration is stable. As we know that for the numerical simulations, it is very easy to divide the original dynamical system into drive and response subsystems (for PC technique) and to choose a drive variable (for feedback technique). However in the real physical system there are certain limitations. In addition, not all the possible combinations of drive and response systems are stable. So it may create problems while implementing the above discussed techniques for the identical synchronization everywhere. To avoid these difficulties Kocarev, Parlitz and their group [30, 31, 60] proposed a general drive response scheme named as Active Passive Decomposition (APD). The basic idea of the active passive synchronization approach consists in a decomposition of a given chaotic system into an active and a passive part where different copies of the passive part synchronize when driven by the same active component. In the following, we explain the Electronic Journal of Theoretical Physics 3, No. 11 (2006) 33–70 57 Fig. 12 Largest conditional Lyapunov exponent (CLE of Eq. (26)) as a function of coupling strength (c) for JD4, JD5, JD6, JD7. basic concept and terminology of the active passive decomposition. Consider an autonomous n-dimensional dynamical system, which is chaotic as Ż = F (Z), (28) where Z = (z1 , z2 , z3 , ..., zn )T and F (Z) = (f1 (Z), f2 (Z), ..., fn (Z))T . Now rewrite this autonomous system as a non-autonomous system that possesses certain synchronization properties as follows: Ẋ = G(X, S), (29) where X = (x1 , x2 , ..., xn )T is the new state vector corresponding to Z and S is some vector valued function of time given by S = H(X), (30) Ṡ = H(X, S), (31) The pair of functions G and H constitutes a decomposition of the original vector field F . The crucial point of this decomposition is that suitable choices of the function H for which any system (32) Ẋ ′ = G(X ′ , S), 58 Electronic Journal of Theoretical Physics 3, No. 11 (2006) 33–70 that is given by same non-autonomous vector field G, the same driving S, but with ′ ′ ′ different variables X ′ = (x1 , x2 , ..., xn )T , synchronizes with the original system (29), i.e. X − X ′  → 0 for t → ∞. More precisely, synchronization of the pair of identical systems (29) and (32) occurs if the dynamical system describing the evolution of the difference E = X ′ − X, Ė = Ẋ ′ − Ẋ = G(X ′ , S) − G(X, S), (33) = G(X + E, S) − G(X, S), (34) Ė = DF (X, S) · E, (35) possesses a stable fixed point at the origin E = 0. Here E = (e1 , e2 , ..., en )T and ′ ei = xi − xi for i = 1 to n. The dynamics of the error system can be understood by studying the following linearized system for small E: In this decomposition of the vector field F into G and H, the system described by vector field G (Eq. (29)) is a passive system whereas the component described by H is an active component and hence this technique is called active passive decomposition (APD). We note that in APD approach the freedom to choose H that defines the drive signal, leads to a large flexibility in applications. Thus APD is different from the Pecora-Carroll synchronization approach, where only a finite number of possible coupling exists that is given by the number of stable subsystems of the dynamical system. Now the question arises that how to decompose the original vector field F into the active H and passive G components, so that the copies of the passive part possess identical synchronization i.e. X = X ′ . To study the stability of identical synchronization in APD approach, one can again make use of the three basic criteria introduced in the Section 3.1: (i) If the largest eigenvalue of the Jacobian matrix DF (X, S)appeared in Eq. (35) is negative then the copies of the passive system driven by same active component will possess identical synchronization [6-8]. (ii) If a suitable Lyapunov function [27] L(E)exists, which satisfies: (a)L(E) ≥ 0, (b) L(0) = 0, (c) dL(E)/dt ≤ 0 and (d) dL(0)/dt = 0. (iii) If all the Lyapunov exponent of the system described by Eq. (35)(conditional Lyapunov exponents) are negative, then the copies of the passive system driven by same active component will possess identical synchronization [24, 25]. Instead of decomposing a given chaotic system one may also synthesize it starting from a stable linear system Ẋ = A · X given by some constant matrix A where an appropriate nonlinear function S = H(X)is added such that the complete system Ẋ = A · X + S, (36) is chaotic [61]. In this case S = H(X) is treated as active component and Ẋ = A · Xis as passive. Now we consider the copy of the passive part with different variables but driven by same active component i.e. Ẋ ′ = A · X ′ + S, (37) In this case the error dynamics is given by Ė = Ẋ ′ − Ẋ = A · (X ′ − X) = A · E, (38) Electronic Journal of Theoretical Physics 3, No. 11 (2006) 33–70 59 Since Ais a constant matrix hence if the eigenvalues of matrix A are all negative then the dynamical system describing the evolution of error (Eq. (38)) will possess a stable fixed point at the origin therefore two chaotic systems described by Eqs. (36) and (37) possess identical synchronization independent of initial conditions and active signalS. The APD approach described above has been applied to demonstrate identical synchronization of chaos in Lorenz, Rossler and Chua circuit [30, 31]. In this study they have also suggested a possible application of APD synchronization approach to communication and demonstrated it through experimental realization in Chua circuit as well as numerical simulations in Lorenz and Rossler systems. They also showed that how this approach can be used to construct high dimensional synchronizing systems in a systematic way using low dimensional systems as building blocks. Later Parlitz et. al. [60] showed that how synchronization using APD can be used for modeling and time series analysis. Later on Jinlan et. al. [62] used APD scheme to realize the spatiotemporal chaotic synchronization. Now, we use APD approach described above to demonstrate the identical synchronization of chaos in jerk dynamical systems. We consider a general form of jerk dynamical system as: x = J(x, ẋ, ẍ), (39) The dynamical system representation of Eq. (39) in terms of first order ODE’s by using the transformations ẋ = yand ẏ = zwill be as: ẋ = y ẏ = z , ż = J(x, y, z) (40) where J(x, y, z)is a jerk function. Now we start with the simplest way to implement the APD scheme, which is described in Eqs. (36)-(38). We rewrite Eq. (40) as: ẋ = y ẏ = z , ż = ax + by + cz + J(x, y, z) − ax − by − cz (41) T which"is in form of # Ẋ = A · X + S, whereX = (x, y, z) , S = J(x, y, z) − ax − by − cz, 0 1 0 A = 0 0 1 and a, band care arbitrary real constants which ensure the desired a b c identical synchronization. In the above we add and subtract the same term ax + by + cz in the evolution of z-variable of Eq. (40). Now by assumingS = J(x, y, z)−ax−by−cz as an active element for the APD scheme, we consider the copy of the functionally identical jerk system as Eq. (40) but with different variables and same active element (drive) S as: ẋ′ = y ′ ẏ ′ = z ′ , (42) ż ′ = ax′ by ′ + cz ′ + J(x, y, z) − ax − by − cz The error between the variables of Eqs. (41) and (42) is governed by the error equation as: (43) Ė ′ = A · E, 60 Electronic Journal of Theoretical Physics 3, No. 11 (2006) 33–70 where E = (x′ − x, y ′ − y, z ′ − z)T . Since Ais a constant matrix hence if the eigenvalues of matrix A are all negative then the dynamical system describing the evolution of error (Eq. (43)) will possess a stable fixed point at the origin therefore two chaotic systems described by Eqs. (41) and (42) possess identical synchronization independent of initial conditions. The characteristic equation of A is: λ 3 − cλ 2 − bλ − a = 0. (44) The eigenvalues (roots of Eq. (44)) λ 1 , λ 2 and λ 3 satisfy the followings: λ1 + λ2 + λ3 = c λ 1 λ 2 + λ 2 λ 3 + λ 3 λ 1 = −b . λ 1λ 2λ 3 = a (45) To ensure the identical synchronization, all the eigenvalues should be negative. Since a, b and c are arbitrary real constant so it is always possible to choose these constants in such a way that all the roots of Eq. (44) be negative. In addition, in the above analysis J(x, y, z)is any jerk function i.e. the above analysis is independent of the jerk function and hence can be applied to all the jerk dynamical systems. It has been shown that many three dimensional dynamical systems can be transformed into jerk dynamical system [15], hence the above analysis can be applied to any three dimensional system that can be transformed into jerky dynamics. Now we consider a special family of jerk dynamical systemx = −Aẍ − ẋ + G(x), where G(x)is a nonlinear function, which has been studied in detail by Patidar and Sud [19]. If we implement the above formalism (Eqs.(40) to (45)) to these systems then the active and passive systems will be as follows: and ẋ = y ẏ = z ż = ax + by + cz − Az − y + G(x) − ax − by − cz (46) ẋ′ = y ′ ẏ ′ = z ′ . ż ′ = ax′ + by ′ + cz ′ − Az − y + G(x) − ax − by − cz (47) In Figure $ 15, we have shown the results of numerical calculation of Euclidean error (i.e.|e| = (x′ − x)2 + (y ′ − y)2 + (z ′ − z)2 ) between the systems described by Eqs. (46) and (47) as a function of time (t) for A = 0.6, G(x) = 0.58(x2 − 1)and three specific sets of eigenvalues of matrix A: (i)λ 1 = −1.2, λ 2 = −1.5and λ 3 = −1.7(ii)λ 1 = −3.6, λ 2 = −4.5and λ 3 = −5.1 (iii)λ 1 = −7.2, λ 2 = −9.0and λ 3 = −10.2, which correspond to (i) a = -3.06, b = -6.39 and c = - 4.4 (ii) a = -82.62, b = -57.51 and c = - 13.2 (iii) a = -660.96, b = -230.04 and c = - 26.4 respectively. In Figure 16, we have shown the results of numerical computation of Euclidean error forG(x) = (|x| − 2), however the other parameters are same as for Figure 15. We observe from Figures 15 and 16 that the Eucledean error decays to zero after a certain time and identical synchronization of active and passive system occurs. We also note that the Electronic Journal of Theoretical Physics 3, No. 11 (2006) 33–70 61 $ Fig. 13 Euclidean error (i.e.|e| = (x′ − x)2 + (y ′ − y)2 + (z ′ − z)2 ) between the systems described by Eqs. (46) and (47) as a function of time (t) for A = 0.6, G(x) = 0.58(x 2 − 1) and three specific sets of parameters (i) a = -3.06, b = -6.39 and c = - 4.4 (ii) a = -82.62, b = -57.51 and c = - 13.2 (iii) a = -660.96, b = -230.04 and c = - 26. $ Fig. 14 Euclidean error (i.e.|e| = (x′ − x)2 + (y ′ − y)2 + (z ′ − z)2 ) between the systems described by Eqs. (46) and (47) as a function of time (t) for A = 0.6, G(x) = (|x| - 2) and three specific sets of parameters (i) a = -3.06, b = -6.39 and c = - 4.4 (ii) a = -82.62, b = -57.51 and c = - 13.2 (iii) a = -660.96, b = -230.04 and c = - 26. results in these two figures are independent of the form of nonlinear function G(x). In addition, we also observe that for the third parameter set (a, b, c) = (-660.96, -230.04, -26.4) for which the eigenvalues are larger in magnitude the decay is faster than the second and first parameter sets. 62 Electronic Journal of Theoretical Physics 3, No. 11 (2006) 33–70 In the above described active passive decomposition the Jacobian matrix appeared in the error equation (Eq. (43)) is a constant matrix and hence the stability of identical synchronization can be analyzed very easily by simply calculating the eigenvalues of the Jacobian matrix. The above described way of active passive decomposition provides us the flexibility to choose the value of constants a, b and c freely to attain the identical synchronization. However many more ways are possible to decompose the given dynamical system into active and passive parts. To illustrate other active passive decompositions, we again consider the same family of jerk dynamical systems as considered above. For example, one of the possible active passive decomposition is given by ẋ = −x + S ẏ = z ż = −Az − y + G(x) (48) S =x+y (49) ẋ′ = −x′ + S ẏ ′ = z ′ . ż ′ = −Az ′ − y ′ + G(x′ ) (50) with and The dynamical system describing the evolution of the difference between systems described by Eqs. (48) and (50) is given as, # #" ′ # " " ′ ẋ − ẋ x −x −1 0 0 y′ − y . ẏ ′ − ẏ = 0 0 1 (51) ′ ′ (G(x )−G(x))/(x′ − x) −1 −A z′ − z ż − ż It is clear that in this case the Jacobian matrix appearing in the error equation is not constant and hence not easy to calculate its eigenvalues easily. For analyzing the stability of identical synchronization, we have used the numerical computation of the Lyapunov exponents of Eq. (51) (i.e. conditional Lyapunov exponents). If we consider G(x) = 0.58(x2 − 1)and A = 0.6then conditional Lyapunov exponents are λ1 = −0.3006E0, λ2 = −0.3039E0, λ3 = −0.9991E0and if we consider G(x) = (|x| − 2) and then conditional Lyapunov exponents are λ1 = −0.3014E0, λ2 = −0.3012E0, λ3 = −0.9999E0. It is clear that for both cases all the three conditional Lyapunov exponents are negative and hence the identical synchronization is stable. Other possible active passive decompositions and their corresponding conditional Lyapunov exponents for the above discussed family of jerk dynamical system with G(x) = 0.58(x2 − 1)and G(x) = (|x| − 2) are given in Tables 4 and 5 respectively. From Tables 4 and 5, it is clear that several active passive decompositions are possible which lead to stable identical synchronization unlike to Pecora-Carroll and feedback techniques where only a few drive-response configurations lead to identical synchronization. Finally, we extend the active passive decomposition approach for identical synchronization of chaos to the jerk dynamical representations of the Sprott’s simple chaotic flows [9] by considering their classification [15] according to the hierarchy of quadratic jerk functions with increasing many more terms. In Table 6, we have given a few possible active Electronic Journal of Theoretical Physics 3, No. 11 (2006) 33–70 63 Table 4: Examples of active passive decompositions of the jerk dynamical system having quadratic nonlinearity and their corresponding conditional Lyapunov exponents (CLE’s) ẋ = y − x + s ẏ = z ż = −Az − y + B(x2 + C) ẋ = −x − z + s ẏ = z ż = −Az − y + B(x2 + C) ẋ = y ẏ = z − x + s ż = −Az − y + B(x2 + C) ẋ = y ẏ = z − y + s ż = −Az − y + B(x2 + C) ẋ = y ẏ = −y + s ż = −Az − y + B(x2 + C) ẋ = y ẏ = −x − y + s ż = −Az − y + B(x2 + C) ẋ = x − x2 + s ẏ = z ż = −Az − y + B(x2 + C) ẋ = y ẏ = y − xz + s ż = −Az − y + B(x2 + C) ... x = −Aẍ − ẋ + B(x2 + C); A = 0.6; B = 0.58; C = −1 ẋ′ = y ′ − x′ + s s=x ẏ ′ = z ′ ż ′ = −Az ′ − y ′ + B(x′2 + C) ẋ′ = −x′ − z ′ + s s=x+y+z ẏ ′ = z ′ ż ′ = −Az ′ − y ′ + B(x′2 + C) ẋ′ = y ′ s=x ẏ ′ = z ′ − x′ + s ż ′ = −Az ′ − y ′ + B(x′2 + C) ẋ′ = y ′ s=y ẏ ′ = z ′ − y ′ + s ż ′ = −Az ′ − y ′ + B(x′2 + C) ẋ′ = y ′ s=y+z ẏ ′ = −y ′ + s ż ′ = −Az ′ − y ′ + B(x′2 + C) ẋ′ = y ′ s=x+y+z ẏ ′ = −x′ − y ′ + s ż ′ = −Az ′ − y ′ + B(x′2 + C) ẋ′ = x′ − x′2 + s s = x2 + y − x ẏ ′ = z ′ ż ′ = −Az ′ − y ′ + B(x′2 + C) ẋ′ = y ′ s = xz + z − y ẏ ′ = y ′ − x′ z ′ + s ż ′ = −Az ′ − y ′ + B(x′2 + C) λ1 λ2 λ3 λ1 λ2 λ3 λ1 λ2 λ3 λ1 λ2 λ3 λ1 λ2 λ3 λ1 λ2 λ3 λ1 λ2 λ3 λ1 λ2 λ3 = −0.1005E0 = −0.3188E0 = −0.1181E + 1 = −0.1798E0 = −0.2354E0 = −0.1184E + 1 = −0.3695E − 1 = −0.1555E0 = −0.4081E0 = −0.9567E − 1 = −0.5749E0 = −0.9308E0 = −0.2313E − 2 = −0.6016E0 = −0.1000E + 1 = −0.4979E0 = −0.5013E0 = −0.6013E0 = −0.2620E0 = −0.3024E0 = −0.1884E + 1 = −0.8211E − 1 = −0.1118E0 = −0.5253E0 Table 5: Examples of active passive decompositions of the jerk dynamical system having absolute nonlinearity and their corresponding conditional Lyapunov exponents (CLE’s) ẋ = y − x + s ẏ = z ż = −Az − y + B(|x| + C) ẋ = −z + s ẏ = z ż = −Az − y + B(|x| + C) ẋ = −x − z + s ẏ = z ż = −Az − y + B(|x| + C) ẋ = y ẏ = z − x + s ż = −Az − y + B(|x| + C) ẋ = y ẏ = z − y + s ż = −Az − y + B(|x| + C) ẋ = y ẏ = −x + s ż = −Az − y + B(|x| + C) ẋ = y ẏ = −y + s ż = −Az − y + B(|x| + C) ẋ = y ẏ = −x − y + s ż = −Az − y + B(|x| + C) ẋ = x − x2 + s ẏ = z ż = −Az − y + B(|x| + C) ... x = −Aẍ − ẋ + B(|x| + C); A = 0.6; B = 1.0; C = −2 ẋ′ = y ′ − x′ + s s=x ẏ ′ = z ′ ż ′ = −Az ′ − y ′ + B(|x′ | + C) ẋ′ = −z ′ + s s=y+z ẏ ′ = z ′ ż ′ = −Az ′ − y ′ + B(|x′ | + C) ẋ′ = −x′ − z ′ + s s=x+y+z ẏ ′ = z ′ ż ′ = −Az ′ − y ′ + B(|x′ | + C) ẋ′ = y ′ s=x ẏ ′ = z ′ − x′ + s ż ′ = −Az ′ − y ′ + B(|x′ | + C) ẋ′ = y ′ s=y ẏ ′ = z ′ − y ′ + s ż ′ = −Az ′ − y ′ + B(|x′ | + C) ẋ′ = y ′ s=x+z ẏ ′ = −x′ + s ż ′ = −Az ′ − y ′ + B(|x′ | + C) ẋ′ = y ′ s=y+z ẏ ′ = −y ′ + s ż ′ = −Az ′ − y ′ + B(|x′ | + C) ẋ′ = y ′ s=x+y+z ẏ ′ = −x′ − y ′ + s ż ′ = −Az ′ − y ′ + B(|x′ | + C) ẋ′ = x′ − x′2 + s s = x2 + y − x ẏ ′ = z ′ ż ′ = −Az ′ − y ′ + B(|x′ | + C) λ1 λ2 λ3 λ1 λ2 λ3 λ1 λ2 λ3 λ1 λ2 λ3 λ1 λ2 λ3 λ1 λ2 λ3 λ1 λ2 λ3 λ1 λ2 λ3 λ1 λ2 λ3 = −0.7373E − 1 = −0.4001E0 = −0.1127E + 1 = −0.2887E − 2 = −0.7248E − 1 = −0.5270E0 = −0.1601E0 = −0.3662E0 = −0.1074E + 1 = −0.2011E − 1 = −0.1577E0 = −0.4227E0 = −0.1468E0 = −0.5344E0 = −0.9160E0 = −0.1138E − 2 = −0.8297E − 3 = −0.6007E0 = −0.1599E − 2 = −0.5985E0 = −0.9991E0 = −0.4997E0 = −0.4992E0 = −0.6017E0 = −0.2601E0 = −0.3005E0 = −0.3616E + 1 64 Electronic Journal of Theoretical Physics 3, No. 11 (2006) 33–70 passive decompositions for the jerk dynamical representations of some of the Sprott’s simple chaotic flows and their corresponding CLE’s (we have given the results for only one model from each class i.e., one each from JD1 to JD6 ). Table 6: Examples of active passive decompositions for the jerk dynamical representations of some of the Sprott’s simple chaotic flows and their corresponding conditional Lyapunov exponents (CLE’s). JD1 Model J ẋ = y − x + s ẏ = z ż = k1 z + k2 x + xy + k3 ẋ = −x + s ẏ = z ż = k1 z + k2 x + xy + k3 ẋ = y ẏ = z − y + s ż = k1 z + k2 x + xy + k3 ẋ = y ẏ = −x − y + s ż = k1 z + k2 x + xy + k3 JD2 Model S ẋ = −x + s ẏ = z ż = k1 z + k2 y + x2 + k3 ẋ = y ẏ = z − y + s ż = k1 z + k2 y + x2 + k3 ẋ = y ẏ = −x − y + s ż = k1 z + k2 y + x2 + k3 JD3 Model F ẋ = y − x + s ẏ = z ż = k1 z + k2 y + k3 x2 +xy + k4 ẋ = −x + s ẏ = z ż = k1 z + k2 y + k3 x2 +xy + k4 ... x = k1 ẍ + k2 x + xẋ + k3 , k1 = −2.0 ; k2 = −4.0 ; k3 = −4.0 ẋ′ = y ′ − x′ + s s=x ẏ ′ = z ′ ż ′ = k1 z ′ + k2 x′ + x′ y ′ + k3 ẋ′ = −x′ + s s=x+y ẏ ′ = z ′ ż ′ = k1 z ′ + k2 x′ + x′ y ′ + k3 ẋ′ = y ′ s=y ẏ ′ = z ′ − y ′ + s ż ′ = k1 z ′ + k2 x′ + x′ y ′ + k3 ẋ′ = y ′ s=x+y+z ẏ ′ = −x′ − y ′ + s ż ′ = k1 z ′ + k2 x′ + x′ y ′ + k3 ... x = k1 ẍ + k2 ẋ + x2 + k3 , k1 = −1.0 ; k2 = −4.0 ; k3 = −16.0 λ1 λ2 λ3 λ1 λ2 λ3 λ1 λ2 λ3 λ1 λ2 λ3 = −0.1603E − 1 = −0.4438E0 = −0.2544E + 1 = −0.1399E0 = −0.9985E0 = −0.1862E + 1 = −0.1496E0 = −0.5205E0 = −0.2332E + 1 = −0.4996E0 = −0.5002E0 = −0.2000E + 1 ẋ′ = −x′ + s λ1 = −0.4971E0 s=x+y ẏ ′ = z ′ λ2 = −0.5007E0 λ3 = −0.1003E + 1 ż ′ = k1 z ′ + k2 y ′ + x′2 + k3 ẋ′ = y ′ λ1 = −0.6668E − 1 s=y λ2 = −0.5307E0 ẏ ′ = z ′ − y ′ + s λ3 = −0.1401E + 1 ż ′ = k1 z ′ + k2 y ′ + x′2 + k3 ẋ′ = y ′ λ1 = −0.4958E0 s=x+y+z λ2 = −0.5004E0 ẏ ′ = −x′ − y ′ + s λ3 = −0.1004E + 1 ż ′ = k1 z ′ + k2 y ′ + x′2 + k3 ... x = k1 ẍ + k2 ẋ + k3 x2 + xẋ + k4 , k1 = −0.5 ; k2 = −2.5 ; k3 = −0.25; k4 = 1 ẋ′ = y ′ − x′ + s ẏ ′ = z ′ ż ′ = k1 z ′ + k2 y ′ + k3 x′2 +x′ y ′ + k4 ẋ′ = −x′ + s ẏ ′ = z ′ ż ′ = k1 z ′ + k2 y ′ + k3 x′2 +x′ y ′ + k4 s=x λ1 = −0.8347E − 1 λ2 = −0.1432E0 λ3 = −0.1275E + 1 s=x+y λ1 = −0.9779E − 1 λ2 = −0.4059E0 λ3 = −0.1000E + 1 Electronic Journal of Theoretical Physics 3, No. 11 (2006) 33–70 ẋ = y ẏ = z − y + s ż = k1 z + k2 y + k3 x2 +xy + k4 ẋ = y ẏ = −y + s ż = k1 z + k2 y + k3 x2 +xy + k4 ẋ = y ẏ = −x − y + s ż = k1 z + k2 y + k3 x2 +xy + k4 JD4 Model O ẋ = y − x + s ẏ = z ż = k1 z + k2 y + k3 x2 +xz + k4 ẋ = −x + s ẏ = z ż = k1 z + k2 y + k3 x2 +xz + k4 ẋ = −x − z + s ẏ = z ż = k1 z + k2 y + k3 x2 +xz + k4 ẋ = y ẏ = z − x + s ż = k1 z + k2 y + k3 x2 +xz + k4 ẋ = y ẏ = z − y + s ż = k1 z + k2 y + k3 x2 +xz + k4 ẋ = y ẏ = −x − y + s ż = k1 z + k2 y + k3 x2 +xz + k4 65 ẋ′ = y ′ λ1 = −0.6420E − 2 ẏ ′ = z ′ − y ′ + s s=y λ2 = −0.4293E0 ′ ′ ′ ′2 ż = k1 z + k2 y + k3 x λ3 = −0.1065E + 1 ′ ′ +x y + k4 ẋ′ = y ′ λ1 = −0.1957E − 2 ẏ ′ = −y ′ + s s=y+z λ2 = −0.5019E0 ′ ′ ′ ′2 ż = k1 z + k2 y + k3 x λ 3 = −0.1000E + 1 +x′ y ′ + k4 ′ ′ ẋ = y λ1 = −0.4964E0 ẏ ′ = −x′ − y ′ + s s = x + y + z λ2 = −0.5003E0 ż ′ = k1 z ′ + k2 y ′ + k3 x′2 λ3 = −0.5039E0 ′ ′ +x y + k4 ... x = k1 ẍ + k2 ẋ + k3 x2 + xẍ + k4 , k1 = −0.5 ; k2 = −1.7 ; k3 = −1.0; k4 = 0.25 ẋ′ = y ′ − x′ + s ẏ ′ = z ′ ż ′ = k1 z ′ + k2 y ′ + k3 x′2 +x′ z ′ + k4 ′ ẋ = −x′ + s ẏ ′ = z ′ ż ′ = k1 z ′ + k2 y ′ + k3 x′2 +x′ z ′ + k4 ′ ẋ = −x′ − z ′ + s ẏ ′ = z ′ ż ′ = k1 z ′ + k2 y ′ + k3 x′2 +x′ z ′ + k4 ẋ′ = y ′ ẏ ′ = z ′ − x′ + s ż ′ = k1 z ′ + k2 y ′ + k3 x′2 +x′ z ′ + k4 ẋ′ = y ′ ẏ ′ = z ′ − y ′ + s ż ′ = k1 z ′ + k2 y ′ + k3 x′2 +x′ z ′ + k4 ′ ẋ = y ′ ẏ ′ = −x′ − y ′ + s ż ′ = k1 z ′ + k2 y ′ + k3 x′2 +x′ z ′ + k4 s=x λ1 = −0.2513E − 1 λ2 = −0.3436E0 λ3 = −0.9209E0 s=x+y λ1 = −0.1325E0 λ2 = −0.1375E0 λ3 = −0.9996E0 s=x+y+z λ1 = −0.1181E0 λ2 = −0.1189E0 λ3 = −0.1032E + 1 s=x λ1 = −0.5036E − 1 λ2 = −0.7334E − 1 λ3 = −0.1455E0 s=y λ1 = −0.1205E0 λ2 = −0.4668E0 λ3 = −0.6832E0 s=x+y+z λ1 = −0.2681E0 λ2 = −0.4996E0 λ3 = −0.5004E0 66 JD5 Model D ẋ = y − x + s ẏ = z ż = k1 z + k2 x2 + k3 y 2 + xz ẋ = −x − z + s ẏ = z ż = k1 z + k2 x2 + k3 y 2 + xz ẋ = y ẏ = z − y + s ż = k1 z + k2 x2 + k3 y 2 + xz ẋ = y ẏ = −x − y + s ż = k1 z + k2 x2 + k3 y 2 + xz JD6 Model P ẋ = y − x + s ẏ = z ż = k1 z + k2 y + k3 x2 +k4 y 2 + xz + k5 ẋ = −x + s ẏ = z ż = k1 z + k2 y + k3 x2 +k4 y 2 + xz + k5 ẋ = y ẏ = z − y + s ż = k1 z + k2 y + k3 x2 +k4 y 2 + xz + k5 ẋ = y ẏ = −x − y + s ż = k1 z + k2 y + k3 x2 +k4 y 2 + xz + k5 Electronic Journal of Theoretical Physics 3, No. 11 (2006) 33–70 ... x = k1 ẍ + k2 x2 + k3 ẋ2 + xẍ, k1 = −1.0 ; k2 = 1.0 ; k3 = −1.0 ẋ′ = y ′ − x′ + s λ1 = −0.8313E − 1 s=x ẏ ′ = z ′ λ2 = −0.2672E0 λ3 = −0.1598E + 1 ż ′ = k1 z ′ + k2 x′2 + k3 y ′2 + x′ z ′ ẋ′ = −x′ − z ′ + s λ1 = −0.5850E0 s=x+y+z ẏ ′ = z ′ λ2 = −0.4168E0 λ3 = −0.2352E + 1 ż ′ = k1 z ′ + k2 x′2 + k3 y ′2 + x′ z ′ ẋ′ = y ′ λ1 = −0.3648E − 1 s=y λ2 = −0.4033E0 ẏ ′ = z ′ − y ′ + s λ3 = −0.1744E + 1 ż ′ = k1 z ′ + k2 x′2 + k3 y ′2 + x′ z ′ ẋ′ = y ′ λ1 = −0.4979E0 s=x+y+z λ2 = −0.5011E0 ẏ ′ = −x′ − y ′ + s λ3 = −0.1184E + 1 ż ′ = k1 z ′ + k2 x′2 + k3 y ′2 + x′ z ′ ... x = k1 ẍ + k2 ẋ + k3 x2 + k4 ẋ2 + xẍ + k5 ; k1 = −1.0; k2 = −1.7; k3 = −0.5; k4 = 1.0; k5 = 0.5 ẋ′ = y ′ − x′ + s ẏ ′ = z ′ ż ′ = k1 z ′ + k2 y ′ + k3 x′2 +k4 y ′2 + x′ z ′ + k5 ẋ′ = −x′ + s ẏ ′ = z ′ ż ′ = k1 z ′ + k2 y ′ + k3 x′2 +k4 y ′2 + x′ z ′ + k5 ẋ′ = y ′ ẏ ′ = z ′ − y ′ + s ż ′ = k1 z ′ + k2 y ′ + k3 x′2 +k4 y ′2 + x′ z ′ + k5 ′ ẋ = y ′ ẏ ′ = −x′ − y ′ + s ż ′ = k1 z ′ + k2 y ′ + k3 x′2 +k4 y ′2 + x′ z ′ + k5 s=x λ1 = −0.2839E − 1 λ2 = −0.2464E0 λ3 = −0.1119E + 1 s=x+y λ1 = −0.1826E0 λ2 = −0.2120E0 λ3 = −0.9995E0 s=y λ1 = −0.3656E − 1 λ2 = −0.4006E0 λ3 = −0.8625E0 s=x+y+z λ1 = −0.3878E0 λ2 = −0.4997E0 λ3 = −0.5007E0 Conclusions In this paper, we have investigated identical synchronization in chaotic jerk dynamical systems by implementing three algorithms: Pecora-Carroll (PC) technique, Feedback (FB) technique and active passive decomposition (APD. We have also investigated the stability of identical synchronization in the aforesaid methods using extensive transversal stability analysis (i.e., extensive conditional Lyapunov exponents (CLE’s) calculation) as well as Lyapunov function construction criterion. We summarize the results of our investigations as follows: • In case of Pecora-Carroll (PC) technique- Our extensive numerical calculation reveals that the x-drive configuration for all the jerky representation of the Sprott’s simple chaotic flows by considering their classification into seven different classes of jerky dynamics (according to the hierarchy of quadratic jerk functions with increasing many more terms), guarantees the identical synchronization except for the models ‘L’, ‘R’ and ‘D’, where no subsystem is found to be stable. However y-drive and z-drive configurations are unstable for all the jerk dynamical systems. • In Case of feedback (FB) technique- we have done the extensive calculation for conditional Lyapunov exponents by implementing FB technique with x-drive configuration in the jerky representations of the Sprott’s chaotic flows by considering Electronic Journal of Theoretical Physics 3, No. 11 (2006) 33–70 67 their classification into seven different classes of jerky dynamics (according to the hierarchy of quadratic jerk functions with increasing many more terms). We have done the extensive calculations of CLE’s for the complete range of coupling strength i.e. 0 ≤ c ≤ 1 for all the jerk systems. We observed that the feedback technique works well for synchronizing all the chaotic jerk dynamical systems considered by us. However y-drive and z-drive configurations do not lead to stable identical synchronization. • In case of active passive decomposition (APD) –we have suggested how it is possible to decompose any jerk dynamical system into active and passive elements to achieve the stable synchronization without being bothered about the specific form of jerk function. Besides this generalized decomposition of jerk dynamical system into active and passive elements, we have also suggested several other active and passive decompositions for the jerky representation of the Sprott’s simple chaotic flows to achieve the stable identical synchronization and calculated their corresponding conditional Lyapunov exponents. As the jerk dynamical systems describe various phenomena in physics and engineering such as electrical circuits, mechanical oscillators, laser physics, solar wind driven magnetosphere ionosphere (WINDMI) model, damped harmonic oscillator driven by nonlinear memory term, biological systems etc. It has been also shown that many three dimensional dynamical systems can be transformed into jerk dynamical system [15], hence results of our investigations for the identical synchronization of chaos are of great use in diverse areas of science, where we would like to behave two or more systems in synchronized manner although maintaining their chaotic nature. 68 Electronic Journal of Theoretical Physics 3, No. 11 (2006) 33–70 References [1] S. Wiggins, Introduction to applied nonlinear dynamical systems and chaos; SpringerVerlag (1990). [2] E. Ott, Chaos in dynamical systems; Cambridge University Press, Cambridge (1993). [3] J. Argyris, G. Faust, M. 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E 58, 3017-3027 (1998). EJTP 3, No. 11 (2006) 71–83 Electronic Journal of Theoretical Physics Second Order Perturbation of Heisenberg Hamiltonian for Non-Oriented Ultra-Thin Ferromagnetic Films P. Samarasekara ∗ Department of Physics, University of Ruhuna, Matara, Sri Lanka Received 25 January 2006 , Accepted 15 March 2006, Published 25 June 2006 Abstract: The second order perturbation of magnetic energy for ferromagnetic thin films of two and three layers has been studied using classical Heisenberg Hamiltonian. According to our model, the film with two layers is equivalent to an oriented film, when anisotropy constants do not vary inside the film. But the energy of films with three layers indicates periodic variation. Introducing second order perturbation induces some sudden overshooting of energy curves, compared with smooth energy curves obtained for oriented ferromagnetic ultra thin films in one of our previous report. After taking the fourth order anisotropy into account, the overshooting part dominates by reducing the smooth part of energy graphs. Several minimums can be observed in last 3-D graph implying that the film with N=3 can be oriented in some preferred directions by applying a certain value of stress. The shape of graphs of energy variation of all sc(001), fcc(001) and bcc(001) ferromagnetic ultra thin films with second (or fourth) order anisotropy is exactly same. Easy and hard directions of these all types with the effect of second order anisotropy only are 34.40 and 124.40 , respectively. The angle between easy and hard directions J ω Ks ω is exactly 900 as expected. Although these simulations were given for only, this same approach can be carried out for any values of (2) J Dm , ω ω , and (2) (4) Dm = 10, Kωs = 10 and Dωm = 5 values ω (4) Dm or any type of ferromagnetic material. ω = 10, Considering the other terms such as dipole interaction and demagnetization factor really complicates the simulation. c Electronic Journal of Theoretical Physics. All rights reserved.  Keywords: Heisenberg Hamiltonian, Ferromagnetic Films, PACS (2006): 75.10.Hk , 75.30.Gw , 75.70.-i 1. Introduction: The research of exchange anisotropy has received a wide attention in last decade, due to the difficulties of physical understanding of exchange anisotropy and to its application in magnetic media technology and magnetic sensors [4] . The magnetic properties of ferromagnetic thin films and multi layers have been extensively investigated because of their ∗ pubudus@phy.ruh.ac.lk 72 Electronic Journal of Theoretical Physics 3, No. 11 (2006) 71–83 potential impact on magnetic recording devices. The magnetic properties of thin films of ferromagnetic materials have been investigated using the Bloch spin-wave theory earlier [5] . The magnetization of some thin films shows an in plane orientation due to the dipole interaction. Due to the broken symmetry uniaxial anisotropy energies at the surfaces of the film, the perpendicular magnetization is preferential. But due to the strain induced distortion in the inner layers, bulk anisotropy energies will appear absent or very small in the ideal crystal. Very thin films indicate a tetragonal distortion resulting in stressinduced uniaxial anisotropy energy in the inner layers with perpendicular orientation of easy axis. The magnetic in-plane anisotropy of a square two-dimensional (2D) Heisenberg ferromagnet in the presence of magnetic dipole interaction has been determined earlier [6] . The long range character of the dipole interaction itself is sufficient to stabilize the magnetization in 2-D magnet. Also the easy and hard axes of the magnetization with respect to lattice frame are determined by the anisotropies. Magnetic properties of the Ising ferromagnetic thin films with alternating superlattice layers were investigated [7] . In addition to these, Monte Carlo simulations of hysteresis loops of ferromagnetic thin films have been theoretically traced [8] . Since the surfaces slightly distort the symmetry of the system under consideration, physical quantities in the vicinity of surfaces generally deviate from those in the bulk. The second order perturbation solution of Heisenberg Hamiltonian for ultra-thin ferromagnetic films has been found without considering the effect of stress induced anisotropy and demagnetization factor in some early reports [1] . The stress of a film arises mainly due to the difference between thermal expansion coefficients of the film and the substrate. When the film is cooled down or heated after annealing or deposition, the stress takes place in the film. The effect of stress on the coercivity and anisotropy of sputtered ferromagnetic thin films has been previously studied by us [2,3] . The contribution of stress to the magnetic energy is given by Ks sin2θ. Here Ks depends on the magnetization and the magnitude of stress, and θ is the angle between the stress and the magnetization. The theoretical values of demagnetization factor within the film plane and normal to the film plane are 0 and 1 in SI units, respectively. 2. The Model The Heisenberg Hamiltonian of ferromagnetic films can be formulated as following. H=−  (2)  (4) n 3(S n ) m .S m .rmn )(rmn .S ω S J   z 2 z 4 Sm .Sn + ( 3 − ) − D (S ) − Dλm (Sm ) m λ m 5 2 m,n 2 m=n rmn rmn m m −  m,n  − (Nd S m − n /μ0 )].S [H  Ks Sin2θm m m and S n are two spins. Above equation can be simplified into following form [1]. Here S Electronic Journal of Theoretical Physics 3, No. 11 (2006) 71–83 E(θ)= - 12 N  m,n=1 − [(JZ|m−n| − ω4 Φ|m−n| ) cos(θm − θn ) − N  73 3ω Φ|m−n| 4 cos(θm + θn )] (2) (4) (Dm cos2 θm +Dm cos4 θm + Hin sin θm + Hout cos θm ) m=1 + N N   Nd cos(θm − θn ) − Ks sin 2θm μ 0 m=1 m,n=1 (2) (1) (4) Here J, Z|m−n| , ω, Φ|m−n| , θ, Dm , Dm , Hin , Hout , Nd , Ks , m, n and N are spin exchange interaction, number of nearest spin neighbors, strength of long range dipole interaction, constants for partial summation of dipole interaction, azimuthal angle of spin, second and fourth order anisotropy constants, in plane and out of plane applied magnetic fields, demagnetization factor, stress induced anisotropy constant, spin plane indices and total number of layers in film, respectively. When the stress applies normal to the film plane, the angle between mth spin and the stress is θm . The azimuthual angles of spins can be given as θm = θ + εm and θn = θ + εn . After substituting these new angles in above equation number 1, the cosine and sine terms can be expanded up to the second order of εm and εn as following. E(θ)=E0 +E(ε)+E(ε2 )+——— If the third and higher order perturbations are neglected, then E(θ) = E0 + E(ε) + E(ε2 ) Here (2) N N  1  ω 3ω cos 2θ Φ|m−n| (JZ|m−n| − Φ|m−n| ) + E0 = − 2 m,n=1 4 8 m,n=1 2 − cos θ N  (2) Dm m=1 4 − cos θ N  m=1 (4) Dm − N (Hin sin θ + Hout cos θ − Nd + Ks sin 2θ) μ0 (3) N N N    3ω (4) (2) 2 sin 2θ Dm εm E(ε) = − Dm εm + 2 cos θ sin 2θ Φ|m−n| (εm + εn ) + sin 2θ 8 m=1 m=1 m,n=1 −Hin cos θ E(ε2 ) = N  εm + Hout sin θ m=1 N  m=1 εm − 2Ks cos 2θ N  εm m=1 N N  ω 3ω 1  cos 2θ (JZ|m−n| − Φ|m−n| )(εm − εn )2 − Φ|m−n| (εm + εn )2 4 m,n=1 4 16 m,n=1 2 2 −(sin θ − cos θ) N  m=1 (2) 2 Dm εm 2 2 2 + 2 cos θ(cos θ − 3 sin θ) N  (4) 2 Dm εm m=1 N N N N    Hin Hout Nd  2 2 2 sin θ cos θ + (εm − εn ) + 2Ks sin 2θ ε2m εm + εm − 2 2 2μ 0 m,n=1 m=1 m=1 m=1 74 Electronic Journal of Theoretical Physics 3, No. 11 (2006) 71–83 After using the constraint N  εm = 0, E(ε) = α  .ε m=1  Here α  (ε) = B(θ) sin 2θ are the terms of matrices with N 3ω  (4) Φ|λ−m| + Dλ (2) + 2Dλ cos2 θ Bλ (θ) = − 4 m=1 (4) Also E(ε2 ) = 21 ε.C.ε Here the elements of matrix C can be given as following, Cmn = −(JZ|m−n| − +δmn { N  λ=1 ω 3ω 2Nd Φ|m−n| ) − cos 2θΦ|m−n| + 4 4 μ0 [JZ|m−λ| − Φ|m−λ| ( ω 3ω (2) + cos 2θ)] − 2(sin2 θ − cos2 θ)Dm 4 4 (4) +4 cos2 θ(cos2 θ − 3 sin2 θ)Dm + Hin sin θ + Hout cos θ − 4Nd + 4Ks sin 2θ} μ0 (5) Therefore, the total magnetic energy given in equation 2 can be deduced to [1] 1 1  · C+ · α   · ε + ε · C · ε = E0 − α E(θ) = E0 + α 2 2 (6) Here C+ is the pseudo-inverse given by C · C+ = 1 − E N (7) Here E is the matrix with all elements Emn =1. 3. Results and Discussion: First the energy will be found for a film with two layers (N=2). Since it is reasonable to assume that the anisotropy energies remain constant within an ultra-thin film with (2) (2) (4) (4) two layers, D1 =D2 and D1 =D2 . From equation 5, C11 =C22 and C12 =C21 . + + + + Therefore from equation 7, C12 = C21 = 2(C211−C22 ) = −C11 = −C22 + Using above results, α  · C+ · α  = (α1 − α2 )2 C11 But from equation 4, for a film with two layers α1 = α2 Therefore α  · C+ · α  = 0, and the total energy in equation 6 is E0 . This means that the energy of a film with two layers is reduced to the energy of the oriented film. Now if the anisotropy constants change within the film, then C12 =C21 and C22 =C11 . + + + + 21 = C221 +C11 ) . = −C22 and C21 = 2(CC1122C+C = −C12 Therefore, C11 2 22 −C ) 21 2(C21 −C11 C22 + +  = (α1 − α2 )(C21 α2 − C12 α1 ) Hence, α  · C+ · α Now the total energy is different from that of an oriented film. The matrix elements of C+ , which were found using equation 7 for a film with three layers (N=3), contained about 20 terms of Cmn elements. Electronic Journal of Theoretical Physics 3, No. 11 (2006) 71–83 75 The C+ 31 has been given as an example below. + A C31 = D Here A=-[(C22 C11 -C12 C21 )-0.67(C11 -C21 )(C12 -C22 )]C31 -0.67C32 (C11 -C21 )2 -0.33(C22 C11 -C12 C21 )(C11 -C21 ) D=C31 [(C11 C23 -C13 C21 )(C12 -C22 )-(C13 -C23 )(C22 C11 -C12 C21 )] -(C11 C23 -C13 C21 )(C11 -C21 )C32 +(C22 C11 -C12 C21 )(C11 -C21 )C33 Therefore, the final result became really complicated. To avoid this problem, the matrix elements were found for special case, C · C + =1.  ·α Under this condition, energy is given by equation 6 only if E  = 0. Therefore, α1 + α2 + α3 = 0. This implies that the average value of first perturbation α1 +α2 +α3 ) is zero under this condition. Now the C+ is the standard inverse of a matrix, ( 3 + nm given by matrix elementCmn = cof actorC . Also the matrix elements Cmn can be given det C as following according to equation 5. 2Nd ω Φ1 (1 − 3 cos 2θ) + 4 μ0 ω 2Nd C31 = −JZ2 + Φ2 (1 − 3 cos 2θ) + 4 μ0 ω 2Nd (2) C33 = J(Z1 + Z2 ) − (Φ1 + Φ2 )(1 + 3 cos 2θ) − + (2 cos 2θ)Dm 4 μ0 (4) 4 cos2 θ(cos2 θ − 3 sin2 θ)Dm + Hin sin θ + Hout cos θ + 4Ks sin 2θ 2Nd ω (2) + (2 cos 2θ)Dm 2JZ1 − Φ1 (1 + 3 cos 2θ) − 2 μ0 2 2 2 (4) 4 cos θ(cos θ − 3 sin θ)Dm + Hin sin θ + Hout cos θ + 4Ks sin 2θ C12 = C21 = C23 = C32 = −JZ1 + C13 = C11 = + C22 = + (8) Because it is reasonable to assume that second or fourth order anisotropy constants (2) (2) (2) (4) (4) (4) do not change inside an ultra thin film, D1 =D2 =D3 and D1 =D2 =D3 . Therefore for the convenience the matrix elements C+ mn will be given in terms of C11 , C22 , C32 , and C31 only. + C11 = + C12 = + C13 = + C22 = 2 C11 C22 − C32 + = C33 2 2 C11 (C11 C22 − C31 ) + 2C32 (C31 − C11 ) C32 C31 − C32 C11 + + + = C21 = C23 = C32 2 2 C11 (C11 C22 − C31 ) + 2C32 (C31 − C11 ) 2 C32 − C22 C31 + = C31 2 2 (C31 − C11 ) C11 (C11 C22 − C31 ) + 2C32 2 2 − C31 C11 2 2 C11 (C11 C22 − C31 ) + 2C32 (C31 − C11 ) (9) Both matrices C and C+ are highly symmetric about both matrix diagonals. From equation 6, + + + + (α12 + α32 ) + C32 (2α1 α2 + 2α2 α3 ) + C31 (2α1 α3 ) + α22 C22 ] E(θ) = E0 − 0.5[C11 (10) 76 Electronic Journal of Theoretical Physics 3, No. 11 (2006) 71–83 Because Cmn terms contain sometimes about nine different terms for N=3, the product of two Cmn terms and hence C+ mn terms will contain about 80 terms. For an example, the term C11 contains nine different terms, and therefore C+ 22 will contain about 81 terms. To simplify the problem, only the magnetic exchange interaction, second order anisotropy and stress induced anisotropy terms have been considered for this simulation. For sc(001) lattice, Z0 =4, Z1 =1, Z2 =0 [1], and from above equations number 4 and 8, (2) α = α1 = α2 = α3 =Dm sin2θ, C12 = C21 = C23 = C32 = −J, C13 = C31 = 0 (2) (2) C11 = C33 = J + (2 cos 2θ)Dm + 4Ks sin 2θ andC22 = 2J + (2 cos 2θ)Dm + 4Ks sin 2θ But from equation 10, E(θ) = E0 − 0.5α2 C11 (2C22 − 4C32 + C11 ) + C31 (4C32 − 2C22 − C31 ) 2 2 C11 (C11 C22 − C31 ) + 2C32 (C31 − C11 ) (2) E(θ) = E0 − 1.5[Dm ]2 sin2 2θ (2) (2 cos 2θ)Dm + 4Ks sin 2θ (2) From equation number 3, E0 = -8J-3[Dm cos2 θ+Ks sin2θ] (2) Therefore, E(θ)= -8J-3[Dm cos2 θ+Ks sin2θ]− (2) 1.5[Dm ]2 sin2 2θ (2) (2 cos 2θ)Dm +4Ks sin 2θ (2) Dm = 10, ω = 10, Kωs = 10, then sin2 2θ = −80 − 30[cos2 θ+sin2θ]− cos7.5 2θ+2 sin 2θ The graph between E(θ) and θ is given in figure 1. If Ks is also a variable, then ω E(θ) sin2 2θ 2 = −80 − 3[10cos θ + Kωs sin2θ]− 20 cos150 ω 2θ+4 Ks sin 2θ J ω E(θ) ω If ω The 3-D plot of E(θ) versus Kωs and θ is given in figure 2. ω = 0. When the energy is minimum, ∂E ∂θ After taking the derivative of above equation, at minimum energy the Kωs satisfies following cubic equation. 2( Kωs )3 cos2θ+10( Kωs )2 [cos2 2θ-sin2θ]+25( Kωs )cot2θ[2cos2θ–sin2 2θ] +125[-2cos2θ+2cos2 2θ-sin2 2θ]=0 Using above equation, the preferential direction of orientation for different values of stress can be calculated. For bcc(001) lattice Z 0 =0, Z1 =4 and Z2 =0 [1], and hence C12 = C21 = C23 = C32 = −4J, C13 = C31 = 0, (2) + 4Ks sin 2θ, C11 = C33 = 4J + (2 cos 2θ)Dm (2) C22 = 8J + (2 cos 2θ)Dm + 4Ks sin 2θ. E0 of this lattice is exactly same as that of sc(001). Finally, we can show that energy E(θ) of bcc(001) lattice is exactly same as that of sc(001) lattice. Using Z0 =4, Z1 =4 and Z2 =0 for fcc(001) lattice [1] C12 = C21 = C23 = C32 = −4J, C13 = C31 = 0, (2) C11 = C33 = 4J + (2 cos 2θ)Dm + 4Ks sin 2θ, (2) + 4Ks sin 2θ. C22 = 8J + (2 cos 2θ)Dm Electronic Journal of Theoretical Physics 3, No. 11 (2006) 71–83 77 (2) But E0 = -14J-3[Dm cos2 θ+Ks sin2θ] (2) Therefore, E(θ)= -14J-3[Dm cos2 θ+Ks sin2θ]− (2) 1.5[Dm ]2 sin2 2θ (2) (2 cos 2θ)Dm +4Ks sin 2θ (2) Dm Using ωJ = 10, ω = 10, Kωs = 10 for this simulation, E(θ) sin2 2θ = −140 − 30[cos2 θ+sin2θ]− cos7.5 ω 2θ+2 sin 2θ The graph between E(θ) and θ in this case is similar to the graph given in figure 1. ω According to graph, the easy and hard directions of this film are 34.40 and 124.40 , respectively. These angles are higher than those of oriented ferromagnetic thin films investigated in one of our early report. Therefore, introducing second order perturbation has increased the angles corresponding to easy and hard directions. Also these angles of easy and hard directions are valid for all three type of ferromagnetic materials described in this report. The separation between any two consecutive minimums or maximums is 1800 , similar to regular sine or cosine curve. When Ks is also a variable, E(θ) sin2 2θ = −140 − 3[10cos2 θ + Kωs sin2θ]− 20 cos150 ω 2θ+4 Ks sin 2θ ω versus Kωs and θ in this case is similar to the graph given in The 3-D plot of E(θ) ω figure 2. This graph indicates a slow variation of energy with the variation of the stress or the angle. The graph corresponding to the minimum energy of bcc(001) and fcc(001) are exactly same as that of sc(001). Taking dipole interaction into account will give an energy equation with at least fifty terms, and therefore the effect of dipole interaction will not be discussed in this report. After taking fourth order anisotropy term into account for sc(001) C12 = C21 = C23 = C32 = −J, C13 = C31 = 0, (4) (2) + 4Ks sin 2θ + 4 cos2 θ(cos2 θ − 3 sin2 θ)Dm C11 = C33 = J + (2 cos 2θ)Dm (2) (4) C22 = 2J + (2 cos 2θ)Dm + 4 cos2 θ(cos2 θ − 3 sin2 θ)Dm + 4Ks sin 2θ (2) (4) E(θ)= -8J-3[Dm cos2 θ+Dm cos4 θ+Ks sin2θ] (2) − If (4) Dm = 5, ω E(θ) = ω (4) 1.5[Dm + 2Dm cos2 θ]2 sin2 2θ (2) (4) (2 cos 2θ)Dm + 4 cos2 θ(cos2 θ − 3 sin2 θ)Dm + 4Ks sin 2θ then 7.5[1+cos2 θ]2 sin2 2θ −80 − 15[2cos2 θ+cos4 θ+2sin2θ] − cos 2θ+cos 2 θ(cos2 θ−3 sin2 θ)+2 sin 2θ and θ is given in figure 3. According to figure 1 and 3, The graph between E(θ) ω introducing the fourth anisotropy has destroyed the smoothness of the curve. Sudden periodical overshooting can be observed in the graph. The separation between any adjacent minimum and maximum is very small. The first and second minimums of overshooting can be observed at 740 and 160.50 . The curved parts with smooth maximums and minimums in figure 1 have been flattened in figure 3. If Ks is a variable, 78 Electronic Journal of Theoretical Physics 3, No. 11 (2006) 71–83 E(θ) ω = −80 − 3[10cos2 θ+5cos4 θ + − Ks sin2θ] ω 150[1 + cos2 θ]2 sin2 2θ 20 cos 2θ + 20 cos2 θ(cos2 θ − 3 sin2 θ) + 4 Kωs sin 2θ The 3-D plot of E(θ) versus Kωs and θ is given in figure 4. According to this graph, ω the energy varies rapidly with angle and stress after taking the effect of fourth order anisotropy into account. Several minimums of energies can be observed at different angles and stresses. The stresses corresponding to the different directions of preferred orientation can be determined from this graph. 4. Conclusion: Although the ultra-thin ferromagnetic film with two layers behaves as an oriented film when anisotropy constants remain same for both layers, the energy of film with three layers indicates periodic variation. Some sudden overshooting of energies can be observed after taking second order perturbation into consideration, compared with energy curves of oriented ferromagnetic ultra thin films described in one of our previous report. The way of energy variation of all sc(001), fcc(001) and bcc(001) ferromagnetic ultra thin films with second (or fourth) order anisotropy are exactly same. Easy and hard directions of these sc(001), fcc(001) and bcc(001) ferromagnetic ultra thin films with the effect of second order anisotropy only are 34.40 and 124.40 , respectively. Here these angles have been measured with respect to the normal drawn to the film. The angle between easy and hard directions is exactly 900 as expected. These angles corresponding to easy and hard directions are higher than those of oriented ultra ferromagnetic thin films studied in our early report. Introducing the fourth order anisotropy reduces the smoothness of energy graphs, and indicates several minimums in 3-D graph. Although (2) (4) these details were given for ωJ = 10, Dωm = 10, Kωs = 10 and Dωm = 5 values only, all these (2) (4) simulations can be performed for any values of ωJ , Dωm , Kωs and Dωm or any other type of ferromagnetic material than sc(001), fcc(001) and bcc(001). Since tedious simulations have to be carried out after considering all the terms, above few terms have been taken into account. According to graph 4, the film can be preferentially oriented in some certain directions by applying some certain stresses. Electronic Journal of Theoretical Physics 3, No. 11 (2006) 71–83 79 References [1] K.D. Usadel and A. Hucht, Phys. Rev. B 66, 024419-1 (2002). [2] P. Samarasekara and F.J. Cadieu: Jpn. J. Appl. Phys. 40, 3176 (2001). [3] P. Samarasekara and F.J. Cadieu, Chinese J. Phys. 39(6), 635 (2001). [4] David Lederman, Ricardo Ramirez and Miguel Kiwi, Phys. Rev. B(70), 18442 (2004). [5] Martin J. Klein and Robert S. Smith, Phys. Rev. 81, 378 (1951). [6] M. Dantziger, B. Glinsmann, S. Scheffler, B. Zimmermann and P.J. Jensen, Phys. Rev. B(66), 094416 (2002). [7] M. Bentaleb, N. El Aouad and M. Saber: Chinese J. Phys. 40(3), 307 (2002). [8] D. Zhao, Feng Liu, D.L. Huber and M.G. Lagally: J. Appl. Phys. 91(5), 3150 (2002). 80 Electronic Journal of Theoretical Physics 3, No. 11 (2006) 71–83 Fig. 1 Graph between E(θ) ω and θs for Ks ω =10 with the effect of second order anisotropy Electronic Journal of Theoretical Physics 3, No. 11 (2006) 71–83 Fig. 2 3-D plot of E(θ) ω versus Ks ω and θs with the effect of second order anisotropy 81 82 Electronic Journal of Theoretical Physics 3, No. 11 (2006) 71–83 Fig. 3 Graph between anisotropy E(θ) ω and θs for Ks ω =10 with the effect of second and fourth order Electronic Journal of Theoretical Physics 3, No. 11 (2006) 71–83 Fig. 4 3-D plot of E(θ) ω versus Ks ω and θs with the effect of second and fourth order anisotropy 83 EJTP 3, No. 11 (2006) 85–109 Electronic Journal of Theoretical Physics Frameable Processes with Stochastic Dynamics Enrico Capobianco ∗ Questlab, Research and Statistics Mestre-Ve, Italy Received 14 February 2006 , Accepted 5 April 2006, Published 25 June 2006 Abstract: A crucial goal in many experimental fields and applications is achieving sparse signal approximations for the unknown signals or functions under investigation. This fact allows to deal with few significant structures for reconstructing signals from noisy measurements or recovering functions from indirect observations. We describe and implement approximation and smoothing procedures for volatility processes that can be represented by frames, particularly wavelet frames, and pursue these goals by using dictionaries of functions with adaptive degree of approximation power. Volatility is unobservable and underlying the realizations of stochastic processes that are non-i.i.d., covariance non-stationary, self-similar and non-Gaussian; thus, its features result successfully detected and its dynamics well approximated only in limited time ranges and for clusters of bounded variability. Both jumps and switching regimes are usually observed though, suggesting that either oversmoothing or de-volatilization may easily occur when using standard and non-adaptive volatility models. Our methodological proposal combines wavelet-based frame decompositions with blind source separation techniques, and uses greedy de-noisers and feature learners. c Electronic Journal of Theoretical Physics. All rights reserved.  Keywords: Frames and Wavelets, Blind Source Separation, Sparse Approximation, Greedy Algorithm, Volatility Smoothing PACS (2006): 02.50.Fz, 05.40.-a,02.50.-r, 02.50.Ey 1. Introduction and Volatility Background Volatility processes have been for almost two decades a cross-disciplinary research subject and have suggested many challenging problems to financial mathematicians and statisticians. The main interest of this work too is in volatility, particularly the approximation of its dynamics and its statistical modelling. It is useful to start from the following general classification. ∗ The author carried out most of this work at CWI, Amsterdam (NL). 86 Electronic Journal of Theoretical Physics 3, No. 11 (2006) 85–109 Volatility can be specified as conditionally dependent on past squared returns and own lags, thus representing the class of Generalized Autoregressive Conditionally Heteroscedastic (GARCH) processes [30, 7], or as a stochastically indipendent process, characterized by markovian structure and a noise source independent from the disturbance term in the conditional mean equation, thus referring to Stochastic Volatility (SV) processes [33]. The main stylized facts about volatility models appear from many empirical studies [47], and among them the most important are (1) heavy-tailed (leptokurtic) marginal return distributions; (2) volatility clustering, and thus tail dependence; (3) second-order dependence, visible in absolute and squared transformed returns; (4) long memory and covariance non-stationarity. The GARCH and SV classes are often selected according to different goals, since they refer to different information sets. In the first model class there is observation-driven 2 2 + . . . + αp yt−p (i.e., ARCH(p) information, with yt | Yt−1 ∼ N (0, σt2 ), for σt2 = α0 + α1 yt−1 case); thus, the information set Ft for the index retun process is formed by past squared observations up to time t-p, i.e., σ{Ys2 : s ≤ t − p}. In the other class of models this is not possible, since they are driven by parameters and include both observable and unobservable variables; the index return process is distributed according to yt | ht ∼ N (0, exp(ht )), and the volatility is specified as ht = γ0 + γ1 ht−1 + ηt , ηt ∼ N ID(0, ση2 ). With ηt Gaussian, ht is autoregressive of orγ0 der one, and covariance stationarity follows if | γ1 |< 1. Then μh = E(ht ) = 1−γ , 1 σ2 E(y 4 ) σh2 = var(ht ) = 1−γη 1 and the kurtosis is (σ 2t)2 = 3exp(σh2 ) ≥ 3, resulting in fatter tails y than the Gaussian ones. Non-parametric and semi-parametric approaches have been introduced with the aim of relaxing assumptions (see for instance the seminal work of [31]) on the error probability density of regression models, or on the unobservable volatility function itself. Usually one may end up with iteratively smoothing the unspecified function, as in log-transformed multiplicative models [34] where backfitting estimation procedures [35] are adopted. In these cases, the flexibility allowed so to account for the unrestricted structure in the volatility function may be not sufficient when dealing with stochastic volatility, for then the consistency of estimation procedures may fail. This fact occurs when a certain orthogonality is built in the model, which relies on the statistical independence between the information sets to which the volatility and the conditional mean dynamics refer. The procedures that we suggest and implement are recursive in nature, being this an important condition for building effective stochastic dynamics models, and are designed for operating under both orthogonal and non-orthogonal conditions. The reference framework is that of non-orthonormal and redundant frame-based systems, together with that of orthonormal systems built on atomic dictionaries o approximating functions. It has been shown in many studies (see also [21] for an excellent review) that from such systems, non-linear estimators can be built and result effective and computationally fast. For inhomogeneous function classes they result superior to linear estimators employed by backfitting methods with some kind of smoother. We illustrate how sparsified frame-based volatility models may be investigated by a novel combination of greedy approximation and Electronic Journal of Theoretical Physics 3, No. 11 (2006) 85–109 87 space dimensionality reduction techniques that lead to near-optimal volatility smoothing and feature detection. The paper is organized as follows: Section 2 reviews frames, while Section 3 and 4 illustrate, respectively, multiresolution and greedy approximation techniques. Section 5 suggests an estimation technique based on blind source separation analysis and reports experiments on modeling and smoothing volatility from a stock index return series. The conclusions are in Section 6. 2. Frames A general way of approximating families of functions belonging to general spaces refers to frames [29, 20], which represent redundant sets of approximating vectors. Throughout the paper, volatility processes will be considered to be non-stationary or piecewise stationary, as in local stationary processes [19, 48]; thus, their realizations belong to spaces that allow for spatial inhomogeneity and a certain complexity of dependence structure. Together with the variance positivity constraint, a local boundedness condition for the volatility paths is also assumed; this allows for preventing the volatility function from explosive behavior. We thus consider frameable volatility functions those ones that can be represented by frames or similarly derived systems. Frame components are not linearly independent, but despite this aspect which seems to penalize them from a computational standpoint, there are advantages in using frames since they lead to numerically stable and robust-to-noise reconstruction algorithms while also allowing for increased feature detection power, due to their flexibility. A formal specification of frames requires the presence of a system {γk }M k=1 and bounds A and B such that: 2 A || x || ≤ M  k=1 |< x, γk >|2 ≤ B || x ||2 (1) ∀x ∈ RN . A frame operator associated to them is F such that: [F x]M k=1 =< x, γk > (2) Noting that the redundancy of the frame comes from M ≥ N , and it is measured by the ratio M/N , the frame operator, when multiplied by its transpose F ∗ , shows two properties: • F ∗ F is invertible and A−1 and B −1 are its bounds; • a dual frame {γ˜k }M k=1 is defined such that: γ˜k = (F ∗ F )−1 γk (3) where now the associated dual operator is: F̃ = F (F ∗ F )−1 (4) 88 Electronic Journal of Theoretical Physics 3, No. 11 (2006) 85–109 A tight frame is defined when A = B, while an orthonormal basis requires a value of 1 for the bounds; this allows for an improved reconstrution power, as in this case F ∗ F is diagonal and col(F ) are orthogonal, thus suggesting that a basis can be formed. Another aspect of interest is given by the reconstruction step, where an inverse or pseudoinverse of the F operator is required. The standard formula is delivered by: F − = (F ∗ F )−1 F ∗ (5) but it can be shown that F − = F˜∗ , which means that one choice for the pseudoinverse is provided by the transpose of the operator associated to the dual frames, and this leads to a reconstruction formula which depends on this last operator as follows: x = F˜∗ F x = M  < x, γk > γ˜k (6) k=1 This linear reconstruction leads with noisy signals to a change in the previous expansion so to include the effects of noise according to: M  [ < x, γk > + ǫk ] γ˜k (7) k=1 given the noise ǫ. Note that a generalized reproducing kernel [42] is found whenever we have a vector xn ⊂ x (i.e., a subset of the frame coefficients [F x]M k=1 =< x, γk >) for which we compute the denoising projection:  xk < γ˜k , γk > (8) P xn = k Since for a sequence of frame coefficients it holds that x = P x, our previous relation is valid for x too, due to the kernel < γ˜k , γk >. The mean square error (MSE) is found as: M 1 1  E || x − x̂ ||2 = σ 2 || γ˜k ||2 N N k=1 (9) with σ 2 the variance of the noise term. Given the definition of the dual frames, and replacing in the MSE formula, one may achieve some optimality results, as when an orthogonal matrix can be used a sequence of uniform frames is obtained which asymptotically approaches a tight frame. And it is precisely for this class of frames that the MSE results optimal, i.e., minimum. The MSE computed with frames is of course related to sparsity and coherence measures; we might expect that more sparsity and coherence bring MSE closer to its minimum, due to the fact that the contribution coming from informative and/or correlated Electronic Journal of Theoretical Physics 3, No. 11 (2006) 85–109 89 structures dominates that related to the noise, and a better performance in terms of minimizing errors is achieved by greedy approximation algorithms. These aspects will be introduced afterwards, while now special cases are illustrated. 3. Multiresolution Feature Learning 3.1 Wavelet Frames Wavelet frames are constructed by sampling from the Continuous Wavelet Transform (W T c ) [20] over time and scale coordinates. For f ∈ L2 (R) (with < ., . > the L2 inner product) and given an analyzing (admissible) wavelet with its doubly-indexed generated family, the W T c is: − 12 c W T (f )jk =< f, ψjk >=| j |  f (t)ψ( t−k )dt j (10) The function f can be recovered from the following reconstruction formula: f= c−1 ψ   W T c (f )jk ψjk djdk j2 (11) and this comes from the ”resolution of identity formula”: 2 ∀f, g ∈ L (R),   c dkdj % W T c (f )jk W T (g)jk 2 = cψ < f, g > j (12) Given a constant cψ < ∞ and integration ranging from −∞ to ∞, the W T c maps f into an Hilbert space, and its image, say L2W T c (R), is a closed subspace and a Reproducing Kernel Hilbert Space too, since from the resolution formula it is enough to replace g with ψ so to get: c % T (g)jk > = < ψj ′ k′ , ψjk > K(j, k, j ′ , k ′ ) =< W (13) More generally, given a scaling function φ, such that its dilates and translates constitute orthonormal bases for all the Vj subspaces obtained as scaled versions of the subspace V0 to which φ belongs, and given a mother wavelet ψ together with the terms indicated with ψjk and generated by j−dilations and k−translations, such that ψjk (x) = j 2 2 ψ(2j x − k), one obtains differences among approximations computed at successively coarser resolution levels. Thus, a so-called Multiresolution Analysis (MRA) [41, 20] is obtained, i.e. a sequence ¯ j∈Z Vj = L2 (R), of closed subspaces satisfying . . . , V2 ⊂ V1 ⊂ V0 ⊂ V−1 ⊂ V−2 ⊂ . . ., with ∪ j ∩j∈Z Vj = {0} and the additional condition f ∈ Vj ⇐⇒ f (2 .) ∈ V0 . The last condition is a necessary requirement for identifying the MRA, meaning that all the spaces are scaled versions of a central space, V0 . 90 Electronic Journal of Theoretical Physics 3, No. 11 (2006) 85–109 An MRA approximates L2 [0, 1] through Vj generated by orthonormal scaling functions φjk , where k = 0, . . . , 2j − 1. These functions allow also for the sequence of 2j wavelets ψjk , k = 0 . . . , 2j − 1 to represent an orthonormal basis of L2 [0, 1]. Signal decompositions with the MRA property have also near-optimal properties in a quite wide range of inhomogeneous function spaces, Sobolev, Holder, for instance, and in general all Besov and Triebel spaces [46]. Generally speaking, with a Discrete Wavelet Transform (DWT) a map f → w from the signal domain to the wavelet coefficient domain is obtained; one applies, through a bank of quadrature mirror filters, the transformation w = W f , so to get the coefficients for high scales (high frequency information) and for low scales (low frequency information). A sequence of smoothed signals and of details giving information at finer resolution levels is found from the wavelet signal decomposition and may be used to represent a signal expansion: f (x) =  cj0 k φj0 k (x) +  j>j0 k djk ψjk (x) (14) k where φj0 k is associated with the corresponding coarse resolution coefficients cj0 k and & & djk are the detail coefficients, i.e., cjk = f (x)φjk (x)dx and djk = f (x)ψjk (x)dx. In short, the first term of the right hand side of (1) is the projection of f onto the coarse approximating space Vj0 while the second term represents the cumulated details. 3.2 De-noising and De-correlation In the wavelet-based representations of signals sparsity inspires strategies that eliminate redundant information, not distinguishable from noise; this can be done in the wavelet coefficients domain, given the relation between true and empirical coefficients: d˜jk = djk + ǫt (15) The wavelet shrinkage principle [24, 25, 26] applies a thresholding strategy which yields de-noising of the observed data; it operates by shrinking wavelets coefficients to zero so that a limited number of them will be considered for reconstructing the signal. Given that a better reconstruction might be crucial for financial time series in order to capture the underlying volatility structure and the hidden dependence, de-noising can be useful for spatially heterogeneous signals. Financial time series are realizations of non-stationary and non-Gaussian stochastic processes; a reason why wavelet systems could be effectively used when dealing with these signals, concerns the ability of wavelets to de-correlate along time and across scales. The de-correlation effect of the wavelet coefficients is one of the main properties that wavelet transforms bring into the analysis [37, 2]; thus, from a structure with long range dependence (LRD), either short range dependence (SRD) or de-correlation are found, when looking at wavelet coefficients at each scale 2j or resolution level j. Electronic Journal of Theoretical Physics 3, No. 11 (2006) 85–109 91 Given a probability space (Ω, F, P), consider a stochastic process and one of its realizations observed with strong dependence structure; following [1], and with a slight variation in the notation compared to before due to the frequency (scale) characterization, let E[dx (j, k)] = 0 and the variance be:  2 E[dx (j, k) ] = Γx (v)2j | Ψ0 (2j v) |2 dv (16) and it represents a measure of the power spectrum Γx (.) at frequencies vj = 2−j v0 , with Ψ0 the Fourier Transform of ψ0 . Given Γx (v) ∼ cf | v |−α and v → 0, then: E[dx (j, k)2 ] ∼ 2jα cf C(α, ψ0 ) for j → ∞, and C(α, ψ0 ) =  | v |−α | Ψ0 (v) |2 dv (17) (18) for α ∈ (0, 1). One can then look at the variance law as follows: var(dx (j, k)) ≈ 2jα (19) for j → ∞. The decay of the covariance function is much faster in the wavelet expansion coefficients domain than in the domain originated by long memory processes. The covariance function of the wavelet coefficients is controlled by M , the number of vanishing moments; when they are present in sufficiently high number they lead to high compression power. The sequence of detail signals or wavelet expansion coefficients is a stationary process if the number of vanishing moments M satisfies the constraint that the variance of dx (j, k) shows scaling behaviour in a range of cut-off values j1 ≤ j ≤ j2 which has to be determined. The sequence no longer shows LRD but only SRD when M ≥ α2 , and the higher M the shorter the correlation left, due to: E[dx (j, k)dx (j, k ′ )] ≈ | k − k ′ |α−1−2M , f or | k − k ′ | → ∞ (20) These assumptions don’t rely on a Gaussian signal, and could be further idealized by assuming E[dx (j, k)dx (j, k ′ )] = 0 for (j, k) = (j, k ′ ). Thus, with an LRD process, the effect of the wavelet transform is clear, bringing back decorrelation or small SRD due to the control of non-stationarity and dependence through the M parameter. Across scales, instead, a certain degree of independence is obtained, so that the detail series might individually contribute to different information content in terms of frequency. 92 Electronic Journal of Theoretical Physics 3, No. 11 (2006) 85–109 With regard to non-stationarity, an almost natural condition of financial time series, especially when measured at very high frequencies; wavelets stationarize the data when they are observed in their transformed wavelet coefficients, and they enable this change in a resolution-wise fashion. As suggested by [16] X(t) is a stationary process if and only if it is stationary at all levels of resolutions: Definition 1: k-stationarity Stationarity at the kth level of resolution if ∀n ≥ 1, t1 , . . . , tn ∈ T and l ∈ Z holds when: d [X(t1 + 2−k l), . . . , X(tn + 2−k )] = [X(t1 , . . . , X(tn )] (21) 3.3 Multiscale Decomposition We face two problems, when approximating the volatility function and estimating the model parameters involved: the role of smoothness and the presence of noise. Following [5], we might rely on quadratic information from the data, leading to non-negative estimators of the following kind: σ̃ 2 (t) =  αi ri2 (t) (22) i   for i αi = 0 and i αi2 = 1. This can just be an initial estimate for a more calibrated and robust procedure, since it can be improved by estimators that better account for smoothness and sparsity. Consider the L2 wavelet decomposition for the volatility function, expressed this time through inner products: 2 σW (t) =  k < σ 2 (t), φj0,k > φj0,k (t) +  j>j0 < σ 2 (t), ψj,k > ψj,k (t) (23) k where a smooth part is combined with a cumulated sequence of details obtained at different scales. We can then apply the same decomposition to σ̃ 2 , being σ2 unobservable, and obtain a perturbed version of the previous approximation: 2 σ̂ 2 (t) = σ̃W (t) + ǫ(t) (24) where ǫ = W ξ represents a wavelet transformed disturbance. In order to sparsify the estimator, we can apply noise shrinkage techniques or use function expansions in different bases or overcomplete dictionaries. If instead of considering the L2 elements we consider other function classes more suitable for inhomogeneous behavior we can build non-linear estimators for the volatility function through wavelet-type families [32]. Electronic Journal of Theoretical Physics 3, No. 11 (2006) 85–109 93 3.4 Overcomplete Representations Function dictionaries are collections of parameterized waveforms or atoms [15] ; they are available for approximating many classes of functions, formed directly from a particular family or from merging two or more dictionary classes. Particularly in the latter case an overcomplete dictionary is composed, with linear combinations of elements that may represent remaining dictionary structures, thus originating a non-unique signal decomposition. Wavelet packets (WP) [17] are one example of overcomplete representations; they represent an extension of the wavelet transform and allows for better adaptation due to an oscillation index f which delivers a richer combination of functions. & +∞ Given the admissibility condition −∞ W0 (t)dt = 1, ∀(j, k) ∈ Z 2 , following [39] we have: − 12 2 ∞  t hi−2k Wf (t − i) W2f ( − k) = 2 i=−∞ (25) where f relates to the frequency and h to the low-pass impulse response of a quadrature mirror filter. Also the following holds: ∞  1 t 2− 2 W2f +1 ( − k) = gn−2k Wf (t − n) 2 n=−∞ (26) where g is this time an high-pass impulse response. For compactly supported wave-like functions Wf (t), finite impulse response filters of a certain length L can be used, and by P -partitioning in (j, f )-dependent intervals Ij,f one finds an orthonormal basis of L2 (R), i.e., a wavelet packet: j {2− 2 Wf (2−j t − k), k ∈ Z, (j, f ) | Ij,f ∈ P } (27) One thus obtains a better domain compared to simple wavelets for selecting a basis to represent the signal and can always select an orthogonal wavelet transform by changing the partition P and defining w0 = φ(t) and Wf = ψ, from the so-called WP transform (WPT). Correspondingly, Cosine Packets (CP) and the related transform (CPT) suggest optimal bases in terms of compression power and sparsity [27], and optimal bases for dealing with non-stationary processes with time-varying covariance operators [44]. The building blocks in CP are localized cosine functions, i.e., localized in time and forming smooth basis functions, and they form almost eigenvectors for certain classes of non-stationary processes, and thus almost diagonal operators for approximating the covariance function. The CPT has an advantage over the classic Discrete Cosine Transform (DCT); the latter defines an orthogonal transformation and thus maps a signal from the time to the 94 Electronic Journal of Theoretical Physics 3, No. 11 (2006) 85–109 frequency domain, but it is not localized in time and thus is not able to adapt well to non-stationary signals. Depending on the taper functions we select, the cosine packets decay to zero within the interval where they are defined and in general determine functions adapted to overcome the limitations of DCT. The DCT-II transform is defined as:  n−1 2  (2i + 1)kπ sk ) (28) gk = fi+1 cos( n i=0 2n for k = 0, 1, . . . , n − 1, and scale factor sk resulting 1 if k = 0 or n, and An inverse DCT is instead given by:  n−1 2 (2i + 1)kπ ) fi+1 = gk sk cos( n k=0 2n √1 2 if k = 0 or n. (29) for i = 0, 1, . . . , n − 1. 4. Greedy Algorithms Optimal algorithms often require adaptive signal approximation techniques based on sparse representations. Sparsity refers to the possibility of considering only few elements of a dictionary of approximating functions selected among a redundant set. In this way, by restricting the search to a sub-set of the original atomic dictionary, one may combine fast convergence through computing less inner products. The MP algorithm [43] is a good example, and it has been successfully implemented in many studies for its simple structure and effectiveness. A signal is decomposed as a sum of atomic waveforms, taken from families such as Gabor functions, Gaussians, wavelets, wavelet and cosine packets, among others. We focus on the WP and CP Tables, whose signal representations are given by:  W P (t) = jf k wj,f,k Wj,f,k (t) + resn (t) and  CP (t) = jf k cj,f,k Cj,f,k (t) + resn (t) In summary, the MP algorithm approximates a function with a sum of n elements, called atoms or atomic waveforms, which are indicated with Hγi and indexed by a dictionary Γ of functions whose form should ideally adapt to the characteristics of the signal at hand. The MP decomposition exists in orthogonal or redundant version and refers to a greedy algorithm which at successive steps decomposes the residual term left from a projection of the signal onto the elements of a selected dictionary, in the direction of that one allowing for the best fit. At each time step the following decomposition is computed, yielding the coefficients hi which represent the projections, and the residual component, which will be then reexamined and in case iteratively re-decomposed according to: Electronic Journal of Theoretical Physics 3, No. 11 (2006) 85–109 f (t) = n  hi Hγi (t) + resn (t) 95 (30) i=1 and (1) (2) (3) following the procedure: inizialize with res0 (t) = f (t), at i=1; & compute at each atom Hγ the projection μγ,i = resi−1 (t)Hγ (t)dt; find in the dictionary the index with the maximum projection, γi = argminγ∈Γ || resi−1 (t) − μγ,i Hγ (t) ||, which equals from the energy conservation equation argmaxγ∈Γ | μγ,i |; (4) with the nth MP coefficient hn (or μγn ,n ) and atom Hγn the computation of the updated nth residual is given by: resn (t) = resn−1 (t) − hn Hγn (t); (5) repeat the procedure until i ≤ n. With H representing an Hilbert Space, the function f ∈ H can thus be decomposed as f =< f, gγ0 > gγ0 + Rf , with f approximated in the gγ0 direction, orthogonal to Rf , such that f 2 =|< f, gγ0 >|2 +Rf 2 . Thus, to minimize the Rf  term requires a choice of gγ0 in the dictionary such that the inner product term is maximized (up to a certain optimality factor. The choice of these atoms from the D dictionary occurs by choosing an index γ0 based on a certain choice function conditioned on a set of indexes Γ0 ∈ Γ. The main aspect of interest for the computational learning power of the MP algorithm has appeared in our study like in many others, and refers to how it is capable of dealing efficiently with the so-called coherent structures compared to the dictionary noise components [22]. 5. Non-parametric Estimation 5.1 Data Figure 1 reports one-day 1m returns compared to the 5m aggregated values, together with their absolute and squared transforms. We deal with stock returns observed at very high frequencies from the Nikkei 225 composite index; the data were collected minute by minute and refer to a certain market activity year, 1990. We have approximately 35,000 observed values, covering intra-day market activity and excluding holidays and weekends; we have transformed the prices in financial returns by taking, as usual, the logarithms of ratios of consecutive time point prices. We have also generated a temporally aggregated (every five minutes) series, which basically smooths the original series, at the price of losing high frequency content. More than 7,000 5-minute observations remain available to conduct a compared analysis. We note a self-similar behavior and the typical function shape conditioned to the different intensity of intra-day activity hours according to usual market technical phases. In defining self-similarity, addressed as the property of self-affinity by [45], from [6] we Electronic Journal of Theoretical Physics 3, No. 11 (2006) 85–109 0 50 100 150 200 250 0.04 0.03 0.02 0.01 0 A. 50 100 150 200 250 Time 50 100 150 200 250 C. 0.06 Time 0 10 20 30 Time 40 50 D. 0.0 0.0 -0.2 0.01 0.05 -0.1 0.02 0.10 0.03 0.0 0.15 0.04 0.05 0.20 0.2 0.1 0 B. 0.25 Time 0.0 -0.2 0.0 -0.1 0.05 0.0 0.10 0.1 0.15 0.2 0.20 96 0 10 20 30 Time 40 50 E. 0 10 20 30 Time 40 50 F. Fig. 1 A) Raw 1m returns. B) Absolute 1m returns. C) Squared 1m returns. D) Raw 5m returns. E) Absolute 5m returns. F) Squared 5m returns. have the following definition: Definition 2: self-similarity Given α ∈ (0, 1), γ > 0, f : Rd → R and x̄ ∈ Rd , a local re-normalization operator γ family, Rα,x̄ can be constructed such that: γ Rα,x̄ f (x) = 1 [f (x̄ + γx) − f (x̄)] γα ∀x ∈ Rd (31) Thus, for instance, a Gaussian process X defined on a probability space (Ω, F, P ), is a d γ X = X], ∀γ ∈ R+ and ∀x̄ ∈ Rd . self-similar process of degree α if [Rα,x̄ As a remark, working with just one long realization of the underlying return process means accepting the limitations that necessarily follow with regard to asymptotic inference, but at the same time represents a de facto typical situation in non-experimental contexts, where the suggested techniques might be used. Dealing with non-stationarity is imposed by real circumstances; the observed series is subject to regime changes and external factors whose impact on the dynamics of returns and structure of volatility is undoubtedly relevant. The observed index return time series and the underlying volatility process represent a challenging context for approximation and estimation techniques. 5.2 Iterative Smoother Instead of using very noisy squared return sequences, we may run the MP greedy algorithm directly over the returns and then look at the computed absolute and squared residuals for investigating the volatility features. The corresponding autocorrelation function delivers a diagnostic measure of the feature detection power of the MP algorithm run over WP and CP dictionaries. Thus, in Electronic Journal of Theoretical Physics 3, No. 11 (2006) 85–109 97 our experiments, a main issue is to control the behavior of the transformed MP residual terms after n approximation steps. We have examined (see Figure 2) wavelet and cosine packets, whose time-frequency partitioning role is complementary. While cosine packets build a partition in the time domain and then run over each interval further frequency segmentation, wavelet packets work the other way around, by first segmenting along frequencies and then along time time. Thus, the former face frequency inhomogeneous smoothness while the latter deal with time inhomogeneity [28]. The within-block coefficients of the WP and CP structures describe the contribution at each resolution level of both time and frequency components in representing the signal features under a varying oscillation index varying from 0 to 2J − 1 (right-wise). The WP Table presents sets of coefficients stored in sequency order according to an increasing oscillation index; at level 0 the original signal is represented and at level 1 the two subsets w1,0 and w1,1 have scale 2, corresponding to c1 and d1 obtained with the DWT. The CP table presents instead blocks ordered by time and the coefficients within the blocks are ordered by frequency. Level 0 Level 0 Level 1 Level 1 Level 2 Level 2 Level 3 Level 3 Level 4 Level 4 Level 5 Level 5 Level 6 Level 6 0 1000 2000 3000 4000 5000 6000 7000 A. B. Fig. 2 CP table (A) and WP table (B) with signal segmentation level-by-level. The way these plots should be interpreted suggests that the low frequency information in the signal is expected to be concentrated on the left side and the high frequency information on the right side of the table. For the CP table, the high frequency part of the signal is now expected on the left side, while the low frequency behavior appears from the right side. We investigate the performance of the MP algorithm when is applied on an ad hoc restricted and selected dictionary, based on a certain range of most independently informative resolution levels. Details are introduced next. 98 Electronic Journal of Theoretical Physics 3, No. 11 (2006) 85–109 5.3 Blind Source Separation The goal of searching for statistically independent coordinates characterizing certain objects and signals, or otherwise for least dependent coordinates, due to a strong dependence in the nature of the stochastic processes observed through the structure of the index series, leads to Independent Component Analysis (ICA) [13, 18]. Combination these goals with the search for sparse signal representations suggests hybrid forms of Sparse Component Analysis (SCA) [40, 23, 49]. With SCA one attempts to combine the advantages delivered by sparsity and independence of signal representations, which transfer to better compression power and estimation in statistical minimax sense. By assuming that the sensor outputs are indicated by xi , i = 1, . . . , n and represent a combination of independent, non-Gaussian and unknown sources si , i = 1, . . . , m, a non-linear system Y = f (X) could be approximated by a linear one AS, where X = AS. Instead of computing f (X) one may now work for estimating the sources S together with the m × m mixing matrix A, where usually m << n, with n the number of sensor signals, but with m = n holding in many cases too. Independent components can be efficiently computed by ad-hoc algorithms such as joint approximate diagonalization of eigenmatrices for real signals (JadeR) [14]. For Gaussian signals, the Independent Components are exactly the known Principal Components; with non-Gaussian signals ICA delivers superior performance, due to the fact that it relies on high order statistical independence information. The JadeR algorithm is the ICA algorithm that we have applied in our experiments to deliver estimates for the separating or de-mixing matrix B = A−1 , and obtain the Y = BX, such that (under a perfect separation) Y = BAS = S. As a matter of fact, the solution holds approximately and up to permutation P and scaling D, i.e., Y = DP S. The de-correlation and rotation steps which have to be implemented will deliver a set of approximate m independent components. 5.4 Volatility Source Separation We set the following system to represent a volatility process: y t = A t xt + ǫ t (32) where the observed returns are indicated by yt , the mixing matrix At is to be estimated, together with the sources or latent variables xt ; the noise ǫt is superimposed to the system 2 the volatility dynamics, with an i.i.d. (0, σǫ,t ) distribution. We indicate with vt = σǫ,t process. Ideally the volatility sources have a sparse representation, represented through the following system: xt = Cjf t Φjf t + ηt (33) Electronic Journal of Theoretical Physics 3, No. 11 (2006) 85–109 99 The compression and decorrelation properties of wavelet transforms can be supported by a more effective search for least dependent components through combined MRA and ICA steps. These steps can be set to work together in an hybrid method, as described below. Since the sources are unobservable, estimating them and the mixing matrix is quite complicated; we can either build an optimization system with a regularized objective function through some smoothness priors, so to estimate the parameters involved, or we can proceed more recursively in the mean square sense. One route is going through the iterations of the MP algorithm, by processing the observed returns with the WP and CP libraries, and by looking at: Yt ≈ Pt Φt + ξt = At Ct Φt + ξt (34) where the noise is including an approximation error from the system equation and residual measurement effects ǫt . We start by considering the detail signals obtained through WP and CP transforms, which refer to different degrees of resolution. Then, we combine an ICA step with the MP algorithm operating on WP and CP tables; through such a joint search for sparsity and statistical independence we are basically adopting an hybrid SCA solution. We aim to optimize sparsity through the choice of ad hoc function dictionaries and optimze the performance of non-linear thresholding estimators. Furthermore, we search least dependent coordinates such that an almost diagonal covariance operator is achieved, helping the interpretation of latent volatility features. 5.5 Mixing Matrix Estimates In Table 1 below we report the two estimated mixing matrices A, where the observed sensor signals are those computed at each resolution levels by the WP and the CP transforms. These signals are passed through the ICA algorithm for the extraction of ”m” possible sources which we set equal to the number of sensors. We look at the results of this table so to extract from each detail level an approximate value indicating its contribution to the global signal features, independently from the other levels. The highest values computed suggest what are the dominant independent components on a scale-dependent basis, without identifying their specific nature or the underlying economic factors, being them system dynamics or pure shocks. From the WP estimated mixing matrix A we note a strong within-level factor always dominating apart from levels 5 and 6, where a mutual cross-influence appears to dominate. From the CP extimated mixing matrix A things change substantially, since each level depends mainly from out-of-level factors, i.e., components belonging to other resolution levels, and only negligibly influenced by within-level factors. ICA selects the finest resolution levels of the WP Table, while for the CP Table it delivers a mix of components which are not concentrated at the finest resolutions. 100 Electronic Journal of Theoretical Physics 3, No. 11 (2006) 85–109 Resol. lev. 0 1 2 3 4 5 6 level 0 0.2218 0.0028 0.0085 0.0047 0.0023 0.0069 0.0085 level 1 0.0002 0.1951 −0.0013 0.0001 −0.0189 −0.0035 −0.0037 level 2 0.0068 0.0003 −0.167 0.0015 0.0007 0.0019 −0.001 level 3 0.0031 −0.0057 −0.0008 −0.1438 −0.0019 −0.0045 0.0059 level 4 0.0012 −0.0125 0.0017 0.0028 −0.1318 0.0117 0.0 level 5 0.0032 −0.0023 0.0014 −0.0045 0.0008 −0.0011 −0.1147 level 6 0.0023 −0.0009 −0.0018 0.0047 −0.0082 −0.121 0.0017 level 3 0.1868 −0.0008 −0.0038 −0.0114 −0.0057 −0.0031 0.006 level 4 −0.0022 0.1832 0.0011 −0.0002 −0.0191 −0.0083 0.0053 level 6 0.0014 0.0046 0.1748 −0.0021 0.0036 −0.0052 0.002 level 2 −0.0089 −0.0023 −0.0031 −0.1712 0.0031 0.0057 −0.0006 level 5 0.006 0.0142 −0.0059 0.002 0.1482 −0.0053 0.0035 level 0 0.0029 0.0062 0.008 0.0031 0.0021 0.1261 0.0033 level 1 0.0012 0.0033 0.0013 0.0013 0.0041 0.0039 −0.1204 WP-A CP-A Table 1 Weights of the estimated ICA mixing matrix A distributed across resolu- tion levels for residual 5m series obtained in WP/CP tables. The diagonalization achieved by ICA with WP delivers a resolution-wise ordered sequence of informative coefficients, with the highest scale delivering the most informative component, while for CP the order is not the same. Lower scale sequences are more Gaussian and stationary, due to aggregation effects, and thus it is likely that a certain loss of efficiency follows from the fact of missing finer detail structure. Conversely, higher scale sequences, being locally more irregular series, might present better efficiency together with faster convergence rates. The coefficient sequences obtained with WP and CP transforms have resolution-wise approximation power and thus perform more accurately, compared to the original series, with regard to the sample path from which the N data are observed. Processes for which a convergence rate N −1 is reached by pointwise continuous real valued functions are called irregular path processes [8] and the rate is called super-optimal. For stationary Gaussian processes with differentiable paths these rates are not achievable; similarly, lower scale sequences, compared to higher scale ones, reflect this aspect. Electronic Journal of Theoretical Physics 3, No. 11 (2006) 85–109 101 5.6 Realised-Integrated Volatility Due to the fact that daily squared returns don’t help too much in forecasting the latent volatility structure because of noise and of different dynamics, a measure has been suggested so to obtain a more accurate estimate of the volatility function with high frequency observations.  The realised volatility function is obtained by σ̂ 2 (t) = Ti=1 ri2 (t), and thus fulfills the scope of approximating the integral of the unobservable variable by averaging a certain number of 1m or 5m intraday values ri2 (t) [3, 4]. & It holds [38] that σ̂t2 → σt2 = σs2 ds, i.e., the integrated volatility is approximated by the realised volatility obtained according to the quadratic variation (QV) principle derived from the following definition: Definition 3: pth variation given Xt and the partition T = {t0 , t1 , . . . , tn } of [0, t], the pth variation of Xt over T is: (p) Vt (T ) = n  k=1 | Xtk − Xtk−1 |p (35) (2) If || T ||= max1≤k≤n | tk − tk−1 | goes to 0, then lim||T ||→0 Vt (T ) =< X >t , where the limit is the QV of Xt . This implies, in turn, that a convergence in probability applies, &1 2  2 i.e., nj=1 rn,t,j →n→∞ 0 σt+τ dτ , where the cumulative squared high frequency returns are employed rather than the daily values, so to improve the volatility prediction power. The QV principle and the realised-integrated volatility relation lead quite naturally [36] towards a non-parametric regression model as a de facto reference setting. With a normalized sampling index t = ni , i = 0, . . . , n, and with yi/n = n(x(i+1)/n − xi/n )2 indi& (i+1)/n 2 σ (x)s ds, from this latter expression one can recover cating noisy estimates of n i/n 2 σ (xt ) and thus the whole procedure leads to a model like yt ≈ σ 2 (xt ) + ǫt . Since both the regressors and the disturbances are not indipendent and identically distributed random variables, the volatility estimator can be found through families of non-parametric estimators where the search for more spatially homogeneous observational points require a re-mapping of the original values to some more regular domain, i.e., a  ν-indexed grid such that σ̂n2 (x̄) = νi Ki,n (xν , x̄). Our next step is to identify efficient and feasible solutions for smoothing volatility. 5.7 Experiments For pointwise volatility estimation a possibility is to use a parametric model such as a GARCH or SV, computed directly on the residuals obtained with the MP algorithm, after a certain number of steps [10, 11, 12]. The focus is on detecting the time varying volatility features. We thus show parametric estimates from a model designed for the series of MP residues obtained after 100 and 500 iterations, i.e. by using 100 or 500 approximating structures. 102 Electronic Journal of Theoretical Physics 3, No. 11 (2006) 85–109 We have adopted an MA(1)-GARCH(1,1) model, thus balancing serial correlation in the returns, and have used a Student’s t as marginal distribution. The model should be written with a conditional mean equation (including an intercept and the MA(1) term in k) like: yt = k + ξt σt , with σt = 2 ht = a1 ht−1 + b1 yt−1 $ ht (36) (37) where yt | Ψt−1 ∼ i.i.d.(0, ht ), ξt ∼ N (0, 1), and given the set of past information Ψt−1 . By the prediction error decomposition, the log-likelihood function for a sample y1 , . . . , yi is given by: li (Θ) = logLT (Θ) = T  i=1 T T  xi 1 loghi + log g( 1/2 ) logp(yi | Ψi−1 ) = − 2 i=1 hi i=1 (38) where g(.) is the Student’s t distribution, an heavy tailed conditional distribution, given by g(x) = c 1 v+1 x2 (1+ v−2 ) 2 , where c = Γ( v+1 ) 2 1 (π(v−2)) 2 Γ( v2 ) . −1 Note that x = yi hi 2 are the standardized residuals, and that the degrees of freedom are estimated together with the other parameters, say Θ, in the model. We then check diagnostic and statistical properties, reported in Table 2, where t-stat is calculated from the estimated standard deviations; Deg. Fr. refers to the return Student’s t distribution; LB stands for Ljung-Box statistics for estimated squared standardized residuals; MaxLik is the estimated value of the likelihood function. Parameters (G)hf100w.res (G)hf500w.res (G)hf100c.res (G)hf500c.res 0.30 0.19 0.33 0.21 (33.87) (18.53) (30.08) (17.7) ARCH 0.012 0.014 0.076 0.078 t-stat (9.95) (8.51) (9.6) (5.96) GARCH 0.973 0.98 0.699 0.20 (341.75) (387.8) (25.89) (1.895) Deg.Fr. 2.88 5.29 2.84 5.16 LB 20.07 10.86 40.25 17.49 5327.87 7229.1 5176.29 6628.33 MA(1) t-stat t-stat MaxLik Table 2 GARCH (G) estimates for the residual series with 100 and 500 runs of MP with WP and CP tables indicated by hf100/500w.res and hf100/500c.res. We observe that in general 500 iteration residues seem to suggest better models, as far as concerns likelihood estimated values and LB statistics. For CP and WP the estimates Electronic Journal of Theoretical Physics 3, No. 11 (2006) 85–109 103 of the parameters change, particularly for the GARCH one, resulting for CP smaller in absolute terms and surprisingly not significant for the 500 iteration residues. Some plots are now reported with regard to the quantiles with respect to the Studentt distributions of the residual sequences obtained by MP after 100 and 500 iterations, together with the ACF of the squared residuals (see Figure 3). With 500 iterations the distributional properties suggested by the QQ-plots improve, for a better alignment with the ideal benchmark distributions. Ideally, coherent structures should be removed and the algorithm should be stopped when dictionary noise is encountered. We observe second order statistical properties since features relate now to the volatility process characterizing the signal. When no structure is found this fact has to be interpreted as the evidence that only pure volatility aspects are left in the residual series, now clean of dependence, seasonal and non-stationary components. Our results and diagnostic plots indicate that volatility is smoother when estimated from 500 MP iterations, compared to only 100. The possibilities are that it is either approximating the true volatility or that the MP algorithm is overfitting. For the CP case the same observation could be done, but due to the behavior of the GARCH parameter estimates, we might conclude that more than possible oversmoothing, MP has produced de-volatilization in the returns process. Figure 4 is about the performance of the MP algorithm when examined through the residues obtained at varying approximation power employed. For the case under study, we consider the L2 and L1 errors, from respectively squared and absolute transformed residual terms. We compare them with the number of MP approximating, possibly coherent, structures employed, up to 500, which corresponds to the L0 norm of the expansion coefficients, i.e., a measure of sparsity. We note from the CP plots (C-D) that MP has a fast convergence; with both L2 and L1 norms, the first minimal turning point is at 100 structures, while the second one is at 200 structures. For the L2 norm the successive decay is smooth, while for the L1 norm is slightly steeper in approaching the new minimum at approximately 500. From the WP plots (A-B) the limit of 100 structures is confirmed, but then convergence is lost. This confirms that by iterating 500 times the MP algorithm surely oversmooths the volatility function with WP approximating atoms, and instead of learning its structure it yields de-volatilization in the CP case. We thus can conclude that the number of 100 iterations represents a fairly good benchmark iteration number of the MP algorithm for exploring the dynamics and learning the features of stock index return volatility, while at the same time preventing from possible numerical instability and overfitting. 6. Conclusions Financial time series are very complex structures and we show that in order to investigate their structure and detect volatility features it might be useful to adopt sparse 104 Electronic Journal of Theoretical Physics 3, No. 11 (2006) 85–109 representations and independent component decompositions, together with greedy approximation techniques. An example is offered by ICA applied to wavelet and cosine packets, and forming an hybrid form of SCA when combined with greedy feature learners. It appears from the experiments that the there is a clear improvement in the feature detection power of the latent volatility structure underlying a stock index return series. The resolution selection operated by ICA, concentrated on high scale signals for WP function dictionaries, eliminates redundant information by keeping highly localized time resolution power without simultaneously losing too much frequency resolution, due to the fact that low scale information can be reproduced by averaging high scales. When ICA is applied to a CP library, it doesn’t really build a sparse representation, since the CP coordinates are already naturally endowed with that property; in terms of decomposing the signal, the advantage of using a CP transform is thus in the inherent diagonalization power with respect to the covariance operator. The hybrid SCA method that we have designed for our application yields least dependent resolution levels which are used for calibrating the MP algorithm. 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Zibulewsky, and B.A. Pearlmutter, Blind Source Separation by Sparse Decomposition in a Signal Dictionary, Neural Computation, 13(4), (2001), 863-882. Electronic Journal of Theoretical Physics 3, No. 11 (2006) 85–109 10 15 108 7040 7040 10 70387039 xres -10 -5 -5 0 0 xres 5 5 7039 1 -20 -10 0 10 20 -10 10 -5 0 A. Quantiles of t(2.88872294767274) distribution 5 10 Quantiles of t(5.2936730230749) distribution 7040 0 xres -4 -5 -2 xres 0 2 5 4 6 7040 B. -6 -10 2 -8 1 -30 -20 -10 0 10 20 Quantiles of t(2.84688052704445) distribution -10 C. -5 0 5 10 Quantiles of t(5.16731001818244) distribution Series : resq500w D. ACF 0.0 0.0 0.2 0.2 0.4 0.4 ACF 0.6 0.6 0.8 0.8 1.0 1.0 Series : resq100w 30 2 1 0 2000 4000 6000 0 2000 Lag 4000 6000 Lag E. F. Series : resq500c ACF 0.0 0.0 0.2 0.2 0.4 0.4 ACF 0.6 0.6 0.8 0.8 1.0 1.0 Series : resq100c 0 2000 4000 6000 0 Lag 2000 4000 6000 Lag G. H. Fig. 3 QQ-plots of standardized residuals vs quantiles of a Student’s t, for 100 (A,B) and 500 (C,D) MP iterations run on respectively WP/CP Tables; correspondent ACF of squared residuals for WP (E,F) and CP (G,H). Electronic Journal of Theoretical Physics 3, No. 11 (2006) 85–109 109 0.08 0.05 0.02 0 100 200 300 400 500 A. 0.3 0.2 0.1 0.0 0 100 200 300 400 500 B. 0.3 0.2 0.1 0.0 0 100 200 300 400 500 C. 0.5 0.3 0.1 0 100 200 300 400 500 D. Fig. 4 L2 error vs number of approximating structures, for WP (A) and CP (C) and L1 error vs L0 norm for WP (B) and CP (D). EJTP 3, No. 11 (2006) 111–122 Electronic Journal of Theoretical Physics Ab-initio Calculations for Forbidden M1/E2 Decay Rates in Ti XIX ion A. Farrag ∗ Physics Department,Faculty of Science, Cairo University, Cairo, Egypt Received 27 February 2006 , Accepted 5 April 2006, Published 25 June 2006 Abstract: The rates of the electric quadrupole E2 and magnetic dipole M1 forbidden transitions in the ground configuration and some excited configurations of the Ti XIX ion have been calculated. The multiconfiguration Hartree - Fock (MCHF) method has been used. The relativistic corrections are included in the Breit - Pauli approximation. A detailed comparison of the present theoretical results with previous calculations and the available data in literature is presented. c Electronic Journal of Theoretical Physics. All rights reserved.  Keywords: Ab-initio Calculations, Multiconfiguration Hartree - Fock, Ti XIX Ion, Breit -Pauli Approximation PACS (2006): 31.15.Ar, 31.15.Ne, 21.10.Ky 1. Introduction Recently there has been considerable interest in the forbidden electric quadrupole and magnetic dipole transitions in ionic systems. These forbidden lines are spectral lines which arise from transitions that are forbidden by selection rules for the electric dipole E1 transitions. The M1 and E2 transitions have been found to be very useful in the diagnostics of astrophysical and laboratory plasmas [1,2] and are necessary for the interpretation of the observed line intensities and may contribute to the width and shape of the spectral line associated with nearby allowed transitions [3,4]. These transitions often represent optically thin lines i.e isolated lines with low transition probabilities, due to low self-absorption effects in the plasma. Moreover, the M1and E2 lines frequently occur at longer wavelengths, if compared with the electric-dipole allowed transitions, since they may connect the levels within the same electron configuration. Thus, by having lines ∗ azza farrag@hotmail .com 112 Electronic Journal of Theoretical Physics 3, No. 11 (2006) 111–122 in the visible or near uv range, high resolution techniques can be employed in order to obtain detailed information including the shape of the lines. Traditionally, the interest in these lines has come from the astrophysicists and has been restricted to light elements. More recently, however, such lines have been observed in fusion plasmas. In particular, Tokamak plasmas have been found to be a rich source of forbidden lines in heavy elements, heavier than those observed in the stars; these lines have been used for diagnostic purposes in Tokamak machines. A current interest in the forbidden transitions of the metallic impurity elements in Tokamak discharges results from their use for certain diagnostic purposes. The intensities of these transitions allow measurements of the concentrations of impurity ions which originate in the high temperature interior of the discharge. Many calculations of the dipole oscillator strengths have been made for the Be- like ions [5, 6], and dipole transition rates are now known quite accurately for many transitions of he Be-isoelectronic sequence. Some studies of the forbidden transition rates have appeared in literature [7, 8, 13, 15]. Given the importance of forbidden transition for the Be-like –Ti ion, it is important to present accurate data for these transition rates. Forbidden transitions for highly ionized Be like ions are expected to play an increasingly important role in fusion plasma diagnostics in future, but until recently, very little was known about these transitions. This paper intends to fill this gap. The theory is presented in section 2. Section 3 displays the results of the present calculations together with a comparison of these results with the available data in literature; finally a conclusion is given in the last section 4. 2. Theoretical Method and Computational Procedure In this study, the atomic state wave functions (ASFs) have been generated by the widely used atomic structure package method, the multiconfiguration Hartree-Fock MCHF method [9]. The theoretical approach employed is sketched below. A more detailed description can be found in [10]. The relativistic effect is included as a correction to the non relativistic Hamiltonian by adding the Breit -Pauli operator. 2.1 The Breit - Pauli Wave Functions In the multiconfiguration approximation, the Breit -Pauli wave functions for a state labeled γJMJ , where γrepresents the configuration and the set of quantum numbers required to specify the state, are expanded in terms of configuration state functions (CSFs) ψ(γJMJ ) = M  i=1 ci φ(γi Li Si JMJ ) (1) Electronic Journal of Theoretical Physics 3, No. 11 (2006) 111–122 113 The configuration state functions φ are antisymmetrized linear combinations of the products of the spin-orbitals functions  LML SMS |LSJMJ φ(LML SMS ) (2) φ(γLSJMJ ) = ML MS φnℓmℓ ms = 1r Pnℓ (r)Yℓmℓ (θ, ϕ)ξms (σ) (3) In the present method, the radial functions building the CSFs are taken from a nonrelativistic MCHF code and only the expansion coefficients are optimized, this leads to the matrix eigen value problem Hψ = Eψ (4) where H is the Hamiltonian matrix with elements Hij = γi Li Si JMJ |HBP |γJ Lj Sj JMJ  (5) The problem of finding the eigenvalues and eigenfunctions of the Breit –Pauli Hamiltonian can be reduced to the evaluation of matrix elements between LSJ coupled CSFs and a matrix diagonalization for each J value. 2.2 The Breit - Pauli Hamiltonian The Breit - Pauli Hamiltonian is a first order perturbation correction to the non – relativistic Hamiltonian. The Breit-Pauli Hamiltonian is given by: HBP = HN R + HRS + HF S where HN R is the ordinary non-relativistic many electron Hamiltonian .   N N   1 2 Z i + HN R = − ∇i − 2 ri r i>j ij i=1 (6) (7) HRS is the relativistic shift operator which commutes with L and S and can be written as HRS = HM C + HD1 + HD2 + HOO + HSSC (8) where HM C is the mass correction term HM C N 1  2 † 2 (∇i ) ∇i =− 2 α i=1 (9) and HD1 and HD2 are the one and two – body Darwin terms HD1 = − N α2 Z  2 1 ∇( ) , 8 i=1 i ri (10) 114 Electronic Journal of Theoretical Physics 3, No. 11 (2006) 111–122 HD2 HOO is the orbit-orbit term HOO N α2  2 1 = ∇( ) . 4 i<j i ri  N  α2  Pi .Pj rij (rij .Pi )Pj =− + 3 2 i<j rij rij (11) (12) and finally HSSC is the spin-spin contact term N 8πα2  =− (si .sj )δ(ri .rj ) 3 i<j HSSC (13) The fine - structure operator HF S describes the interactions between the spin and orbital angular momenta of the electrons, and does not commute with L and S but only with the total angular momemtum J=L+S. The fine –structure operator HF S consists of three terms: HF S = HSO + HSOO + HSS (14) where HSO is the spin-orbit term HSO N α2 Z  1 = ℓi .si 2 i=1 ri3 (15) HSOO is the spin-other-orbit term HSOO N α2  rij × Pi (si + 2sj ) =− 3 2 i<j rij (16) and HSS is spin –spin term HSS   N  (si .rij )(sj .rij ) 1 =α si .sj − 3 3 2 r rij ij i<j 2 (17) 2.3 Electric Quadrupole Decay Rates: E2 The transition probabilities are calculated using the following formula AE2 = 1.120 × 1018 SE2 gi λ5 in(sec−1 ) (18) where λ is the transition wavelength in Å, gi is the statistical weight of the initial state and SE2 is the line strength given by SE2 = |Ψf |OE2 | Ψi |2 (19) Electronic Journal of Theoretical Physics 3, No. 11 (2006) 111–122 115 OE2 is the electric quadrupole operator OE2 =  ri2 Cq(2) (i) (20) i Because this operator has even parity, the electric quadrupole transitions involve no change in parity . The selection rules on the total angular momentum are ΔJ=0, ±1, ±2 J+J’≥ 2 where J and J’ are the total angular momentum for the initial and final states, respectively. 2.4 Magnetic Dipole Decay Rates: M1 The transition probabilities are calculated using the following formula AM 1 = 2.697 × 1013 SM 1 gλ3 (sec−1 ) (21) where λ is the transition wavelength in Å, gi is the statistical weight of the initial state and SM 1 is the line strength given by: SM 1 = |ψf |OM 1 | ψi |2 3. Results and Discussion The MCHF method has been used with the Breit-Pauli relativistic corrections to calculate the transition energies, wavelengths and the electric quadrupole and magnetic dipole transition rates for the Ti XIX ion of the Be-isoelectronic sequence and that among each of the configurations 2s2p, 2p2 and the transitions between 2s2 and 2p2 . It should be mentioned here that there are two alternative forms of the electric dipole matrix element, namely the length form and the velocity form. The two forms are equivalent for exact solutions of the Hamiltonian. It is customary to compute both forms and use the agreement between the two results as one of the quality criteria for the calculation. Moreover that the agreement between the length and velocity form is a necessary condition for accurate wavefunctions. These wavefunctions have been used to calculate the magnetic dipole and the electric quadrupole transitions between the n=2-2 levels. It should be noticed that the dominant correlation effect is in the n=2 intrashell correlation using the three configuration basis composed of 2s2 , 2s2p and 2p2 . To achieve a better agreement between experiment and theory for the transition energies between the n=2-2 levels, additional correlation functions have to be included in the basis set including most of the n=3 configurations. No attempt has been made to include configurations with n=4 and n=5 in the basis functions in the present work. In these ab-initio calculations, the wavefunctions with configuration interactions (CI) between the levels with the same parity and the same total angular momentum have been obtained. The even configurations considered are (2s2 , 2p2 , 2s3s , 2s3d , 2p3p and 3p2 ) 116 Electronic Journal of Theoretical Physics 3, No. 11 (2006) 111–122 while the odd configurations are (2s2p ,2s3p ,2p3s ,2p3d , 3s3p and 3p3d), outside the1s2 core. Tables (I), (II) and (III) present the transition energies ΔE, the wavelengths (λ), the electric quadrupole transitions rates A(E2) and the magnetic dipole transitions rates A(M1) for the transitions among the 2s2p, 2p2 and the transition from 2s2 to 2p2 configurations. The present calculated values are referred to (a) while (b) refers to Glass results [11] which presented very sophisticated relativistic intermediate-coupling calculations of the wavefunctions and (c) stands for the calculations of Bhatia et al [12] which also used relativistic intermediate coupling wavefunctions, while (d) stands for the semiempirical wavelengths of Edlén [14], (e) gives the NIST values [16] and (f) are the values for the multiconfiguration Dirac –Fock (MCDF) of Froese-Fischer [15]. The NIST values are followed by a capital letter denoting the uncertainties in the atomic transition probability data which is C for uncertainties within 25%, D for uncertainties within 50% and E for uncertainties greater than 50%. 3.1 M1/E2 Transitions Among the 2s2p Levels The 2s2p configuration of Ti XIX has 4 fine structure levels 1 P1 and 3 P0,1,2 . These levels are separated into two groups by the different multiplets. The possible E2 transitions among these 2s2p levels include the 2s2p (1 P1− 3 P1,2 ), 2s2p(3 P1− 3 P2 ) and (3 P0− 3 P2 ) lines, while for the possible M1 transitions they include the 2s2p (1 P1− 3 P0,1,2 ), 2s2p(3 P0− 3 P1 ) and 2s2p (3 P1− 3 P2 ) lines. As seen from table I, the calculated transition energies from the singlet state to the triplet states are in good agreement with the calculations (b) of Glass and (f) of Froese- Fischer, both calculations were done performing purely relativistic calculations, the relative percentage is a maximum of 1% with Glass and NIST values and up to 0.2% with Froese –Fischer, the discrepancy with the calculation of Bathia et al. is slightly large, it is up to 3.4% relative percentage difference, while with the semi –empirical values of Edlén, the agreement is good and is less than 1% relative percentage difference. For the triplet-triplet transitions, the (3 P0− 3 P1 ) line show the biggest discrepancy, the difference is 787.5 cm−1 which is 4.8% relative percentage difference as compared to Glass and 670.13 cm−1 or 4% as compared to Froese–Fischer. For all the other presented data, the relative percentage difference with the other calculations does not exceed 0.2%. The electric quadrupole E2 and the magnetic dipole M1 transition probabilities show good agreement with the calculations of Froese-Fischer, it is 3.4% maximum of the relative percentage difference and 13.7% maximum of the relative percentage difference with all other calculations. 3.2 M1/E2 transitions among the 2p2 levels The 2p2 configuration of Ti XIX has 5 fine structure levels 3 P0,1,2 , 1 D2 , 1 S0 , which are separated into three groups by the different multiplets. The possible E2 transitions among Electronic Journal of Theoretical Physics 3, No. 11 (2006) 111–122 117 the 2p2 levels include seven transitions, while the M1 transitions include five transitions which are presented in table II. Our calculated values for the transitions between 2p2 levels are close to the semi empirical values of Edlén and the (MCDF) calculations of Froese-Fischer. They have up to 1% relative percentage with Edlén and up to 0.2% with the calculations of FroeseFischer. The present calculated transition energies among the triplet states show a large difference with that of Glass and Bathia et al. for the (3 P1 −3 P2 ) transition, the difference is 710 cm−1 or 2.5% with Glass and 2268 cm−1 or 7.5% with Bathia et al., but it is 116.2 cm−1 or 0.4% with Edlén and 41.04 cm−1 or 0.14% with Froese-Fischer. The value of the E2 decay is weak among the triplet states, in these transitions the M1 decay is about four to five orders of magnitude faster than the E2 decay. In the singlet D and triplet P transitions, our calculated transition energies show the best agreement with the (MCDF) calculations of Froese-Fischer, it is 156.8 cm−1 or 0.1%, 78.8 cm−1 or 0.07% and 119.9 cm−1 or 0.13 % for the transitions (1 D2 −3 P0,1,2 ), respectively. The transitions to the singlet S show a good agreement with all the calculations as well as with the semi empirical values of Edlén and the NIST values, it is 1% of the relative percentage difference for all transitions except for the (3 P1− 1 S0 ) there is a difference of 7852.1 cm−1 or 2.5% of the relative percentage difference and for (3 P2− 1 S0 ) a difference of 5577.7 cm−1 or 2% with the calculations of Bathia et al.. For the transition rates, the A(E2) agree to within 15% and within 9% for the A(M1) of the relative percentage difference with all other calculations. 3.3 M1/E2 transitions between the 2s2 − 2p2 levels In table III the even-even transitions between 2s2 − 2p2 levels arise in two transitions (1 S0 −3 P2 ) and (1 S0 −1 D2 ) for the electric quadrupole, they have prominent transition probabilities A(E2), and have one transition (1 S0 −3 P1 ) for the magnetic dipole. In these transitions the relative percentage difference for the transition energies agrees with all other calculations and with the semi empirical values of Edlén as well as with the NIST values, there is less than 9% as maximum of relative percentage difference. The quadrupole transition rates present a relative percentage of 50 % and 45 % with the values of Glass and Bathia et al., respectively but show a better agreement with that of Froese-Fischer, it is 16% with the same transition (1 S0 −1 D2 ) and for the (1 S0 −3 P2 ) transition the relative percentage difference is 39%, 32% and 40% with that of Glass, Bathia et al and the NIST, respectively, but it is close to the calculation of Froese-Fischer, it is 7.5% of a relative percentage difference. The magnetic dipole decay rates for (1 S0 −3 P1 ) transition show a good agreement between all calculations, it has a difference of 4.65% with that of Glass, 8.7% with Bathia et al, 6.8% with the NIST values and 1.75% with Froese-Fischer. 118 Electronic Journal of Theoretical Physics 3, No. 11 (2006) 111–122 Conclusion In summary, the multi-configuration Hartree-Fock (MCHF) method has been used to study the E2 and M1 transitions among some configurations of TiXIX, the inclusion of the configuration interaction and relativistic corrections by using the Breit-Pauli approximation in the calculation of the wave functions yielded satisfactory results as compared with the available theoretical data in literature. It should be mentioned here that while progress is being made at the theoretical level, there is a lack of experimental data to check this material. Experimental measured transition energies and transition probabilities are highly desirable for the highly ionized atomic system. Electronic Journal of Theoretical Physics 3, No. 11 (2006) 111–122 119 References [1] Jonsson, P., Froese Fischer, C. and Träbert, E., J. Phys. B: At. Mol. Opt. Phys. 31, 3497(1998). [2] Biemont, E. and Zeippen, C. J., Physica Scripta T 65, 192(1996). [3] Beauchamp, A., Wesemael, F., Bergeron, P. and Liebert, J., Astrophy. J. 44 L 85(1995). [4] Liebert, J., Beaver, E. A., Robertson, J. W. and Strittmatter, P. A., Astrophy. J. 204 L 119(1976). [5] Tachier, G., Fischer,C. F., J. Phys. B: At. Mol. Opt. Phys. 32, 5805(1999). [6] Kingstone, A. E., Hibbert, A., J. Phys. B: At. Mol. Opt. Phys. 33,693(2000). [7] Kingstone, A. E., Hibbert, A., J. Phys. B: At. Mol. Opt. Phys. 34,81(2001). [8] Koc, K., J. Phys. B: At. Mol. Opt. Phys. 36,L93(2003). [9] Fischer, C. F., Comput. Phys. Commun. 128, 635(2000). [10] Fischer, C. F., Brage, T., and Per JÖnsson, “ Computational Atomic Structure: An MCHF Approach,” edited by Institute of Physics, Bristol, 1997. [11] Glass, R., Zeitschrift Fur Phys. A. 320, 545(1985). [12] Bhatia, A. K., Feldman, U. and Doschek, G. A., J. Appl. Phys. 51,1464(1980). [13] Tachier, G., Fischer, C. F., ADNDT. 87, 1(2004) [14] Edlén, B. Phys.Scr. 20,129(1979). [15] http://atoms.vuse.vanderbilt.edu [16] Martin, G. A., Fuhr, J. R., and Wiese,W. L., “ Atomic Transition Probabilities Scandium through Manganese” in Journal of Physical and Chemical Reference Data ,volume 17, 1988 supplement No 3. 120 Electronic Journal of Theoretical Physics 3, No. 11 (2006) 111–122 Table 1 Transition energies (cm−1 ), wavelength (Å) and E2,M1 transition probabilities (sec−1 ) within the 2s2p configuration of Ti XIX Transition 1P − 3P 1 0 1P 1 1P 1 − 3 P1 − 3 P2 3P 3 0− P1 3P 3 0− P2 3 3P 1− P2 a b c d e f a b c d e f a b c d e f a b c d f a b c d f a b c d f ∆Eij (cm −1 ) 303821 303683 312470 30152 301504.5 303110.9 286691 286779 295394 285095 285095.2 286651.03 244459 244678.9 253267 242453 242453.6 244025.31 17130 16342.5 17073.5 16433.8 16459.87 59362 58997 59206.6 59066.7 59085.59 42232 42662.1 42122.9 42643.9 42625.72 λ ( Å) 329.14 329.29 320.03 331.65 331.67 329.91 348.8 348.02 338.53 350.76 350.76 348.85 409.06 408.7 394.84 412.45 412.45 409.79 5837.71 6119 5857 6085 6075.32 1684.6 1695 1689 1693 1692.44 2367.8 2344 2374 2345 2345.98 A (E2) 9.69E+00 9.87E+00 9.90E+00 9.80E+00 1.89E+00 1.49E+00 1.84E+00 A (M1) 5.03E+03 4.46E+03 4.94E+03 4.50E+03 D 4.93E+03 3.21E+03 2.81E+03 3.10E+03 2.80E+03 D 3.17E+03 3.32E+03 2.92E+03 3.29E+03 2.90E+03 D 3.25E+03 8.20E+01 7.81E+01 8.87E+01 7.93E+01 3.86E-02 3.65E-02 3.77E-02 3.77E-02 1.67E-02 1.60E-02 1.65E-02 1.04E+03 1.04E+03 9.99E+02 1.03E+03 Electronic Journal of Theoretical Physics 3, No. 11 (2006) 111–122 121 Table 2 Transition energies (cm−1 ), wavelength (Å) and E2,M1 transition probabilities (sec−1 ) within the 2p2 configuration of Ti XIX Transition 3P 3 0− P1 3P 3 0− P2 3P 3 1− P2 1D 2 − 3 P0 1D 2 − 3 P1 1D 2 − 3 P2 3P 1 1− S0 3P 1 2− S0 1D 2 − 1 S0 a b c d e f a b c d f a b c d e f a b c d f a b c d e f a b c d e f a b c d e f a b c d f a b c d f ∆Eij (cm −1 ) 29069 28776.98 28563.27 29044.44 29088.37 29001.24 56695.8 57110.2 58445.35 56561 56668.59 27626.8 28336.6 29895.3 27510.3 27527.73 27667.35 142806 144408.5 149423.2 141799.7 142657.99 113737 115625.7 120863.4 112756.1 112688.8 113656.75 86110.2 87260.03 90991.81 85251.49 85171.62 85989.4 301837.74 304053 309.4693 299320.5 299275.8 301596.32 274210.94 275710 279790.7 271805.6 273928.97 188100.74 188423.3 188828.9 186560.2 187939.57 λ ( Å) 3440.09 3475 3501 3443 3437.8 3448.09 1763.8 1751 1711 1768 1764.63 3619.7 3529 3345 3635 3632.7 3614.33 700.25 692.48 669.24 705.22 700.97 879.22 864.86 827.38 886.87 887.4 879.83 1161.3 1146 1099 1173 1174.1 1162.92 331.3 328.89 322.9 334.09 334.14 331.57 364.68 362.7 357.41 367.91 365.05 531.63 530.72 529.58 536.02 532.08 A (E2) A (M1) 3.44E+02 4.19E+02 4.08E+02 4.19E+02 C 4.25E+02 3.54E-02 3.32E-02 3.79E-02 3.20E-02 1.55E-03 1.89E-03 1.64E-03 3.30E-03 1.48E-03 2.37E-03 7.72E-03 2.44E-01 2.43E-01 2.62E-01 3.90E-01 3.74E-01 4.09E-01 2.45E+02 2.77E+02 3.23E+02 2.77E+02 C 2.50E+02 2.14E+03 2.14E+03 2.50E+03 2.10E+03 D 2.42E+03 2.45E+03 2.48E+03 2.87E+03 2.50E+03 D 2.77E+03 3.90E+04 4.01E+04 4.27E+04 3.80E+04 D 4.17E+04 9.48E+01 8.71E+01 9.84E+01 1.01E+02 2.36E+02 2.14E+02 2.19E+02 2.05E+02 122 Electronic Journal of Theoretical Physics 3, No. 11 (2006) 111–122 Table 3 Transition energies (cm−1 ), wavelength (Å) and E2,M1 transition probabilities (sec−1 ) within the 2s2 - 2p2 transitions for Ti XIX Transition 1S − 3P 0 2 1S 0 1S 0 − 3 P1 − 1 D2 a b c d e f a b c d e f a b c d f ∆Eij (cm −1 ) 832671.8 833263.9 834794.2 832362.2 832431.5 832488.52 805045 804893.8 804893.8 804829 804893.8 804821.16 918782 920556 925754.5 917599.6 918477.92 λ ( Å) 120.09 120.01 119.79 120.14 120.13 120.12 124.21 124.24 124.24 124.25 124.24 124.25 108.83 108.63 108.02 108.98 108.87 A (E2) 6.74E+02 4.84E+02 5.10E+02 A (M1) 4.80E+02 E 7.28E+02 2.05E+03 2.15E+03 2.25E+03 2.20E+03 E 2.09E+03 8.46E+03 5.62E+03 5.83E+03 7.28E+03 EJTP 3, No. 11 (2006) 123–132 Electronic Journal of Theoretical Physics Some Properties of Generalized Hypergeometric Thermal Coherent States Dušan Popov ∗ University ”Politehnica” of Timişoara, Department of Physics, Piaţa Regina Maria No. 1, 300004 Timişoara, Romania Received 1 April 2006 , Accepted 2 June 2006, Published 25 June 2006 Abstract: The generalized hypergeometric coherent states (GHCSs) have been introduced by Appl and Schiller [1]. In the present paper we have extended some considerations about GHCSs to the mixed (thermal) states and applied, particularly, to the case of pseudoharmonic oscillator (PHO). The Husimi’s Q distribution function and the diagonal P - distribution function, in the GHCSs representation, have been deduced for these mixed states. The obtained distribution functions were used to calculate thermal averages and to examine some nonclassical properties of the generalized hypergeometric thermal coherent states (GHTCSs), particularly for the PHO. We have also defined and calculated the thermal analogue of the Mandel parameter and the thermal analogue of the second-order correlation function. By particularizing the parameters p and q of the hypergeometric functions, we recover the usual Barut-Girardello coherent states and their main properties for the PHO from our previous paper [9]. All calculations are performed in terms of the Meijer’s G-functions [2], which are related to the hypergeometric functions. This manner provides an elegance and uniformity of the obtained results and so the GHCSs become a new field of application for these functions. Moreover, this mathematical approach can be used also for other kind of coherent states (e.g. Klauder-Perelomov, Gazeau-Klauder or nonlinear coherent states [10], [12]). c Electronic Journal of Theoretical Physics. All rights reserved.  Keywords: Coherent states, Hypergeometric Functions, Pseudoharmonic Oscillator PACS (2006): 03.65C, 03.65.-w, 02.30, 42.50.Ar 1. Introduction By considering the so-called generalized lowering and raising operators ∗ dusan popov@yahoo.co.uk 124 Electronic Journal of Theoretical Physics 3, No. 11 (2006) 123–132 p Uq = ∞  pf q (n)|λ;n >< λ;n + 1|, (1) n=0 + p Uq = ∞  pf q (n)|λ; n + 1 >< λ; n| (2) n=0 where |λ; n > are the vectors of an orthonormal basis (usually, the Fock basis indexed by a real parameter λ which can play the role of the degeneracy parameter), Appl and Schiller have defined the generalized hypergeometric coherent states (GHCSs) as the eigenvalues of the lowering operator [1]: p Uq |p; q; λ; z >= z|p; q; λ; z > . (3) The numbers p and q are natural and the positive functions p f q (n) were defined as follows: pf q (n) = (n + 1) (b1 + n)(b2 + n) . . . (bq + n) . (a1 + n)(a2 + n) . . . (ap + n) (4) Consequently, the expansion of the GHCSs in the Fock basis is [1]: |p; q; λ; z >= ∞ − 12  zn 2 $ |λ; n > N (|z| ; λ) p q ρ (n; λ) p q n=0  (5) where the strictly positive parameter functions of the discrete variable n are defined as: 'q j=1 (bj )n p ρq (n; λ) = Γ(n + 1) 'p i=1 (ai )n (6) and where (x)n = Γ(x + n)/Γ(x) is the Pochhammer’s symbol [2]. The appellation ”generalized hypergeometric coherent states” becomes from the normalization function which is given by generalized hypergeometric functions: 2 2 p Nq (|z| ; λ) = p Fq (a1 , a2 , . . . , ap ; b1 , b2 , . . . , bq ; |z| ) = ∞ ∞  (a1 )n (a2 )n . . . (ap )n (|z|2 )n  1 2 n = n=0 (b1 )n (b2 )n . . . (bq )n n! = ρ (n; λ) n=0 p q (7) (|z| ) , where, generally, we consider that ai = ai (λ) and bj = bj (λ) are the complex functions. In the cited paper [1] were examined the main properties of the GHCSs, including the resolution of unity. Let us here we express the weight function of the integration measure through the Meijer’s G-functions (whose definition and main properties can be found, e.g. in [2]): Electronic Journal of Theoretical Physics 3, No. 11 (2006) 123–132 125 d2 z 2 p wq (|z| ; λ) = π ⎞ ⎛ ⎞ ⎛     {1 − ap } ; / d2 z 1,p ⎜ ⎟ q+1,0 ⎜ 2  /; {ap − 1} ⎟  = Gp,q+1 ⎝−|z|2  |z| G ⎠ ⎠ p,q+1 ⎝    π 0, {bq − 1} ; / 0 ; {1 − bq } dμ(p; q; λ; z) = (8) where, in order to simplify the formulae writing, we have used the following notation: {cl } ≡ c1 , c2 , . . . , cl . Below we have also used the connection between the generalized hypergeometric functions and the Meijer’s G-functions [2]: ⎛ ⎞  'q  ⎜  {1 − ap } ; / ⎟ j=1 Γ(bj ) 1,p ' F ({a }; {b }; x) = G −x  ⎝ ⎠. p q p q p p,q+1  i=1 Γ(ai ) 0 ; {1 − b } (9) q 2. GHCSs for Thermal States In [1] were examined some properties of the GHCSs for the case of pure quantum states. One of the aims of present paper is to extend this examination also to the case of the mixed quantum states. As a typical example of mixed states we consider the thermal states described by the normalized canonical density operator: ∞ 1  −β En,λ ρλ = e |λ; n >< λ; n| Zλ n=0 (10) where En,λ are the eigenvalues of the Hamiltonian operator of the examined quantum system and the normalization constant Zλ = Zλ (β) is the partition function. In the present paper we will limiting only to the case of quantum systems with linear energy spectra with respect to the energy quantum number n and we will to describe the corresponding method for examine some properties of generalized hypergeometric thermal coherent states (GHTCSs). For systems with more complicated energy spectra it must elaborate specifical methods (e.g. for the Morse oscillator, see [3]). Let us we assume that the energy of linear spectra (where n = 0, 1, . . . , ∞ and ω is the angular frequency) is En,λ = E0,λ + nω (11) and then the partition function becomes: Zλ = ∞  e−β En,λ = e−β E0,λ (< n > +1) n=0 where we have used the expression of the Bose-Einstein distribution function: (12) 126 Electronic Journal of Theoretical Physics 3, No. 11 (2006) 123–132 1 . (13) −1 The first characteristic of the GHTCSs we want to examine is the thermal Husimi’s distribution function: < n >= eβω ∞ 2 n  1 1 −β En,λ (|z| ) Q|p;q;λ;z> (ρλ ) ≡< p; q; λ; z|ρλ |p; q; λ; z >= . e Zλ p Fq ({ap }; {bq }; |z|2 ) n=0 p ρq (n; λ) (14) For the above examined linear spectra this expression becomes: <n> 2 1 p Fq ({ap }; {bq }; <n>+1 |z| ) = Q|p;q;λ;z> (ρλ ) = < n > +1 F ({a }; {bq }; |z|2 ) ⎛ p q p ⎞   {1 − ap } ; / ⎜ <n> ⎟ 2 G1,p |z| −  ⎝ ⎠ p,q+1 <n>+1  0 ; {1 − bq } 1 ⎞ . ⎛ =  < n > +1  {1 − ap } ; / ⎟ ⎜ 2 G1,p ⎠ p,q+1 ⎝−|z|   0 ; {1 − bq } (15) Using the integration measure expression and the properties of the integral of Meijer’s G-functions products [2], it is not difficult to prove that the thermal Husimi’s distribution function is positive, normalized to unity with the measure dμ(p; q; λ; z) and provides a two-dimensional probability distribution over the complex z = |z| exp (iϕ) plane:  dμ(p; q; λ; z)Q|p;q;λ;z> (ρλ ) = 1. (16) After these considerations let us we perform the diagonal expansion of the density operator ρλ over the projector of GHTCSs:  ρλ = dμ(p; q; λ; z)|p; q; λ > p Pq (|z|2 ; λ) < p; q; λ; z| (17) where the function p Pq (|z|2 ; λ) is called P-function or P-distribution function, even if is in fact a quasi-probability distribution, because of their negative values on certain domains. By inserting the expressions for the GHTCSs and the integration measure, performing the angular integration and using the notation x = |z|2 , we lead to the following equation: ⎞ ⎛  'p  2 ∞  Γ(ai )  |λ; n >< λ; n| R ⎜  /; {ap − 1} ⎟ ρλ = 'qi=1 dx xn Gq+1,0 x  ⎠ p Pq (x; λ) ⎝ p,q+1  p ρq (n; λ) 0 j=1 Γ(bj ) n=0 0, {b − 1} ; / q (18) where R is the convergence radius of the GHTCSs [5]. Electronic Journal of Theoretical Physics 3, No. 11 (2006) 123–132 127 If we perform the function change ⎡ ⎛ ⎞⎤−1   ⎢ q+1,0 ⎜  /; {ap − 1} ⎟⎥ P (x; λ) = G x  ⎣ ⎝ ⎠⎦ p hq (x; λ) p q p,q+1  0, {bq − 1} ; / (19) and consider the expression of strictly positive parameters functions p ρq (n; λ) and the energy En,λ , in order to obtain the expression (10) for the normalized canonical density operator ρλ , we must solve the following equation where p hq (x; λ) is, for the moment, an unknown function which must be determined:  0 R2 'q 1 1 j=1 Γ(bj + n) e−β E0,λ βω n Γ(n + 1) 'p . dx xn p hq (x; λ) = Zλ (e ) i=1 Γ(ai + n) (20) The above problem is just the moment power problem (if R is a finite quantity we have the Haussdorff moment problem, while if R is infinite we have the Stieltjes moment problem [4]) and, in order to solve it, we extend the real values n to the complex ones s = n + 1 [5] and so we lead to the definition of the Meijer’s G-functions [2]. In this stage the problem can be solved. Finally, for the P-function we obtain: ⎞   /; {ap − 1} ⎟  q+1,0 ⎜ <n>+1 Gp,q+1 ⎠ ⎝ <n> |z|2   0, {bq − 1} ; / 1 2 ⎛ ⎞ p Pq (|z| ; λ) =  <n>  /; {ap − 1} ⎜ 2 ⎟ Gq+1,0 ⎠ p,q+1 ⎝|z|   0, {bq − 1} ; / ⎛ (21) Using the properties of Meijer’s G-functions [2] it is not difficult to prove that the P-function is normalized to unity:  dμ(p; q; λ; z)p Pq (|z|2 ; λ) = 1. (22) By particularizing the parameters p and q of the hypergeometric functions, it must examine the positivity of the weight function of the integration measure p wq (|z|2 ; λ) (see, also [1]) and of the Husimi’s function Q|p;q;λ;z> (ρλ ). Implicitly from their definition, the Husimi’s function Q|p;q;λ;z> (ρλ ) is always positive, while the P-function can bring also negative and singular values (for non-classical fields or states). For these reasons, the P-function is sometimes called the quasi-distribution function. If the P-function becomes negative for a certain state, then this state has a non-classical character [6]. One of the practical utility of the P-function of a density operator (which can be measured in experiments) consists in its role in the calculation of thermal averages of an observable A which characterize the quantum system: 128 Electronic Journal of Theoretical Physics 3, No. 11 (2006) 123–132 < A >p;q;λ = T r(ρλ A) =  dμ(p; q; λ; z)p Pq (|z|2 ; λ) < p; q; λ; z|A|p; q; λ; z > . (23) On the other hand, if the P-function has non-classical character (e.g. for a paircoherent state [7]), then the state is entangled. This will be a good test for the inseparability of the quantum states, particularly for the coherent states. In many cases the calculations in the GHTCSs representation are much simpler that in other representations (e.g. in the coordinate or in the momentum representations), which constitute a good reason for using the GHTCSs formalism. An important class of observables is represented by the diagonal operators in the Fock Basis |λ; n >. As an example we examine the thermal averages of integer powers of the number operator < N s >, where s = 1, 2 . . . In this reason we adopt an original ansatz. If we calculate the average of the operator eεN in a pure GHTCS |p; q; λ; z >, where ε is a small positive parameter, i.e. ⎛ < eεN ⎞   {1 − ap } ; / ⎜ ε 2 ⎟ G1,p ⎠ p,q+1 ⎝−e |z|   0 ; {1 − b } ε 2 q p Fq ({ap }; {bq }; e |z| ) ⎞ , ⎛ >|p;q;λ;z> = = 2  p Fq ({ap }; {bq }; |z| )  {1 − ap } ; / ⎟ ⎜ 2 G1,p −|z|  ⎠ ⎝ p,q+1  0 ; {1 − bq } (24) we observe that s < N >|p;q;λ;z> = lim ε→0  ∂ ∂ε s < eεN >|p;q;λ;z> . (25) Then the corresponding thermal averages are s < N >=  1 lim dμ(p; q; λ; z)p Pq (|z| ; λ) < N >|p;q;λ;z> = < n > ε→0 2 s  ∂ ∂ε s 1 , − eε (26) where we have used the integral properties of the Meijer’s G-functions and also the particular expression for the function G11 11 (x| . . . ) [2]. We can observe that these thermal averages are independent of the parameters p, q and λ, which was to be expected, due to the fact that the thermal averages are independent on the representation. The first two power thermal averages can be expressed through the Bose-Einstein distribution function (13): < N >=< n >, < N 2 >=< n > (2 < n > +1). <n>+1 <n> (27) Electronic Journal of Theoretical Physics 3, No. 11 (2006) 123–132 129 With these thermal averages we can define and calculate the thermal second-order correlation function g (2) and the thermal Mandel parameter Q(< n >) [3], which are defined as the thermal analogues of the corresponding quantities for the pure GHCS |p; q; λ; z > [6]: g (2) = < N2 > − < N > = 2, (< N >)2 Q(< n >) =< N > g (2) − 1 =< n > . (28) So, all GHTCSs, independently of the values of integer parameters p, q and λ, have the same (constant) thermal second-order correlation function g (2) and the same thermal Mandel parameter Q(< n >) (which are dependent of the equilibrium temperature T , through the quantity β = (kB T )−1 ). 3. Particularization for the Pseudoharmonic Oscillator In the following, we particularize the GHTCSs formalism for the case of the pseudoharmonic oscillator (PHO), whose effective potential is [8] mred ω 2 2 r0 VJ (r) = 8  r r0 − r0 r 2 + 1 2 J (J + 1) 2 2mred r (29) where mred is the reduced mass of the quantum system (e.g. the diatomic molecule), r0 is the equilibrium distance between the diatomic molecule nuclei and J = 0, 1, 2, . . . is the rotational quantum number, while n will be the vibrational quantum number. The importance of the PHO consists in the fact that this potential also admits the exact analytical solution of the rotational-vibrational Schrödinger equation, being in a certain sense an intermediate potential between the three-dimensional harmonic oscillator potential (HO-3D)(an ideal potential) and other much anharmonic oscillator potentials (the more realistic potentials, e.g. Pöschl-Teller, or Morse [3]). Earlier we have showed [9] that the dynamical group associated with the PHO is SU (1, 1) quantum group, whose lowering operator is K− = ∞  $ (n + 1)(2k + n)|k; n >< k; n + 1| (30) n=0 where k is the Bargmann index which labels the irreducible representations of this group. By comparing this expression with that of generalized hypergeometric operator p Uq and also with the positive functions p f q (n) [1], we observe that the above expression is a particular case, if we take p = 0, q = 1, b1 = 2k and λ = k. So, the GHCSs denoted as |0; 1, k; z >= |k; z >, defined as the eigenstates of the lowering operator K− , i.e. K− |k; z >= z|k; z >, (31) are called the Barut-Girardello coherent states for the PHO and their expansion over the Fock vector basis |k; n > is [9] 130 Electronic Journal of Theoretical Physics 3, No. 11 (2006) 123–132 ∞ |z|2k−1  zn $ |k; n > . I2k−1 (2|z|) n=0 n!Γ(2k + n) |k; z >= (32) In the following we look over all the quantities which characterize the CSs of the PHO, previously obtained in [9], but now regarded only as the particular case of the GHCSs, with the above mentioned values of the parameters p, q, b1 and λ. The weight function of the integration measure finally is expressed as a product of two Bessel functions of the first and second kind: 2 0 w1 (|z| ; λ) 2,0 2 2 = G1,0 0,2 (−|z| |0; 1 − 2k) G0,2 (|z| |0, 2k − 1) = 2I2k−1 (2|z|) K2k−1 (2|z|). (33) On the other hand, the PHO energy eigenvalues are mred ω 2 2 r0 + ω n ≡ E0,k + ω n. (34) 2 while the connection between the Bargmann index k and the rotational quantum number J is [9] En,k = ω k − 1 1 k = k(J) ≡ + 2 2 " 1 J+ 2 2 + m red ω 2 r0 2 2 # 12 . (35) Accordingly to this, the partition function for a fixed quantum number J becomes Zk ≡ ZJ = ∞  e−βEn,k = e−βE0,k (< n > +1) . (36) n=0 Similarly, the total partition function (by considering also the degeneration of the rotational states) is Z= ∞  β (2J + 1)ZJ = e mred ω 2 2 r0 4 (< n > +1) J=0 ∞  (2J + 1) J=0  <n> < n > +1 k(J) . (37) The total partition function is a quantity of exceptional information importance because it enables the calculations of all statistical properties of a PHO quantum canonical gas (for details, see [9]). For the thermal Husimi’s distribution function for the PHO it follows: Q|0;1;k;z> (ρk ) = 1 < n > +1 = G1,0 0,2 1 < n > +1    <n> |z|2 0; − <n>+1  1 − 2k    = 2 0; 1 − 2k G1,0 −|z|  0,2    <n>  k− 12 I2k−1 2|z| <n>+1 < n > +1 <n> I2k−1 (2|z|) (38) . Electronic Journal of Theoretical Physics 3, No. 11 (2006) 123–132 131 On the other hand, for the P-function of the PHO we obtain:   2k − 1 1 2    = 0 P1 (|z| ; k) = <n> 2 0, 2k − 1 |z| G2,0  0,2    <n>+1  k− 21 K 2k−1 2|z| <n> 1 < n > +1 . = <n> <n> K2k−1 (2|z|) G2,0 0,2   <n>+1 |z|2 0, <n> (39) In both equations, in order to express the particular values of the Meijer’s G-functions, we have used the book of Mathai and Saxena [2]. Because, as we have seen, the thermal averages of the integer powers of the number operator are independent of the parameters p and q of the hypergeometric functions, the indicated values in the previous section are identical both for the CSs of the PHO and for the GHCSs with arbitrarily parameters p and q . This property also pass to all statistical averages of the diagonal operators in the Fock basis. 4. Concluding Remarks In the paper we have shown that the formalism of the GHCSs, previously introduced by Appl and Schiller for the pure quantum (coherent) states [1], can be extended also to the mixed (thermal) quantum states and in this manner it can be connected with more practical, quantum statistical, problems. We have showed that by particularizing the parameters of the hypergeometic functions (p = 0 and q = 1) and also for the particularly value of parameter λ (i.e. λ = k and b1 = 2k) we recover the Barut-Girardello coherent states (BG-CSs) for the pseudoharmonic oscillator (PHO), deduced in a previous paper [9], with all statistical properties evinced therein. It is important to point out that we have performed all calculations in terms of the Meijer’s G-functions [2], which are more generally functions and whose particular cases are also the hypergeometric functions. This manner provide an elegance and uniformity in the use of the GHTCSs formalism and, implicitly, indicates a new field of applications of these functions in theoretical physics. Moreover, the mathematical approach of the GHTCSs can be used also for other kind of thermal coherent states, e.g. Klauder-Perelomov, Gazeau-Klauder (for PHO these CSs were deduced in [10]) or for nonlinear CSs [11]. This approach can be also applied in the theory of quantum information [12]. 132 Electronic Journal of Theoretical Physics 3, No. 11 (2006) 123–132 References [1] Appl T and Schiller D 2004 J. Phys. A: Math. Gen. 37 2731-2750; arXiv: quantphys/0308013 (2003) [2] Mathai A M and Saxena R K 1973 Generalized Hypergeometric Functions with Applications in Statistics and Physical Sciences (Lecture Notes in Mathematics vol. 348) (Berlin: Springer) [3] Popov D 2003 Phys. Lett. A 316 369-381 [4] Quesne C 2000 Phys. Lett. A 272 313-321 [5] Klauder J R Penson K A and Sixderniers J-M 2001 Phys. Rev. A 64 013817 [6] Mandel L, Wolf E 1995 Optical Coherence and Quantum Optics (Cambridge: University Press) [7] Agarwal G S and Biswas A 2005 Quantitative measures of entanglement in pair coherent states, arXiv: quant-ph/0501012 [8] Sage M, Goodisman M J 1985 Am. J. Phys 53 350-355 [9] Popov D 2001 J. Phys. A: Math. Gen. 34 5283-5296 [10] Popov D 2004 Density Matrix - General Properties and Applications in the Physics of Multiparticle Systems (in Romanian) (Timişoara: Editura Politehnica) [11] Dodonov V V 2002 J. Opt. B: Quant. Semiclass. Opt. 4 R1 [12] Fujii K 2002 Coherent States and Some Topics in Quantum Information Theory: Review, arXiv: quant-ph/0207178 EJTP 3, No. 11 (2006) 133–142 Electronic Journal of Theoretical Physics Space-Filling Curves for Quantum Control Parameters Fariel Shafee ∗ Department of Physics, Princeton University, Princeton, NJ 08544, USA Received 28 April 2006 , Accepted 8 June 2005, Published 25 June 2006 Abstract: We consider the use of space-filling curves (SFC) in scanning control parameters for quantum chemical systems. First we show that a formally exact SFC must be singular in the control parameters, but a finite discrete generalization can be used with no problem. We then make general observations about the relevance of SFCs in preference to linear scans of the parameters. Finally we present a simple magnetic field example relevant in NMR and show from the calculated autocorrelations that a SFC Peano-Hilbert curve gives a smoother sequence than a linear scan. c Electronic Journal of Theoretical Physics. All rights reserved.  Keywords: Quantum Control, Space-Filling Curves, Peano-Hilbert Curve, Control Parameters PACS (2006): 02.30.Yy , 95.30.Ft, 98.38.Bn, 46.25.Cc, 02.70.Ns 1. Introduction A quantum system is represented by a state vector in Hilbert space, and the system has a Hamiltonian operator that determines how the system evolves by itself. The system may be in the ground state, in which case the evolution is simply a phase change with time of the state vector. If it is in an excited state then sooner or later it will drop to a lower state with either simply an emission of radiation, or more fundamental changes in the system. In a second quantized theory there is provision for considering annihilation and creation of particles; in a first quantized theory the transition of the dynamical system in terms of changes in its Hamiltonian description is more complicated. Feynman’s path integral method is second quantized and yet it avoids explicit use of creation and annihilation operators. The transition in that case is implemented by a similar Hamil∗ f shafee@yahoo.com 134 Electronic Journal of Theoretical Physics 3, No. 11 (2006) 133–142 tonian as the second quantized case, and though an obvert operatorial interpretation is avoided the path integral actually achieves the same purpose of a discrete change in time from one set of fields to another. In particle physics, at relativistic energies this is a simple procedure. In quantum chemistry usually we do not worry about changes in the nuclei, and hence with particle creation or destruction. The system is described by fixed nuclei and a number of electrons, all with the constant masses and charges. It is fairly easy to set up a Schrodinger equation, though not to solve it. If a large molecule dissociates, or if two molecules synthesize into a bigger one, no new particles are created or destroyed, they all correspond to the same Hamiltonian with the same nuclei and electrons as the components. However, the states vary. Like ionization being the free state of an electron, dissociation is also a new state of the system for the same Hamiltonian. But it is possible to change the environmental factors of a system. That can be discrete or continuous. We introduce a control Hamiltonian that changes the overall development of the system. Populations can be inverted from the original ground state to some other desired state. Or other properties of the original state may be changed, e.g. expectation values of an observable. If this control Hamiltonian is a function of two variables, and these variables are continuous, then the control Hamiltonian space may be described by a two dimensional surface. But the process of control takes place as a temporal sequence and any variation of these parameters must also be a temporal sequence. If we try to map a two-dimensional surface by a one-dimensional sequence, we run into the concept of space-filling curves. There is another way space-filling curves can enter quantum chemical systems. If the state space is continuous and described by two parameters, like the parameters of the control Hamiltonian, then too we have a surface of states. In certain cases it may also be possible to relate a two-dimensional state space with a two parameter control space, i.e. two relate the two sets of parameters, So that by changing the control space parameters we achieve an equivalent change in the state space. Evolution of the system in state space can be ill-defined when we try to view the process in a continuous way, but using the concept of Bell’s “beables” Rabitz et al [1] have shown that a satisfactory picture can be formed in a statistical sense. Then the concept of SFC can apply to the state space too in terms of “beables”. In this work we shall first show that in the mathematical sense a true SFC must a singular object in terms of the control parameters. However, its discrete finite versions do not pose the problems of the singularity. We shall then consider a comparison of different ways of scanning the parameter space, including a finite representation of the Peano-Hilbert curve. Electronic Journal of Theoretical Physics 3, No. 11 (2006) 133–142 2. 135 SFC and Singular Parameters A SFC must have a Hausdorff (fractal) dimension more than one; otherwise it cannot fill a two-dimensional space even partly. Let us take a small segment of the curve f (x, y) = c (1) where x , y are the (control) parameters. So anywhere on the curve the tangent is given by dy/dx = −[∂f /∂x]/[∂f /∂y] (2) Now if we consider only a small section of the curve ds = dx However, for a fractal of dimension n $ 1 + (dy/dx)2 ds = ds0 (Λ0 /Λ)n Λ (3) (4) So this goes to infinity as scale parameter Λ goes to zero. Hence dy/dx must be singular at every point of the curve. Hence either ∂f/∂y = 0 or ∂f/∂x is singular at every point. The first option makes the curve trivial. The second option simply states that physically it is impossible to span a two-dimensional parameter space completely using only a single parameter [e.g. time] which remains within a finite, and hence practically realizable range. Hence a practical control experiment involving a two-parameter space must involve discretization of the space, such as choosing a lattice [in parameter space]. The experiment then would correspond to a finite number of steps through the lattice, and any particular sequence will correspond to a specific trajectory, and the choice of trajectory should be such as to optimize the output expected or desired. 3. Control Parameter Space and Sfc Approximations In quantum control processes the effect of a sequence of laser pulses needed to achieve any given program of operations has been studied by Ramakrishna, Rabitz et al. [2]. They considered a one-dimensional model with a one-dimensional amplitude. In general a randomly polarized signal, or one with a given polarization, will have two components of the electrical vector, so the field space would be two-dimensional. We can proceed in a similar fashion and obtain formulas which are generalizations involving both components. The unitary operator which gives the evolution of the system can now be described by similar unitary operators U I = exp[iΓ]Vk ... V1 (5) 136 Electronic Journal of Theoretical Physics 3, No. 11 (2006) 133–142 with k being the pulse number in the sequence, ⎞⎤ ⎛ ⎡ sin φ ⎟⎥ ⎜0 ⎢ Vk = exp ⎣Ck ⎝ ⎠⎦ − cos φ 0 (6) in the two-dimensional vector space formed by |m> and |m+1> and tk+1   μ Ck = dt E. tk E and the dipole µ being two-dimensional vectors. Now, with an unknown orientation of the dipole vector μ it may be necessary to try out different components of E to get the optimal Ck and hence it may be necessary to find the best algorithm to hit on the right value. If it is (the discrete practical limit of) a SFC it may offer certain advantages over a random choice or a linear scan, as we shall see later with a particular numerical example. The point to remember is that on a SFC two dimensional neighborhood values [with small variations of both components] are tried in sequence before moving out of a region, and physically that may offer the possibility of locating the right Ck quicker than varying one parameter at a time as in a linear scan, because such a trajectory is likely to have a better autocorrelation [and hence a smoother variation] than any other option. We may also be interested in a system with an arbitrarily large number of neighboring states each attainable by small changes in control parameters. For example a simple Bohr atom has energy levels going like ∼ 1/n 2 , where n is the principal quantum number. So if we want to jump from any state |n 1 ¿ to another |n 2 ¿, we need a frequency ω(n 1 , n2 ) ∼ 1/n 22 − −1/n21 and this will be nondegenerate for different n1 and n2 , but almost continuous for n1 and/or n2 large. Now supposing we need a particular sequence of jumps (n1 → n 2 ) in an ensemble of such atoms, we would need to use ω( n 1 , n2 ) in the given sequence, i.e. we would need to trace out a trajectory in the (1/n21 , 1/n22 )space (we use the reciprocals to emphasize the approximation to a continuum). There is a solved problem in mathematics, called the ETSP [Euclidean traveling salesman problem] which states that the optimum joining a very large random sequence of cities within a disc is given in the infinite limit by the SFC called MNPeano [3], which is similar to the SFC Peano curve [Figure 1]. The path length is given by [in the units for A below] √ Loptimum = 0.72 N A (7) where N is the number of states to be excited and A is the area of the control parameter space. This expression is similar to a random walk problem, except for the coefficient, which is somewhat smaller than 1. Electronic Journal of Theoretical Physics 3, No. 11 (2006) 133–142 137 So the quickest way to make a large sequence of quantum jumps for an ensemble of Bohr atoms would be given by the discrete approximation of such a MNPeano curve in the control parameter space which would need to be somehow translated into the corresponding frequency of the exciter. Similar situations of greater practical utility may be found involving electron energy bands of materials and continuous spectra of molecules. The former may be relevant to electronic devices. 4. Magnetic Environment: A Numerical Example NMR spectroscopy is a very useful tool in augmenting our knowledge of the structure of large molecules when X-ray diffraction data or analyses are insufficient. The magnetic environment of a nucleus with nonzero magnetic moment, such as a proton, or P31 may be affected by the electron orbits in the neighborhood contributing their own magnetic fields. As a result nuclear magnetic resonance would occur for a shifted field compared to the lone nucleus, which may be expressed as a chemical shift in the Lande g factor.  N HI = −g s . Hμ (8) where s is the nuclear spin, H is the magnetic field at the nucleus, and μN is the nuclear magneton. In nuclear magnetic imaging used in medical diagnosis one displaces a component of the magnetic field spatially as a raster scan and observes the absorption of the electromagnetic waves for the resonant change of spin state. In theory it is possible to combine these two uses of nuclear resonance to get a spatial picture of the magnetic environment in a given space of the molecule, e.g. the active site of an enzyme. This is apparently a more direct and theoretically more convenient way, involving no ill-posed inversion problem, than another novel method using the Ehrenfest theorem and a classical correspondence proposed by Rabitz and Vivie-Riedle [4]. Let us take a crystal of the molecule and inundate it slow neutrons. Let us use two orthogonal narrow strips [one along x and the other along y say] of movable magnetic fields in the z−direction [Figure 2] such that their combined value, which occurs at the intersection of the two strips and hence corresponds to a unique (x, y) point is near the resonance value for the r.f. radiation used for the neutron. Then, if the r.f. is varied over a relevant range for each intersection of (x, y) and the absorbed frequency observed, we get the magnetic field at that point. This may allow us to draw a magnetic map inside the molecule and hence can give us useful ideas about its structure. The problem we ask now is what should be the trajectory of the control parameters (x, y) for an optimal scan. Again, it would appear that the trajectory with the highest autocorrelation would be the best, because it would identify the two-dimensional region in space with smoothness and hence reliability of measurement. We performed a simulation with four dipoles, all with equal x, y components at four corners of a square, with the field measured in a smaller [with half the sides] square inside 138 Electronic Journal of Theoretical Physics 3, No. 11 (2006) 133–142 the larger one. The field is of course  3  = H [μ/(4π)][3 m  i . ri ri /ri2 − mi ]/ri (9) i where r i is the unit vector from the magnetic moment vector m i to the observation point and ri is the magnitude of the distance vector etc. We calculated the autocorrelations for the i−th value in the sequence with the (i +k)th value, both for Hx and Hy , using a linear scan and then using a Hilbert-Peano scan, which is a SFC [finite approximation at a given scale, in this case with 32 points each way]. Table I gives the autocorrelations for Hx (Hy is similar). It is obvious that the HilbertPeano trajectory gives the much better correlated sequence than a linear scan. 5. Conclusions We have argued in general terms about the relevance of SFCs in choosing trajectories of control parameters, and have also shown by a simple simulation example that a SFC is better autocorrelated and hence smoother when magnetic data are sequenced. This is similar to the finding in image processing [5] but much more pronounced. More realistic examples and details are being investigated. Quantum control has previously [2] been identified as useful in three contexts: (a) transferring population from an initial state to a given state, (b) making transitions through a given sequence of states, and (c) forcing the system to have a given expectation value for an observable. We add here the possibility of also using quantum control not to make transitions (the resonant nucleon returns to ground) but to probe the system as it is. The author thanks Professor H. Rabitz for useful discussions and Ignacio Sola and Andrew Tan for encouragement. Electronic Journal of Theoretical Physics 3, No. 11 (2006) 133–142 139 References [1] Denis E. and H. Rabitz, Beable trajectories for revealing quantum control mechanisms, quant-ph/0208109 (2002) Meier C., Mixing quantum and classical dynamics using Bohmian trajectories, ITAMP Workshop paper (May 9-12, 2002) Universite Paul Sabatier Preprint [2] Schirmer S.G., A Greentree, V. Ramakrishna and H Rabitz, Quantum control using sequence of simple control pulses, quant-ph-0105155 (2001) [3] Norman M.G., and P Moscato, The Euclidean Traveling Salesman Problem and a Space-Filling Curve (CeTAD, Universidad Nacional de La Plata preprint) [4] Kurtz L, H Rabitz and Regina de Vivie-Riedle, Optimal use of time-dependent probability density data to extract potential energy surfaces, physics/0109043 (2001) [5] Dafiner R, D Cohen-Or and Y Matias, Context-based Space-Filling Curves, submitted to EUROGRAPHICS 2000 (vol. 19, No. 3) 140 Electronic Journal of Theoretical Physics 3, No. 11 (2006) 133–142 TABLE I Autocorrelation <Hx (i) Hx (i+k)> k Linear scan Hilbert-Peano 0 1.00 1.000 1 0.86 0.995 2 0.73 0.990 3 0.60 0.984 4 0.49 0.980 5 0.38 0.973 6 0.29 0.967 7 0.19 0.960 8 0.11 0.955 9 0.04 0.949 10 -0.01 0.943 Electronic Journal of Theoretical Physics 3, No. 11 (2006) 133–142 Fig. 1 Peano-Hilbert curve [finite approximation] 141 142 Electronic Journal of Theoretical Physics 3, No. 11 (2006) 133–142 Fig. 2 using two orthogonal fields as control to locate point of NM resonance. EJTP 3, No. 11 (2006) 143–149 Electronic Journal of Theoretical Physics The Spectrum of the Lagrange Velocity Autocorrelation Function in Confined Anisotropic Liquids Sakhnenko Elena I∗ . and Zatovsky Alexander V. Department of Atmosphere Physics, Odessa State Environmental University, 15 Lvovskaya Str., Odessa, 65016, Ukraine Received 6 May 2006 , Accepted 5 June 2006, Published 25 June 2006 Abstract: The results of our further analysis of the thermal hydrodynamic fluctuations in an anisotropic liquid under heterogeneous conditions are represented. The heterogeneity is modeled in the form of a plane-parallel layer, the liquid is considered is taken to be incompressible, and the rapid processes of the angular momentum relaxation to equilibrium are ignored. The extended system of hydrodynamics equations is linearized for small deviations from the equilibrium values. For the case of spontaneous fluctuation fields being present in the system of equations for the velocity and inertia tensor components, the boundary problem solution is found in the form of an expansion in the harmonic functions. The spectral densities of the fluctuation correlation functions are obtained by using the fluctuation dissipation theorem (FDT). A special attention is paid to the correlation functions (CFs) for the velocity field in the anisotropic liquid. The spectrum of the Lagrange velocity autocorrelation function (LVACF) and the collective part of the self–diffusion coefficient of the molecules are determined as functions of the coordinate normal to the confining planes. c Electronic Journal of Theoretical Physics. All rights reserved.  Keywords: Confined Liquid, Anisotropic Liquid, Velocity Fluctuations, Autocorrelation Function, Self-Diffusion Coefficient, Lagrange Particle PACS (2006): 61.25-f , 82.70-y 1. Spectrum, Introduction The investigation of thermal hydrodynamic fluctuations in heterogeneous systems is an urgent question at present [1-8]. The heterogeneity can be caused by different external ∗ geophys@ogmi.farlep.odessa.ua 144 Electronic Journal of Theoretical Physics 3, No. 11 (2006) 143–149 factors. Under natural conditions, such factors as geometrical restrictions on the system and imposition of external electromagnetic and gravitation fields are most frequently encountered. Despite the different nature of these factors, their influences are identical and reduce to some orientation ordering of the liquid’s anisotropic molecules. In this connection, the need for studying heterogeneous anisotropic liquids, including confined ones, arises. The influence of a geometrical restriction on the dynamic properties of the system is taken into account by supplementing the system of hydrodynamics equations with proper boundary conditions. The anisotropy of the liquid’s molecules can incorporated by introducing into the above system the anisotropy tensor, which serves as an additional field variable and characterizes the deviations of the anisotropic molecule axes within a liquid volume element from the isotropic distribution. Such an approach was first offered and justified in [9], and then refined in [10]. Recently, with the participation of one of us [11], a complete system of hydrodynamics equations was developed in which along with the local traverse velocity, the internal angular momentum is taken into account and the inertia tensor is considered as the anisotropy tensor. The equations were used for the study of the space–time CF and the spectral density of the inertia tensor fluctuations of the liquid Lagrange particle in a homogeneous liquid under equilibrium and nonequilibrium conditions – against the background of a liquid flow with constant velocity gradient [11, 12]. In this report, the results of our further analysis of the thermal hydrodynamic fluctuations in an anisotropic liquid are presented. The fluctuations are studied for the heterogeneous conditions modeled, for the sake of simplicity, in the form of a planeparallel layer. The extended system of hydrodynamics equations is linearized for small deviations from the equilibrium values, the liquid is taken to be incompressible, and the rapid processes of the angular momentum relaxation to equilibrium are ignored. The solution of the boundary problem with arbitrary initial conditions is found for the system of linked equations for the velocity and inertia tensor components by developing it in an expansion in the harmonic functions. The spectral densities of the fluctuation CFs for the expansion amplitudes are obtained by using FDT. The results obtained for the field variable CFs for the heterogeneous anisotropic liquid are compared with those for the bulk case. The LVACF spectrum for the anisotropic liquid and the collective part of the molecule self–diffusion coefficient are determined as functions of the coordinate normal to the restrictive planes. The transformation from the Euler to the Lagrange variables is carried out on the basis of the Malomuzh approach [13-15]. 2. Analysis The complete system of hydrodynamics equations for the anisotropic liquid was obtained in [11]. Let us assume that the liquid occupies the region between two parallel planes separated by distance d, and the z axis of the Cartesian coordinate system to be directed along the normal to the planes, so that 0 ≤ z ≤ d. For the sake of simplicity, Electronic Journal of Theoretical Physics 3, No. 11 (2006) 143–149 145 we consider the anisotropic liquid to be incompressible and ignore the rapid processes of the internal angular momentum relaxation to equilibrium [11]. Now, let us linearize the equation of motion for the velocity, pressure, and inertia tensor fluctuation fields, and add to them spontaneous sources of the fluctuations. As a result, we obtain the system of linked differential equations 1 ∂ υα = − ∇α p + ν Δυα + ν12 ∇β δ Iαβ + σα , ∂t ρ0 1 ∂ δ Iαβ = − δ Iαβ + D Δδ Iαβ + ν21 (∇α υβ + ∇β υα ) + Rαβ , ∂t τ div υ = 0, νab = ηab/ρ0 , τ = ρ0/η2 , D = μ2/ρ0 . (1) Here ρ0 is the equilibrium density, ν is the kinematics viscosity, τ is the relaxation time, D is the diffusion coefficient for the inertia tensor, δI is the deviation of the inertia tensor from its equilibrium value, and σ and R are the spontaneous fluctuation sources. The velocity field and the inertia tensor components, υα and δIαβ , vanish at the restrictive planes: υ (t, x, y, z = 0) = 0, δIαβ (t, x, y, z = 0) = 0, υ (t, x, y, z = d) = 0, δIαβ (t, x, y, z = d) = 0. (2) In equations (1) and boundary conditions (2), let us pass to the Fourier–transforms with respect to the time and the transversal radius-vector components:   Φ(t, r⊥ , z) = e−i(ωt−k⊥r⊥ ) Φωk (z). (3) dω 4 dk⊥ Retaining the former designations for the amplitudes of the velocity and the inertia tensor fields, and taking into account that the Laplace operator in representation (3) is of the form ∂2 2 2 , k⊥ = kx2 + ky2 , (4) Δ = 2 − k⊥ ∂z after simple transformations we obtain these independent inhomogeneous equations: 1 fˆυα (z) = ν12 ∇β Rαβ + (−iω + − DΔ)σα , τ Here Fαβ fields (5) 1 1 fˆ(−iω + − DΔ)δIαβ= = fˆRαβ + (Fαβ + Fβα ). τ 2 is the result of transformation of the following combination of the spontaneous Fαβ (t, r) = ν12 ν21 ∇β ∇β ′ Rαβ ′ + ν21 ( 1 ∂ + − DΔ)∇α σβ , ∂t τ (6) and the operator ν12 ν21 1 Δ]. fˆ = [(−iω − νΔ)(−iω + − DΔ) − τ 2 (7) 146 Electronic Journal of Theoretical Physics 3, No. 11 (2006) 143–149 It is convenient to represent the solution of the ordinary differential equation system (5) in the form of expansions in the harmonic functions. In view of the properties of these functions and boundary conditions (2), these expansion assume the form  υα (z) = Aαm sin μm z, (8) m δIαβ (z) =  Bαβm sin μm z, m where the eigenvalues are μm = πm/d. Let us represent the components of the spontaneous sources in equations (5) in the form of similar expansions with the expansion 0 . With the orthogonality property and the normality condition for coefficients A0αm , Bαβm the harmonic functions, the amplitudes in (8) are simply expressed from (5) through the expansion amplitudes for the spontaneous fields. Based on these expressions and using the FDT, we find the spectral densities for the amplitudes of the velocity and inertia tensor fluctuations: 2 + μ2m )]−1 , A∗αm Aα′ m′ ω = Θδαα′ δmm′ Re[−iω + νωλ (k⊥ where (9)  . ∗ 1 2 + μ2m )]−1 , Bαβm Bα′ β ′ m′ ω = 2Θδαα′ δββ ′ δmm′ Re[−iω + + D(k⊥ τ k ? T0 1 ν12 ν21 2 [−iω + + D(k⊥ , νωλ = ν + + μ2m )]−1 . (10) 2 2π 2 τ The spectral density of the amplitudes for the inertia tensor fluctuations in (9) is obtained under the assumption that the velocity field fluctuations and the inertia tensor field fluctuations are independent. The difference of the result (9) from the those for the bulk case is only in the discrete values of the wave number k and the appearance of a frequency dependence of the kinematics viscosity coefficient. The spectral densities of the thermal fluctuations of the Euler velocity and inertia tensor hydrodynamic fields are respectively given by Θ= < υα∗ (r⊥ , z, t)υα′ (r′ ⊥ , z ′ , t′ ) >ω = 1 (2π)3 ∗ (r⊥ , z, t)Iα′ β ′ (r′ ⊥ , z ′ , t′ ) >ω = < Iαβ & 1 (2π)3  ′ dk⊥ eik⊥ (r⊥ −r ⊥ )  A∗αm Aα′ m′ ω sin μm z · sin μm′ z ′ ,  &  ∗  ′ Bαβm Bα′ β ′ m′ sin μm z · sin μm′ z ′ . dk⊥ eik⊥ (r⊥ −r ⊥ ) m m′ m m′ ω (11) Let us explore the spectral density of the autocorrelation function for the velocity thermal fluctuations in more detail. In the limiting case of coincident spatial arguments, the expression for the velocity CF from (11) reveals the logarithmic divergence ∼ ln ρ (ρ = ρ→0 ′ |r⊥ − r⊥ |) and, because of this, cannot be used to estimate the self-diffusion coefficient. However, we can analyze the spectrum of the velocity autocorrelation function by changing from the Euler variables to the Lagrange ones. Moreover, since the velocity of the Lagrange particle is usually considered to equal the collective component of the velocity of the molecule [13, 14], the realization of such a change is actually equivalent to obtaining Electronic Journal of Theoretical Physics 3, No. 11 (2006) 143–149 147 the hydrodynamic asymptotic for the spectrum of the velocity autocorrelation function for the velocity of the molecule. To change to the LVACF, we take advantage of the Malomuzh approach [13], whose essence consists in taking into account nonlocal relation between the Lagrange and Euler variables. The derivation of the relation between the Lagrange and Euler velocity CFs is given in [13]. It is considerably simplified if we ignore small drifts of the centre of mass of the Lagrange particle, as in [12]; in this case, it assumes the form   1 (12) ϕL (t) = 2 dr1 dr2 < υ (r(t) + r1 , t)υ (r(0) + r2 , 0) > . VL VL VL where VL is the volume of the Lagrange particle. The spectral density of the Lagrange velocity hydrodynamic field has, according to (11), (12), and in view of the presence of the delta-functions in the spectral densities of the velocity fluctuation amplitudes, the form ∞ RL4  J12 (k⊥ RL ) 2 ϕLω (z) = 2 < A∗αm Aαm >ω S̄m (z)dk⊥ . (13) VL m k⊥ 0 Here J1 (y) is the Bessel function, RL = 2 (z) S̄m 1 = 2 RL 3 V 4π L 1/3 , and RL RL dz1 dz2 sin μm (z + z1 ) sin μm (z + z2 ) = 0 (14) 0 = (μm RL )−2 · (cos μm z − cos μm (z + RL ))2 . After integration over the k⊥ and in view of (9), we get: ϕLω (z) = 9Θ   I1 (RL γml )K1 (RL γml ) 2 al · S̄m (z). 16π 2 ν m l=1,2 (RL γml )2 (15) Here I1 (y) and K1 (y) are the modified Bessel functions, and al and γml are the solutions of these simple algebraic equations: 2 γm1 = μ2m + C1 , 2 = μ2m + C2 , γm2  C1 = λ − λ2 + C1 + C2 = 2λ, a1 = a2 = ω 2 +iω/τ , Dν λ= C2 −C3 , C2 −C1 C3 −C1 , C2 −C1 −iω(D+ν)+ν/τ +ν12 ν21 /2 , 2Dν (16) C3 = D−1 (1/τ − iω) . Since RL ≪ d, using the series expansion for the functions I1 (y) and K1 (y), result (15) is represented in the form    9Θ   1 1 1 RL γml −2 2 ϕLω (z) = CL − · S̄m ln + (RL γml ) + (z), (17) al 32π 2 ν m l=1,2 2 2 2 4 148 Electronic Journal of Theoretical Physics 3, No. 11 (2006) 143–149 where CL is the Euler constant. After carrying out the summation, we finally derive: / 0   1 s (2Cl ) 2 9Θ  1 RL2 Cl 2s (Cl ) 1 ϕLω (z) = + + 3 + CL − ln al , + 128π 2 ν l=1,2 RL 4 2RL 4 RL2 (2Cl )1/2 RL Cl RL4 Cl3/2 (18) where √ √ 2 √ s (x) = cth (d x) (ch ((z + RL ) x) − ch (z x)) + √ √ 2 √ √ √ + (sh ((z + RL ) x) + ch (z x)) − ch (RL x) (ch ((2z + RL ) x) + sh ((2z + RL ) x)) . (19) The derived spectrum of the LVACF allows us to find the expression for the Lagrange self-diffusion coefficient, which is defined as 1 DL (z) = ϕL,ω=0 (z). 3 (20) In the limit ν12 ν21 → 0, when the fluctuations of the inertia tensor and velocity fields are independent, with the use of (15), (16) we obtain: 9Θ d DL (z) = 2 64π 2 ν RL / z d / 1+o  RL d 2 0 −  z 2 d / 1+o  RL d 2 0 RL + d  37 1 8πd − ln 12 2 RL   RL +o d (21) Expression (21) is given with an accuracy of up to the second order in the small ratio RL /d; depending on the distance d between the parallel planes, this relation takes on values from −∞ to 10−1 . Conclusion The spectrum of the Lagrange velocity hydrodynamic field as a function of the coordinate z normal to the restrictive planes is derived. It represents the hydrodynamic asymptotic form for t → ∞ of the spectrum of the velocity autocorrelation function for the anisotropic molecules of a liquid confined by two parallel planes. The distance between the planes is a parameter for this dependence and can be taken to range from d ∼ 10RL to d → ∞. With the use of the expression for the spectrum, the Lagrange self-diffusion coefficient is determined as a function of z and dimensionless relation RL /d. It is in good agreement with results [6], where the effect of confinement on the modecoupling contribution to the self-diffusion coefficient (in the direction parallel to the walls) in a fluid slab was computed. In [6], these finite-size corrections are shown to reduce the bulk diffusion constant near the walls by an amount of − (σ/d) log (σ/d), where d is the thickness of the fluid slab and the physical interpretation of σ is given, based on intuitive considerations, as the atomic size. Similar finite-size corrections near the walls with RL instead of σ are obtained for the Lagrange self-diffusion coefficient in the present paper. In the limit d → ∞, the results obtained transform into the self-diffusion coefficient for the bulk case. 2 0 . Electronic Journal of Theoretical Physics 3, No. 11 (2006) 143–149 149 References [1] J. M. Ortiz de Zarate, L. M. Redondo (2001), Finite-size effects with rigid boundaries on nonequilibrium fluctuations in a liquid. Eur. Phys. J., 21, 135-144. [2] T. G. Sokolovska, R. O. Sokolovskii, M. F. Holovko (2000), Orientational ordering in fluids with partially constrained molecule orientations. Phys. Rev., E62, N5, 6771-6779. [3] I. Pagonabarraga, M. H. J. Hagen, With. P. Lowe, and D. Frenkel (1999), Shorttime dynamics of colloidal suspensions in confined geometries. Phys. Rev., E59, N4, 4458-4469. [4] I. Pagonabarraga, M. H. J. Hagen, With. P. Lowe, and D. Frenkel (1998), Algebraic decay of velocity fluctuations near a wall. Phys. Rev., E58, N6, 7288-7295. [5] J. Teixeira, J.-M. Zanotti, M.-C. Bellissent-Funel, S.-H. Chen (1997), Water in confined geometries. Physica, B234/236, 370-374. [6] L.Bocquet, J.-L. Barrat (1996), Hydrodynamic properties of confined fluids. J. Phys: Condens. Matter., 8, 9297-9300. [7] K. P. Travis, B. D. Todd, D. J. Evans (1997), Departure from the Navier-Stokes hydrodynamics in confined liquid. Phys. Rev., 55, N4, 4288-4295. [8] F. Benmouna and D. Johannsmann (2002), Hydrodynamic interaction of AFM cantilevers with solid walls: An investigation based on AFM noise analysis. Eur. Phys. J., E9, 435-441. [9] M.A. Leontovich (1941), Relaxation in liquids and scattering of light. J. Phys. USSR, 4, 499-518. [10] I.L. Fabelinski (1997), Spectra of molecular scattering of light. Progress in Optics, 37, 97-184. [11] A.V. Zatovsky, A.V. Zvelindovsky (2001), Hydrodynamic fluctuations of a liquid with anisotropic molecules. Physica A, 298, 237-254. [12] O.I. Sakhnenko (2005), The spectrum of the correlation function for fluctuatios of the anisotropy tensor of a Lagrangian liquid particle. Ukr. J. Phys., 50, N7, 714-719. [13] T. V. Lokotosh, N. P. Malomuzh (2000), Lagrange theory of thermal hydrodynamic fluctuations and collective diffusion in liquids. Physica A., 286, 474-488. [14] T. V. Lokotosh, N. P. Malomuzh (2001), Manifestation of the Collective Effects in the Rotational Motion of Molecules in Liquids. J. Mol. Liq., 93, N1-3, 95-108. [15] T. V. Lokotosh, N. P. Malomuzh, K. S. Shakun (2002), Nature of oscillations for the autocorrelation functions for translational and angular velocities of a molecule. J. Mol. Liq., 96/97, 245-263. EJTP 3, No. 11 (2006) 151–158 Electronic Journal of Theoretical Physics On the Quantum Correction of Black Hole Thermodynamics Kourosh Nozari and S. Hamid Mehdipour ∗ Department of Physics, Faculty of Basic Sciences, University of Mazandaran, P. O. Box 47416-1467, Babolsar, IRAN Received 20 May 2006 , Accepted 7 June 2006, Published 25 June 2006 Abstract: Bekenstein-Hawking Black hole thermodynamics should be corrected to incorporate quantum gravitational effects. Generalized Uncertainty Principle(GUP) provides a perturbational framework to perform such modifications. In this paper we consider the most general form of GUP to find black holes thermodynamics in microcanonical ensemble. Our calculation shows that there is no logarithmic pre-factor in perturbational expansion of entropy. This feature will solve part of controversies in literatures regarding existence or vanishing of this pre-factor. c Electronic Journal of Theoretical Physics. All rights reserved.  Keywords: Quantum Gravity, Generalized Uncertainty Principle, Black Holes Thermodynamics PACS (2006): 04.70.-s, 04.70.Dy 1. Introduction Quantum geometry, string theory and loop quantum gravity all indicate that measurements in quantum gravity should be governed by generalized uncertainty principle[15]. As a result, there is a minimal length scale of the order of Planck length which can not be probed. In the language of string theory, this is related to the fact that a string can not probe distances smaller than its length. Therefore, it seems that a reformulation of quantum theory to incorporate gravitational effects from very beginning is necessary to investigate Planck scale physics. Introduction of this idea, has drawn considerable attentions and many authors have considered various problems in the framework of generalized uncertainty principle[6-20]. Quantum gravitational induced corrections to black hole thermodynamics as a consequence of GUP are studied with details in liter∗ knozari@umz.ac.ir 152 Electronic Journal of Theoretical Physics 3, No. 11 (2006) 151–158 atures. Adler and his coworkers[21] have argued that contrary to standard viewpoint, GUP may prevent small black holes total evaporation in exactly the same manner that the ordinary uncertainty principle prevents the Hydrogen atom from total collapse. They have considered these black holes remnants as a possible source of dark matter. Medved and Vagenas[22], have recently formulated the quantum corrected entropy of black holes in terms of an expansion and have claimed that this expansion is consistent with all previous findings. Bolen and Cavaglia, have obtained thermodynamical properties of Schwarzschild anti-de Sitter black holes using GUP [23]. They have considered two limits of their equations, quantum gravity limit and usual quantum mechanical regime and in each circumstances they have interpreted their results. Action for the exact string black hole has been considered by Grumiller and he has found exact relation for entropy of a string black hole[24]. Existence or vanishing of logarithmic prefactor in the expansion of black hole entropy has been considered in details by Medved. He has argued in [25] that ”the best guess for the prefactor might simply be zero” regarding to mutual cancelation of canonical and microcanonical contributions. But later, considering some general considerations of ensemble theory, he has argued that canonical and microcanonical corrections could not cancel each other to result in vanishing logarithmic pre-factor in entropy[26]. Meanwhile, Hod has employed statistical arguments that constrains this prefactor to be a non-negative integer[27]. There are other literatures considering logarithmic corrections to black hole entropy[28,29], but there is no explicit statement about the ultimate value of this prefactor. Here, using generalized uncertainty principle in its most general form as our primary input, we find explicit perturbational expansion of black hole entropy in microcanonical ensemble. By computing the coefficients of this expansion, we will show that there is no logarithmic prefactor in expansion of microcanonical entropy. 2. Generalized Uncertainty Principle Usual uncertainty principle of quantum mechanics, the so-called Heisenberg uncertainty principle, should be re-formulated regarding to non-commutative nature of spacetime. It has been indicated that in quantum gravity there exists a minimal observable distance on the order of the Planck length which in the context of string theories, this observable distance is referred to GUP[1-5],[30-33]. A generalized uncertainty principle can be formulated as  + const.Gδp, (1) δx ≥ 2δp which, using string theoretical arguments regarding the minimal nature of lp [4], can be written as δp  + α2 lp2 (2) δx ≥ 2δp 2 The corresponding Heisenberg commutator now becomes, [x, p] = i(1 + α′ p2 ). (3) Electronic Journal of Theoretical Physics 3, No. 11 (2006) 151–158 153 Note that commutator (3) is not the direct consequence of relation (2), but can be considered as one of its consequences[11]. α is positive and independent of δx and δp but may in general depend to the expectation values of x and p. In the same manner one can consider the following generalization,   β2 2 δxδp ≥ (4) 1 + 2 (δx) , 2 lp which indicates the existence of a minimal observable momentum. It is important to note that GUP itself can be considered as a perturbational expansion[11]. In this viewpoint, one can consider a more general statement of GUP as follows   α2 lp2 β2  2 2 (5) δxδp ≥ 1 + 2 (δp) + 2 (δx) + γ , 2  lp where α, β and γ are positive and independent of δx and δp but may in general  depend to the expectation values of x and p. Here, Planck length is defined as lp = G . Note c3 that (5) leads to nonzero minimal uncertainty in both position (δx)min and momentum (δp)min . In which follows, we use this more general form of GUP as our primary input and construct a perturbational framework to find thermodynamical properties of black hole and their quantum gravitational corrections. It should be noted that since GUP is a model independent concept[6], the results which we obtain are consistent with any promising theory of quantum gravity. 3. Black Holes Thermodynamics Consider the most general form of GUP as equation (5). A simple calculation gives, " #  2 (γ + 1) lp2 δp  δx  2 1 ± 1 − β 2 α2 + 2 . (6) β  lp (δp)2 Here, to achieve standard values (for example δxδp ≥ ) in the limit of α = β = γ = 0, we should consider the minus sign. One can minimize δx to find (δx)min  ±αlp (1 + γ) . 1 − α2 β 2 (7) The minus sign, evidently has no physical meaning for minimum of position uncertainty. Therefore, we find (δx)min  αlp (1 + γ) . 1 − α2 β 2 (8) This equation gives the minimal observable length on the order of Planck length. Since in our definition, α and β are dimensionless positive constant always less than one(extreme quantum gravity limit), (δx)min is defined properly. Equation (5) gives also " #  2 (γ + 1) l δx p δp  2 2 1 ± 1 − α2 β 2 + . (9) α lp (δx)2 154 Electronic Journal of Theoretical Physics 3, No. 11 (2006) 151–158 Here to achieve correct limiting results we should consider the minus sign in round bracket. From a heuristic argument based on Heisenberg uncertainty relation, one deduces the following equation for Hawking temperature of black holes[21], TH ≈ δpc 2π (10) Based on this viewpoint, but now using generalized uncertainty principle in its most general form, modified black hole temperature in GUP is, "  # 2 (γ + 1) l cδx p THGU P ≈ . (11) 1 − α2 β 2 + 2 2 1− (δx)2 2πα lp Now consider a quantum particle that starts out in the vicinity of an event horizon and then ultimately absorbed by black hole. For a black hole absorbing such a particle with energy E and size R, the minimal increase in the horizon area can be expressed as [34] (ΔA)min ≥ 8πlp2 ER , c (12) then one can write 8πlp2 δpcδx , (13) c where E ∼ cδp and R ∼ δx. Using equation (9)(with minus sign) for δp and defining A = 4π( δxmin )2 , we find 2 (ΔA)min ≥ (ΔA)min " 8A  2 1− α 1 − α2  πlp2 (γ + 1) β2 + A # . (14) Now we should determine δx. Since our goal is to compute microcanonical entropy of a large black hole, near-horizon geometry considerations suggests the use of inverse surface gravity or simply twice the Schwarzschild radius for δx. Therefore, δx ≈ 2rs and defining 4πrs2 = A and (ΔS)min = b = constant, then it is easy to show that, (ΔS)min dS   dA (ΔA)min " 8A 1 − bα2  2 1 − α β2 + πlp2 (γ+1) A # . (15) Three point should be considered here. First note that b can be considered as one bit of information since entropy is an extensive quantity. Considering calibration factor of Bekenstein as ln2 , the minimum increase of entropy(i.e. b), should be ln2. Secondly, 2π (∆S)min dS note that dA  (∆A)min holds since this is an approximate relation and give only relative changes of corresponding quantities. As the third remarks, our approach considers microcanonical ensemble since we are dealing with a Schwarzschild black hole of fixed mass. Now we should perform integration. There are two possible choices for lower limit of integration, A = 0 and A = Ap . Existence of a minimal observable length leads Electronic Journal of Theoretical Physics 3, No. 11 (2006) 151–158 155  2 to existence of a minimum event horizon area, Ap = 4π (δx)2min . So it is physically reasonable to set Ap as lower limit of integration. This is in accordance with existing picture[21]. Based on these arguments, we can write S  A Ap bα2  1 − α2 β 2 + " 8A 1 − πlp2 (γ+1) A # dA. (16) Integration gives,  #  "   −2$A(ζA + η) + A(ζ + 1) + η  $  η + 2ζA + 2$ζA(ζA + η)      $ S  μ ln  $  + ζ ln    η + 2ζAp + 2 ζAp (ζAp + η)   −2 Ap (ζAp + η) + Ap (ζ + 1) + η  (17) where, b μ = 2, η = −πα2 lp2 (γ + 1), ζ = 1 − α2 β 2 , 8β Ap = πα2 lp2 (1 + γ) (1 − α2 β 2 ) (18) This is the most general form of the black hole entropy which can be obtained from perturbational approach based on GUP. Expansion of (17) gives ∞  Dn (A − Ap )n . (19) S n=1 The coefficients of this expansion have very complicated form. The first coefficient is D1 = μ / − √ 2ζAp +η Ap (ζAp +η) +ζ +1 $ + −2 Ap (ζAp + η) + Ap (ζ + 1) + η $ √2ζ 2 A +ζη p ζAp (ζAp +η) + 2ζ 0 . (20) ζ $ 2 ζAp (ζAp + η) + 2ζAp + η The matter which is important in our calculations is the fact that expansion (19) has no logarithmic term. In other words, since expansion (19) contains only integer power of A − Ap , we conclude that in microcanonical ensemble, there is no logarithmic corrections due to quantum gravitational effects for thermodynamics of black holes. Adler et al have found vanishing entropy for remnant in their paper[21]. In other words, their result for entropy vanishes when one considers Planck mass limit. In our framework, when A = Ap , one finds S = 0 and therefore remnant has zero entropy. A result which physically can be acceptable since small classical fluctuations are not allowed at remnant scales because of the existence of the minimum length. 4. Summary In this paper, using generalized uncertainty principle in its most general form as our primary input, we have calculated microcanonical entropy of a black hole. We have 156 Electronic Journal of Theoretical Physics 3, No. 11 (2006) 151–158 shown that in perturbational expansion there is no logarithmic pre-factor, which has been the source of controversies in literatures. Actually in calculation of entropy we should compute the number of possible microstates of the system and there are two possible choices for corresponding ensemble: canonical and microcanonical ensemble. We have shown that the contribution of microcanonical ensemble itself is vanishing. If there is any contribution related to canonical ensemble, it cannot cancel vanishing contribution of microcanonical ones. 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