ISSN 1070-3632, Russian Journal of General Chemistry, 2011, Vol. 81, No. 1, pp. 220–233. © Pleiades Publishing, Ltd., 2011. Original Russian Text © S.F. Timashev, 2009, published in Rossiiskii Khimicheskii Zhurnal, 2009, Vol. 53, No. 6, pp. 50–61. Phenomenology of Complexity: Information in Chaotic Signals S. F. Timashev Karpov Research Physicochemical Institute, ul. Vorontsovo Pole 10, Moscow, 105064 Russia e-mail: serget@mail.ru Received December 1, 2009 Abstract―Inner phenomenological essence was revealed for flicker-noise spec-troscopy (FNS), a methodology for extracting information from multifactor dynamic systems on the basis of primary digital information, which was presented as “pheno-menology of complexity” and “materialization” of Edmund Husserl’s phenomenology. The basic FNS equations used for analysis of experimental data were given. At the present time, the FNS method can be used for solving three types of problems: (1) determination of parameters or patterns that characterize the dynamics or structural features of complex systems; (2) identification of precursors of abrupt changes in the state of various complex systems based on a priori information about the system dynamics; and (3) assessment of the flow dynamics in distributed systems based on analysis of dynamic correlations in stochastic signals that are measured simultaneously at different points in space. DOI: 10.1134/S1070363211010397 … natural philosophy … carries weight only if its every detail can be subjected to the inexorable test of experiment W. Heisenberg (cited after [1], p. 366) And the most exact of the sciences are those which deal most with first principles; and the first principles and the causes are most knowable; for by reason of these, and from these, all other things come to be known… Aristotle, Metaphysics ([2], pp. 9–10) concept [5] and the principle of computational equivalence [6] on this basis have provided conceptual understanding of the physical essence of evolution of open complex systems. It has been shown that the main feature intrinsic of this dynamics consists in intermittency of evolution: Time intervals on which the dynamic variable V(t) (t is time) changes fairly slowly (“laminar phases”) alternate with relatively brief intervals on which it changes sharply and chaotically, so that a transition to the subsequent “laminar phase” is defined with another characteristic V(t) value. Computer calculations revealed a powerlaw distribution for the dynamic variable V(t) in this intermittent evolution. Specifically the concept of essentially irregular “non-Darwinian” evolution has led to understanding of the well-known Gutenberg– Richter and Zipf–Pareto scale-invariant (scaling) INTRODUCTION The discovery of strange attractor and deterministic chaos in the 1960s [3–6] and the computer revolution in the last quarter of the XX century (predetermined by creation of cybernetics by Norbert Wiener in the late 1940s [7], when the basic notion of the state of a (sub) system as discretely varying within the given finite set of states was introduced) have initially brought a hope for gaining insight into “complexity” (the term “science of complexity” was introduced) as being possible in principle, which offered a promise for not only “listening to” but also “hearing” what the “earth is telling us,” carrying on “new dialog” with nature [8–12]. Computer modeling and elaboration of the methodology of creation of cellular automata, as well as the development of self-organized criticality 220 PHENOMENOLOGY OF COMPLEXITY relationships [5], so that flicker noise ceased to be a mystery. It was found that the above-mentioned features of complex system dynamics (in particular, power laws which are indicative of correlations, extended both in space and in time and “infinite” for flicker noise, that arise in examined systems under seemingly chaotic evolution or structural rearrangement during evolution) have internally originated from complex (“multiparticle,” nonlinear) interactions, inevitable dissipative processes and manifestations of inertia. The first two of the mentioned factors were identified as preconditions to formation of a single evolutionary system in a complex conglomerate of subsystems almost twenty-four centuries ago by Aristotle ([3], p. 224) who, being unaware of the inertia property, expressed the multparticle and dissipation aspects in “contact” and “stickiness” terms, respectively. Computer modeling studies disclose the principal features of the dynamics of complex model systems but do not allow, in principle, specific issues of evolution of a real particular system to be addressed. These issues include determining the parameters that unambiguously characterize the dynamic state of each of such systems, elucidating the nature of nonstationarity of evolution, and finding precursors of abrupt (possibly, catastrophic) changes in the state of the examined systems. However, specifically this information (“answers”) would have been derived from analysis of the dynamics of a specific system under some fixed or varied conditions (“questions”), if it have been the case of a real “dialog.” The theory of deterministic chaos and the methods of “nonlinear time series analysis” it underlies [12] also demonstrate their limitedness in application to studies of the dynamics of real systems. The reason is that the Takens’ theorem (which is of fundamental importance for such analysis) describing the conditions under which certain properties of a strange attractor in phase space can be reconstructed from a time series of one component [3, 4] is valid for stationary evolution solely. Clearly, conventional methods of analysis do not seek to identify physically understandable characteristics of complex nonstationary processes, especially those set on finite time intervals. This refers not only to Fourier but also to wavelet analysis. The latter, by virtue of the specific mathematical aspects of the relevant procedures (they leave only a small proportion of the largest in absolute values expansion coefficients for the signal analyzed), demonstrated 221 good results in solving practical problems of data “compression” (packing) and fuzzy image “decoration.” However, it is inapplicable to parameterization of real signals comprising both regular (“resonant”) and chaotic components. In this connection, of great relevance is the opinion expressed by G. Berkeley [13, p. 142]: “… the difference … betwixt natural philosophers and other men, with regard to their knowledge of the phenomena, …consists not in an exacter knowledge of the efficient cause that produces them, for that can be no other than the will of a spirit, but only in a greater largeness of comprehension, whereby analogies, harmonies, and agreements are discovered in the works of Nature, and the particular effects explained, that is, reduced to general rules…” These views have found further conformation and should not embarrass researchers. To author’s knowledge, a similar opinion was expressed by Einstein in a latter to M. Solovine: “We can't solve problems by using the same kind of thinking we used when we created them” (cited after [14, p. 530]) and also by Landau. The inherent complexity of open systems, nature objects in their infinitely diverse interrelationships, makes inevitable accepting the realist phenomenology proposed by German philosopher Husserl with its basic principle “back to things themselves” (“zurück zu den Sachen selbst”) against all “premature systematization,” so as “to return to “things themselves” and to investigate them without … violation of what is given” [15, 16]. Basic Principles of Flicker–Noise Spectroscopy Taken as the starting point for “disclosing reality precisely as it shows itself before scientific inquiry” ([17], p. 8), phenomenology was involved in development of flicker-noise spectroscopy (FNS). The latter underlies methodologically the analysis of chaotic signals produced by complex open systems based on the primary information defined by a set of digital data and extraction of information contained in these signals in the practically needed amount [18–23]. This study will be focused on disclosing the inner essence of FNS as “phenomenology of complexity” and presenting FNS as “materialization” of Husserlian realist phenomenology. It will be demonstrated that FNS logics corresponds to that of Husserl’s phenomenology regarded as the science of entities, contemplation of entities, cognizing of I. Kant’s “thing RUSSIAN JOURNAL OF GENERAL CHEMISTRY Vol. 81 No. 1 2011 222 TIMASHEV in itself” (“das Ding an sich”), science of “world construction, structure of being” [24, p. 477]. All these general definitions and estimations derived form the primary digital information about the phenomena examined (about the dynamics of evolution of multifactor systems, complex systems being formed) can be made more specific so that the relevant calculations will be brought to fairly specific algorithms. The question of cognition, according to Husserl, is “a question of how that consciousness “reaches” its objects, that is, how it is that the “contents” of consciousness correspond to the objects which they represent” [15, p. 7]. Our subsequent discussion will be organized around FNS as a method for extraction of phenomenological information from various chaotic signals, as the “phenomenology of complexity.” In the case of interest, the object (phenomenon) of perception is a signal, a time series V(t) set on the time interval T, and the evolution it describes is realized as intermittency. The information contained in the signal analyzed can be extracted from the autocorrelation function image (which is of fundamental importance for statistical physics) characterizing the interrelation between the values taken by the dynamic variable V(t) at larger and smaller values of the argument: ψ(τ) = <V(t)V(t + τ)>, (1) where τ is the time lag parameter (we presume 0 ≤ τ ≤ T/2). The angular brackets in expression (1) denote averaging over the time interval T: T/2 <(...)> = 1 ⌠ (...)dt. T ⌡ (2) −T/2 Averaging means that all the characteristics extractable from analysis of ψ(τ) functions should be considered as averages over the interval indicated. If T is a subinterval of a larger interval Ttot (T < Ttot), the value of the ψ(τ) function may depend on the position of the interval T inside this larger interval Ttot. If there is no such dependence and ψ(τ) is determined by the difference of the arguments of the dynamic variables from Eq. (2) solely, the evolution process analyzed will be identified as stationary process: ψ(τ) = ψ(–τ). To extract the information contained in ψ(τ) [we presume <V(t)> = 0], it is convenient to analyze certain transforms (“projections”) of this function, specifically the S(f) cosine transform (f is frequency): T/2 S(f) =⌠ <V(t)V(t = t1)>cos (2πft1)dt1. ⌡ (3) −T/2 and second-order difference moments (“structural functions”) Φ(2)(τ): Φ(2)(τ) = <[V(t) – V(t + τ)]2>. (4) Clearly, for a stationary process we have Φ(2)(τ) = 2[ψ(0) – ψ(τ)]. (5) As known, for stationary processes the cosine transform is a positively defined value over extended integration intervals (formally, T → ∞) and, in accordance with the Wiener–Khinchin theorem, it is proportional to the power spectrum SP(f) of the signal, determined from the squared modulus of its Fourier component: S(f) = T/2 1 S (f), S (f) = |v(f)|2, P P T v(f) = ⌠ V(t)exp (−2πift)dt at t→∞. ⌡ (6) (7) −T/2 The signal “energy” ET, defined on the interval T, can be calculated as follows: fmax T/2 2 ET = ⌠ [V(t) dt = ⌠ |v(f)| df, ⌡ ⌡ 2 −T/2 (8) fmin so that |v(f)|2 is a measure of the signal energy in the fmin to fmax frequency range. The FNS methodology considers the S(f) dependences formed by nonstationary signals set on limited time intervals. Such dependences are only conditionally termed “power spectra,” since there can exist frequency areas for which S(f) < 0. As known, the S(f) dependences are characterized by the most prominent manifestations of the frequencies specific for the signal examined, which can be associated with the resonances intrinsic for the signal sources and the interference contributions from such resonances. Below, these specific components will be defined as “resonant” contributions. Higher-frequency chaotic components of the signal give monotonically varying contributions to the S(f) dependences [4]. Isolation of such contributions to S(f), made by the chaotic components, should take into account the abovementioned features of the “intermittent” signal in which two frequency ranges are necessarily manifested: a lower-frequency range (“laminar phases”) and a higher-frequency range (sharp spikes, RUSSIAN JOURNAL OF GENERAL CHEMISTRY Vol. 81 No. 1 2011 PHENOMENOLOGY OF COMPLEXITY 223 accompanied by changes in the characteristic signal values V(t) in the subsequent “laminar phase”). Specifically the above-mentioned feature of “intermittent” evolutionary dynamics makes it convenient to introduce the structural function Φ(2)(τ). This statement will be exemplified by the process of one-dimensional random walk (Fig. 1), i.e., chaotic jumpwise variations of V(t) at a fairly low “kinematic viscosity” of the system νV. A small νV value implies the following: When passing from position Vj to adjacent position Vj+1, which are |Vj+1 – Vj| apart (in value), the system first “overleaps” owing to inertia and attains a somewhat larger V(t) value, whereupon it relaxes (the relaxation time is presumed to be brief compared to the residence time in each of the “walk” positions). Evidently, if the number of walks is large, the Φ(2)(τ) functions calculated for such processes will be determined only by the algebraic sum of the differences of the walks (“jumps”), being independent of the “inertial overleaps” of the system. At the same time, the S(f) functions characterizing the “energy aspect” of the process, will be determined by both these factors (jumps and spikes). This conclusion was consistently substantiated in [18, 19]. It should be emphasized that such separation of the information contained in series of various irregularities is typical specifically for the intermittent evolutionary dynamics [3]: The areas characterized by relatively small changes in the dynamic variable (in our case, in the absence of changes) alternate with short-term sharp significant changes in the variable. It should be clearly understood that any abrupt changes in the signal examined will inevitably be “blurred” because of inertia intrinsic for material objects and constraints inherent in the measurement procedure proper (delays, systematic instrumentation errors, etc.). Via effectively “blurring” the primary features of the dynamics analyzed, these factors lead to differences between the true essence of the processes examined and their real manifestation in measurements [25]. In fact, studies of the dynamics of real objects reveal the situation described by Plato ([26], pp. 255– 258), in which, based on observing “blurred” shadows on the cave wall, the observer has to draw conclusions about the nature of the objects that throw these shadows. The above information concerning the complex dynamics of processes occurring in open systems forms the initial knowledge obtained via “unbiased contemplation” (N. Hartmann). It is necessary for Fig. 1. Schematic of one-dimensional random walk at a low “kinematic viscosity” of the system. developing a phenomenological method of analysis of evolutionary dynamics and extracting the information contained in these signals. Husserl defined this information as the starting point for phenomenological reduction, which implies “bracketing of experience” and transforming it into transcendental, beyond experience, images that adequately reflect the essential features of real objects (in the case of interest, of the signal observed). This is basically the Plato’s “purifying experience,” according to which “knowledge is the correspondence of thought and reality” ([27], p. 79), the search for the Kantian “thing in itself.” Specifically such ideal images, which are not taken directly from experiment, form the deductive principles of any natural science ([28], p. 175). “These principles cannot be obtained by inductive generalization of experimental data; they always rely on guesswork and intuition, inspired by the experiment” [29]. The cumulative experience of science suggests that specifically this approach is of universal applicability in construction of scientific theories for which the basis should be formed by ideal images that reflect the basic essence of the phenomenon or structure. A question arises concerning the application of these general ideas in extraction of the information contained in chaotic signals and “generation” and use of the necessary transcendental images for deductively constructing the “phenomenology” of complexity on their basis. This question was answered through realization of J.S. Nicolis’ concept [30] which implies consideration of the hierarchical levels of the examined evolution and adoption of recursive laws generating information at a given hierarchical level and subsequently compressing it at a higher cognitive RUSSIAN JOURNAL OF GENERAL CHEMISTRY Vol. 81 No. 1 2011 224 TIMASHEV level. Objectively, an insight into a virtually infinite set of hierarchical levels can be gained from measurements of dynamic characteristics of the system with different sets of discretized temporal frequencies (or discrete movements along the spatial coordinate). Introduction of an infinite number of hierarchical levels for dynamic variables characterizing the features of the structure or evolutionary dynamics of the studied system may formally indicate the possibility of “probing” the spatiotemporal organization of this system at all its hierarchical levels with spatial and temporal digitization intervals being widely varied. Clearly, the very consideration of the intermittent dynamics at an individual hierarchical level is beyond experience, and this is the case of an ideal image of such dynamics in terms of Husserlian theory. The information corresponding to such dynamics, for whose introduction C.F. von Weizsäcker’s “TriesteTheory” will attracted, also seems to be transcendental [31]. This theory treats any evolutionary process as a time-irreversible jump from one state to another, with each state in the considered evolutionary sequence (in fact, this refers to “macrostates,” which are essentially large sets of “microstates”) having its intrinsic structural organization (determined by the system of “inner links”). The evolutionary dynamics in this case is regarded as a sequence of step-events and is associated with transitions of the system from one informative δ-interval at ith hierarchical level to an adjacent interval, from which it is some “uninformative” (for the given hierarchical level) interval apart. The very fact of the occurrence of each of these “events,” as evidenced by changes in the values of the dynamic variable V(t), is fixed by the instant of factualization of irreversibility in such transitions. Moreover, the “uninformative” gaps between the small time δ-intervals within which the irreversibility has not yet become a fact are associated with an introduced concept of “now” as the factor linking “the past” that has already taken place to “the future” that potentially exists. It should be emphasized that the key concept in such a deductively introduced image of evolution at each selected hierarchical level is specifically the time interval “now” between two “events,” rather than moments along a continuous time axis, traditionally considered for evolutionary dynamics. Clearly, the introduced “now” intervals should not be “empty;” they should comprise smaller-scale intervals corresponding to the next, ever-smaller-scale hierarchical level, so that the entire hierarchy of possible time intervals will be covered. To what extent the introduction of such a deductive “construction” of evolutionary dynamics of an open complex system is justified can only be judged from how closely the conclusions made on this hypothetical basis fit the experimental results. Specifically such concept of evolution was used in the development of FNS with the introduction of Weizäcker’s real irreversibility with respect to time at each “time” step and with inclusion of “uninformative” “now” intervals. “Materialization” of these philosophical and physical ideas and development on their basis of a methodology of time-series analysis of dynamic variables require idealizing the images introduced and “converging” all the informative δ-intervals belonging to different ith spatiotemporal levels into points. Each of such points, an “instant,” should carry information about the structural and energy state of the system at this point in time, i.e., serve as a marker of irregularities of different types for the system. Zero length of each “instant” means that, at each of these points, the value of the function must contain a singularity (actual or potential), i.e., be represented as a sum of generalized function with zeroth carrier (expressed as the sum over Dirac’s δ-functions and their derivatives) and functions with different types of discontinuities: Heaviside’s step θ-functions and functions with discontinuity in the first, second, and higher-order derivative. It should be noted that a wellknown mathematical formalism, the theory of generalized functions [32], is suitable for treating singular, physically abstract functions. The FNS methodology is underlain by specifically such construction corresponding to a single hierarchical level in the “intermittent” evolution dynamics and fits the Kantian interpretation of transcendental. It can be regarded as the Kantian image of “a thing in itself” beyond the limits of experience [33], characterizing the underlying essential links arising in the evolutionary dynamics of complex systems. Subsequent deduction is to be underlain by the pattern of evolutionary dynamics of complex systems (see Fig. 2.3 in [18]), corresponding to changes in the dynamic variable Vi(t) at ith level of spatiotemporal hierarchy, with spikes and jumps represented via Dirac’s δ-functions and Heaviside’s θfunctions, respectively. Based on the postulated hypotheses (greater generality principles), more special RUSSIAN JOURNAL OF GENERAL CHEMISTRY Vol. 81 No. 1 2011 PHENOMENOLOGY OF COMPLEXITY theoretical relations (lesser generality case) are to be derived [7]. The first step in implementation of the relevant procedures with the use of the generalized functions formalism consists in deriving expressions for Sci(f) cosine transform of autocorrelator and the secondorder difference moment Φс(i2)(τ) (for details, see [18]). Specifically these expressions were subsequently considered as the “generated” source of information characterizing the evolutionary dynamics at ith hierarchical level, provided that the analyzed evolutionary dynamics contains chaotic components solely. The next step, as prescribed by Husserlian phenomenology, should consist in “reduction” (according to Husserl), or “compression” (according to Nicolis), of the information at a higher cognitive level attainable in the experiment. For implementing such procedures, the self-similarity hypotheses were introduced, which imply taking the Sci(f) and Φс(2) i (τ) dependences for stationary evolutionary dynamics as respectively identical for each hierarchical level in the entire set of spatiotemporal hierarchical levels. The data “compression” in the case of stationary dynamics required introduction of certain invariant frequencies [18]. Thereby, a multiparameter self-similarity was postulated, as the Sci(f) and Φс(i2)(τ) dependences yielded by such reduction could be generally characterized by a set of parameters, in contrast to a single scaling factor appearing in the theory of fractals [34, 35] or the theory of renormalization group [36]. (The derived expressions for cosine transform Sci(f) of the autocorrelator and the difference moment Φс(2) i (τ) are presented below.) Although these Sci(f) and Φс(i2)(τ) dependences are valid for a stationary process, in the framework of the phenomenological FNS methodology they may be regarded as the simplest (threeparameter) interpolation expressions for the chaotic components of cosine transform of the autocorrelator and the second-order difference moment, respectively. These expressions are suitable for handling the experimentally measured signals with the aim to determine the FNS parameters in arbitrary processes. This is suggested by analysis of the evolutionary dynamics of diversified nonstationary processes (see examples in [18, 37]). The nonstationarity of a process is typically evident from changes in the low-frequency “resonant” components of a signal, as well as from variations of the introduced FNS parameters calculated on different time intervals of the nonstationary process under invariant functional representation of the Sci(f) 225 and Φс(i2)(τ) dependences. In cases where these relationships proved to be inadequate, more complex phenomenological relations are to be introduced [38]. Basic FNS Expressions Let us write the basic interpolating expressions for the chaotic components of the dynamic variables employed in analysis of the experimentally derived time series. It will be presumed that the parameters characterizing the evolutionary dynamics at all spatiotemporal levels are identical, which suggests a stationary process. In the simplest case, when a unique characteristic scale is introduced for each sequence of spike irregularities and jump irregularities, we will have [18, 19]: Φс(2)(τ) ≈ 2σ2 [1 – Γ–1(H1)Γ(H1, τ/T1)]2, (9) Γ(s, x) = ∫exp(–t)ts–1dt, Γ(s) = Γ(s, 0). Here, Γ(s) and Γ(s, x) are the complete and incomplete gamma functions (x ≥ 0 and s > 0), respectively; σ, rms deviation of the measured dynamic variable; and H1, Hurst constant, which describes the rate at which the dynamic variable “forgets” its values on the time intervals lesser than T1, which, in turn, is the time at which the dynamic variable completely “forgets” its value measured at a certain moment. In special cases we have (2) −2 2 τ Φc (τ) = 2Γ (1 + H1)σ T1 2H1 τ (2) 2 −1 Φc (τ) = 2σ 1 − Γ (H1) T1 τ , if T1 << 1, H1−1 τ exp − T1 if τ >> 1. T1 (10) 2 , (11) The interpolating function for the power spectrum components ScS(f) and ScR(f) formed by spike irregularities and jump irregularities, respectively, can be written as follows: ScS(f) ≈ ScR(f) ≈ ScS(0) 1 + (2πfT0) n0 , ScR(0) 1 + (2πfT1) 2H1+1 (12) . (13) Here, ScS(0) and ScR(0) are the parameters characterizing the low-frequency limits of ScS(f) and ScR(f), and n0 describes the rate of correlation loss in the sequence of spike irregularities on the time intervals T0 . RUSSIAN JOURNAL OF GENERAL CHEMISTRY Vol. 81 No. 1 2011 226 TIMASHEV The interpolating expressions (12) and (13) representing the contributions to the power spectrum from spike irregularities and jump irregularities, respectively, have identical functional forms. At the same time, the parameters in these expressions may be generally different: ScS(0) ≠ ScR(0), T1 ≠ T0, and 2H1 + 1 ≠ n0. This is in line with the previous conclusion concerning the difference in the information content between the parameters derived from power spectra and second-order structural functions in analysis of experimental time series V(t). Jump irregularities have a more regular character than do spike irregularities, for which reason the former are manifested at lower frequencies. Hence, the characteristic flicker-noise power function Sc(f) ~ 1/f n in the high-frequency range of the analyzed power spectra is primarily associated with spike irregularities. τmax (2) (2) Sr(f) = ⌠ cos (2πfτ) Φc (τmax) − Φr (τ) dt. ⌡ (15) 0 The resonant component of the autocorrelator function ψr(τ) can be determined from the “resonant” contribution Sr(f) by incomplete inverse cosine transform fmax ψr(τ) ~ 2 ⌠ Sr(f)cos (2πfτ)df, fmax = 0.5fd. ⌡ (16) 0 For the resonant component Φr(2)(τ) in this case we have Φr(2)(τ) = 2[ψr(0) – ψr(τ)], which allows the chaotic component Φr(2)(τ) of the second-order difference moments, described by Eq. (9), to be represented as As mentioned above, in chaotic dynamics of complex open systems the “nonspecific” correlation dynamic patterns intrinsic to series of informationsignificant irregularities coexist with system-specific slow-varying components having their characteristic sets of frequencies. These frequencies correspond to resonances intrinsic to evolutionary dynamics of systems subject to external impacts, whose frequency spectrum can also contain a set of characteristic frequencies. Also, the S(f) spectra can exhibit interferential frequencies. In the course of evolution of open systems the totality of the above-mentioned resonant and interferential frequencies can be rearranged. From here on, the specific frequencies and their interferential contributions manifested in the oscillatory nature of the dynamic variable V(t) analyzed, irrespective of the genesis of such frequencies fixed in the S(f) dependences, will be termed “resonant” frequencies for convenience in discussion. This will allow representing the V(t) signal as a linear superposition of the high-frequency chaotic component Vc(t) and the low-varying resonant component Vr(t): Let us discuss how the second-order difference moments Φ(2)(τ) which generally characterize random walks of the system state (see Fig. 1) are related to anomalous diffusion (see below). For the latter process, the rms deviation of system states V during time τ over the whole set of possible states (–∞ < V < ∞) from the average value can be represented as V(t) = Vc(t) + Vr(t). <(ΔV)2>pdf = 2Dt0(τ/t0)2H1. In this case the autocorrelator and power spectrum functions can be represented as [18, 20, 37] Here, D is the diffusion coefficient; t0, characteristic time; and H1, Hurst constant. ψ(τ) = ψr(τ) + ψc(τ), S(f) = Sc(f) + Sr(f ). (14) Under presumption of stationarity of the resonant components of the signals analyzed (the “resonant” part of the autocorrelator ψr(τ) depends on the argument τ difference solely), the Sr(f) and Φ(r2)(τ) functions corresponding to Vr(t) will be interrelated as Φc(2)(τ) = Φ(2)(τ) – Φr(2)(τ). (17) Using Eqs. (14)–(17) it is possible to successively separate contributions from the resonant and chaotic components with complete parameterization of signals [18, 20, 37]. Experience suggests that such representation is fairly justified in cases when specific tasks of parameterization as applied to complex signals of different nature are to be accomplished. There can be a large number of resonant components to be taken into account in analyzing the dynamics of complex systems; they are formed both by the resonant frequencies intrinsic to the system examined and the corresponding interferential contributions. Interrelation of FNS and Diffusion Parameters (18) The averaging <(…)>pdf is performed via introducing the probability density function (pdf) W(V, t) describing the probability for the system state to occur within the given interval of states. It is presumed that, at the initial time τ = 0, the system occurred in the vicinity of the V = 0 state (“point”). RUSSIAN JOURNAL OF GENERAL CHEMISTRY Vol. 81 No. 1 2011 PHENOMENOLOGY OF COMPLEXITY 227 Fickian diffusion (H1= 1/2) corresponds to random walks of system states for which the “elementary jumps,” associated with transitions of the system between adjacent states at the characteristic residence time δτ for every state, have some characteristic scale δV. However, if these random walks alternate stochastically with anomalous jumps exceeding δV at the same characteristic residence times δτ for the given state, this is the case of the so-called superdiffusion (“Levy diffusion,” “Levy flights”), for which H1 > 1/2. If random walks stochastically alternate with jumps having anomalously long times of residence in some states (“stability islands” [42]), significantly exceeding δτ for the same characteristic values of jumps δV, this is the case of the so-called subdiffusion for which H1 < 1/2. average value in random walks described by the Fickian equation or the anomalous diffusion equations. Anomalous (non-Fickian) diffusion can be described by diffusion equations comprising constant diffusion coefficients in which the partial derivatives with respect to time and coordinate are replaced by fractional-order derivatives [39–41]. In this case, the subdiffusion processes are represented via introducing an α-order fractional derivative (0 < α < 1) instead of a first-order partial derivative with respect to time. In the case of a superdiffusion process, a β-order fractional derivative is introduced (1 < β < 2) instead of a secondorder partial derivative with respect to coordinate. The parameter H1 varies within 0 < H1 < 1/2 for subdiffusion and within 1/2 < H1 < 1 for superdiffusion. It should be noted, however, that the Hurst constant H1 can exceed unity in the case of more complicated processes, e.g., those described by coordinate-dependent diffusion coefficient. and the “initial” condition The flicker-noise spectroscopic approach is that of a purely phenomenological nature; the parameters introduced in FNS have a certain physical meaning and are estimated via comparison of the data calculated by relationships (4)–(13) with those derived from dependences (1)–(3) obtained with the use of the experimental V(t) values constituting the time series. For a stationary process, in which the autocorrelator ψ(τ) = <V(t)V(t + τ)> depends on the difference in the arguments of the dynamic variables solely and the ergodicity condition is presumably met, the procedure of averaging (1) over time, introduced in FNS, is equivalent to that performed with the use of the probability density function W(V, t) for measuring the dynamic variable on the interval from V to V + dV at the time t. In this case, Eq. (4) can be treated as the generalized expression for the rms deviation from the In [44], the simplest case of Fickian diffusion (H1 = 1/2) for a stationary process was considered, and the phenomenological FNS parameters σ, T1, and H1 were interrelated with the parameters characterizing the diffusion dynamics. In that study it was presumed that the dynamics of the probability density W(V, τ) for the random variable V on the “segment” [–L, +L] over time τ can be described by the diffusion equation: 2 ∂W = D ∂ W ∂V2 ∂τ (19) provided that the “reflection” conditions ∂W = 0, ∂V at V = –L и V = +L, (19a) W(V, 0) = δ(V) (19b) are satisfied at the end points of the segment indicated. The sought solution was found, and the rms deviation <V 2>pdf of this variable from the average value (<V 2>pdf = 0) was calculated +L ⌠ V2W(V, τ)dV ⌡ <V2>pdf = −L +L . (20) ⌠ W(V, τ)dV ⌡ −L Also, the asymptotic expressions were derived: L2 , π 2D L2 2 → L at τ >> 2 . πD 3 <V2>pdf → 2Dτ at τ << (21) <V2>pdf (22) Based on comparison of expressions (21), (22) with (10), (11) for H1 = 1/2, the parameters of the diffusion problem were interrelated with phenomenological FNS parameters: 2 D = 4 σ ; L2 = 6σ2. π T1 (23) Although the asymptotic values coincide, the values for the compared <V2>pdf and Φ(2)(τ) dependences significantly (by ~20%) differ in the region of intermediate values of the parameter τ. This follows from comparison of curve 1 with the dotted line in Fig. 2 demonstrating the normalized φ1(τ) and φ2(τ) RUSSIAN JOURNAL OF GENERAL CHEMISTRY Vol. 81 No. 1 2011 228 TIMASHEV at V = +L: − D ∂W = χW(+L, t) − λw+L(t), ∂V ∂w+L(t) = χW(+L, t) − λw+L(t), ∂t (26) (26a) where, χ and λ are the rate constants for the forward and back transitions of the system between the boundary “diffusion” state and an “adstate,” respecttively (these parameters are presumed to be identical for the both boundaries). Fig. 2. Normalized functions (solid lines 1, 2) φ1(τ) corresponding to boundary conditions (1) Eq. (19a) and (2) Eqs. (27), (28) (see below) at χ = 0.4 and λ = 0.04 and (dotted line) φ2(τ). dependences corresponding to expressions (9) and (20), for which the asymptotic values at x <<1 and x >>1 coincide: 1 (2) 3 (24) ϕ1(x) ≡ 2 <V2>pdf , ϕ2(x) ≡ 2 Φ (Τ1x), x = τ/T1, 2σ L where x = τ/T1. As suggested by analysis of numerous natural signals [18, 21–23, 37, 44], the phenomenological expression (9) more adequately describes the chaotic component of the structural function Φ(2)(τ) than does the <V2>pdf dependence based on expressions (19)– (19b). This fact was associated earlier [45] with the model restrictions imposed on boundary condition (19a) and with the need to use for the problem of interest the generalized boundary conditions [46]. The latter take into consideration the fact that the measured values of the dynamic variable V(τ) can “go beyond” the limits of the [–L, + L] interval and to “reside” there for a certain period. As shown in [46], such “lags” at the boundary can be taken into account by introducing “adstates” of the +L and –L boundaries, in which the system can occur for a finite residence time. Upon introducing the probability densities w–L(τ) and w+L(τ) for the system to occur in such boundary states, the boundary conditions (19a) will be replaced by the following expressions: at V = –L: D ∂W = χW(−L, t) − λw−L(t), ∂V ∂w−L(t) = χW(−L, t) − λw−L(t), ∂t (25) (25a) Upon elimination of the w–L(t) and w+L(t) probabilities from (25a) and (26a) we will obtain new boundary conditions for the problem of interest: t D ∂W ∂V = χW(−L, t) − λχexp (−λt) ⌠W(−L, ξ)exp (ξt)dξ, ⌡ 0 (27) at V = −L, t −D ∂W ∂V = χW(L, t) − λχexp (−λt)⌠W(L, ξ)exp (ξt)dξ, ⌡ at V = L, 0 (28) [The boundary conditions (19a) can be deduced from Eqs. (27), (28) at χ = 0]. Diffusion Eq. (19) with integrodifferential boundary conditions (27), (28) under the initial condition (19b) was numerically solved using the iterative procedures described in [45]. As seen from Fig. 2, the normalized curve φ1(τ) for the rms deviation in the case of diffusion as calculated for boundary conditions (27), (28) at χ = 0.4 and λ = 0.04 approaches much closer the interpolation dependence (9) than does the curve obtained for boundary conditions (19a). The relative error for intermediate values of τ in this case does not exceed 5%. In other words, interpolation (9) is virtually equivalent to the rms deviation function calculated by solving diffusion Eq. (19) with integrodifferential boundary conditions (27), (28) under the initial condition (19b). Similar comparison for the <V2>pdf and Φ(2)(τ) functions in the case of anomalous diffusion (H1 ≠ 1/2) presents certain difficulties. This is due to the fact that the equations for the probability density W(V, τ) of random variable V varying on the segment [–L, +L] over time τ are cumbersome and can be solved by numerical methods solely [24, 29]. Nevertheless, the sought relationship can be derived from comparison of expressions (10) and (18) corresponding to anomalous diffusion at small τ values (in which case it will be natural to choose T1 as the characteristic time t0) under RUSSIAN JOURNAL OF GENERAL CHEMISTRY Vol. 81 No. 1 2011 PHENOMENOLOGY OF COMPLEXITY reasonable presumption that, in the case of anomalous diffusion, the FNS parameter σ should have a corresponding model parameter La defining the “localization area” of the states of the system in the space of values of the dynamic variable analyzed: D= 1 σ2 ; L 2 = bσ2, a Γ (1 + H1) T1 2 (28a) where b is a dimensionless parameter. It was shown previously [37, 45] that the introduced representation of anomalous diffusion, based on interpolation expressions (9), may fairly adequately “fit” the dynamics of chaotic changes of diverse signals measured. Dynamics of Nonstationary Processes: Precursors of System State Changes The nonstationarity effects in processes can be revealed by analyzing the dynamics of changes in the Φ(2)(τ) function, observed when the test interval [tk, tk + T], where k = 0, 1, 2, 3… and tk = kΔT, is successively shifted over the total interval Ttot of the available experimental data series (tk + T < Ttot). The choice of the time intervals T and ΔT should be based on analyzing the physical aspects of the problem of interest, with taking into consideration the expected typical time of the process, most important for the system evolution involving structural rearrangements. In the case of some secondary processes with typical times τi, slightly affecting the basic nonstationary structure rearrangement, the chosen interval T should satisfy the condition τi << T. It is natural to associate the phenomenon of a “precursor” of structural rearrangement of the system with sharp variations in Φ(2)(τ) when the upper boundary of the averaging time interval tk approaches the moment tc of structural rearrangement on various spatial scales. Clearly, reasons to speak about “precursor” of such structural rearrangements appear only if the time tk of the precursor is separated from the moment tc by no less than the interval ΔT, i.e., ΔTsn = tc – tk ≥ ΔT, when inequality ΔTsn << Ttot is valid. As “precursors” of structural rearrangement the flicker-noise spectroscopy treats the spikes of the values of nonstationarity indicators to be calculated using the difference moments Φ(2)(τ) [18]: Qk+1 − Qk C(tk+1) = 2 Q + Q k+1 k αT ΔT ; Q =⌠ (2) k ⌡ [Φ (τ)]kdτ. T (29) 229 This expression takes into account the fact that the Φ(2)(τ) dependences can be reliably calculated only in the case when the [0, αT] range of variation of the argument τ is less than the half of the averaging interval T. The introduced expressions characterize the measure of nonstationarity of the process when the averaging interval T is shifted along the time axis by ΔT. In particular, this is the case when the upper boundary of the averaging time interval tk approaches the moment tc characterizing the change in the state of the system, that can be associated with the structural rearrangement. Correlation Links in Complex System Dynamics Flicker-noise spectroscopy opens new prospects for analyzing various flows (mass, electric, magnetic) in complex systems whose fragments (subsystems) are characterized by significantly “cross-correlated” evolution because of arising of complex nonlinear links among them. The information about the dynamics of the correlation links between the dynamic variables Vi(t) of the same nature, measured at different points i, or between variables of different nature, can be extracted by analyzing the time dependences for the corresponding correlators. Below, the simplest “two-point” correlation expressions, or cross-correlators, characterizing the links between the dynamic variables Vi(t) and Vj(t) will be presented (out of the entire diversity of the correlation expressions introduced [18]): qij(τ, θij) = Vi(t) − Vi(t + τ) Vj(t + θij) − Vj(t + θij + τ) √2σi √2σj , (30) where τ is the lag time, and θij, “time shift” parameter to be determined for the required maximal possible value of the qij (τ, θij) correlator. The sign and magnitude of correlator (30) are indicative not only of the cause-and-effect relation (“flow direction”) between signals Vi(t) and Vj(t) but also of the characteristic time of information transfer between the two points (events) i and j. Under condition θij > 0 the flow moves from point i to point j (the lag of event j relative to event i), and for θij < 0, from j to i (the lag of event i relative to event j). A convenient choice of the rms deviations σi and σj for 0 RUSSIAN JOURNAL OF GENERAL CHEMISTRY Vol. 81 No. 1 2011 230 TIMASHEV dynamic variables Vi(t) and Vj(t) can be found in “reduced rms deviations:” σi(τ) = {<[Vi(t) – Vi(t + τ)]2>}1/2. (31) FNS Applications The FNS methodology is applicable to extraction of information contained both in experimentally measured time series and spatial chaotic series. In the case of “spatial” series, with the dynamic variable (surface relief height, variation of nucleotide bases in genome, etc.) chaotically varying along the configuration coordinate x, the following substitutions in the above-presented equations should be made: t → x and f → fx ≡ k/2π (the “spatial frequency” fx and wavenumber k have the dimensionality [x]–1). For examples illustrating the capabilities of the FNS phenomenology as applied to analyzing the dynamics of chaotic processes and complex structures in physical, geophysical, astrophysical, physicochemical, electrochemical, and biomedical systems, see [18, 20–23, 37, 44, 47–55], as well as papers available in the present issue [56–58]. CONCLUSIONS To conclude, several remarks will be made concerning the FNS phenomenology. The above discussion demonstrates the potentialities of the FNS phenomenology, which was termed by N. Hartmann “the first stage in systematization of the work of thought” in studying complex phenomena, revealing general connections for the systems examined. The possibility of using the autocorrelation function ψ(τ) = <V(t)V(t + τ)> as the basic image for deriving phenomenological information about the evolutionary dynamics of complex systems seems to be a new epistemological paradox (epistémé means knowledge, and logos, study). Formerly, the probability was regarded as an “outstanding example of epistemological paradox, when we can successfully apply our basic concepts without really understanding them” [59]. It is possible even to speak of the “probabilistic revolution in science” because “its incorporation into perception has radically changed the scientific worldview, style of scientific thinking, and basic models of the universe and its perception” [60]. The level of information contained in the autocorrelation function ψ(τ) and in the correlation links of signals on different spatiotemporal intervals lies much deeper compared to the case of the probability density function. Nevertheless, the correlation functions ψ(τ), as well as higher-order correlators employed in various theoretical models, received very little use in analysis of measured signals. Autocorrelation and cross-correlation functions were successfully applied for deriving the information contained in complex signals based on the “prescriptions” of Husserlian classical phenomenology and specific recipe rules proposed by J.S. Nicolis. To author’s knowledge, no similar attempts have been made to develop a practical method for deriving knowledge from a set of measured signals, based on a purely philosophical tradition, including the use of transcendental images of “a thing in itself” and “now,” as well as of deductive schemes. Certainly, classics of natural sciences, in particular, the creators of the two most prominent physical theories of XX century, the general theory of relativity (Einstein) and quantum mechanics (W. Heisenberg and E. Schrödinger), have demonstrated examples of how a theory can be deductively constructed on the basis of physically clear, though essentially transcendental, postulates which are beyond experience due to the ideality of the introduced images, “purified” from Nature’s “noise” (Plato’s idea!). Examples of such theories can be found in Bianchi space (general theory of relativity), as well as in quantum states with their corresponding zero-width energy levels, the “classical instrument” (in quantum mechanics). According to Schrödinger, “… the final word that all the transcendental has to disappear once and for ever cannot be implemented…in the field of knowledge…,” we … cannot dispense of the metaphysical guide” [61]. Heisenberg [62] argued that phenomenology should be underlain by philosophic thinking, Plato's ideas, as only they facilitate “understanding real connections.” In this context, the following fragment extracted from memories about Einstein is also highly indicative: “Once Einstein said that the problem of the “now” worried him seriously. He explained that the experience of the “now” means something special for man, something essentially different from the past and the future, but that this important difference does not and cannot occur within physics.” (cited after [8], p. 189]). Naturally, these quotations from classics date back to more than half a century ago, when a “serious worry” associated with introduction of ideal images into phenomenology in the most general sense, rather than into specific physical models, could not be materialized. RUSSIAN JOURNAL OF GENERAL CHEMISTRY Vol. 81 No. 1 2011 PHENOMENOLOGY OF COMPLEXITY Certainly, every deductive scheme is “a reflection or a model view of reality outside a man” [7]. If subsequent experimental verification would require rejecting the theoretical scheme presented here, the necessary adjustments could be made to the FNS methodology developed, or it could be replaced by new theories. In this connection, a comprehensive analysis of the dynamics and structure of diversified complex systems is highly demanded for the establishment or correction of the phenomenology of “complexity” presented here. One of the new opportunities offered by the FNS phenomenology lies in rejecting the above-mentioned image of a “classical instrument” in the orthodox quantum mechanics. As noted previously [18] in connection with the classical works undertaken by Voss and Clarke in the 1970s [63], analysis of noise (fluctuations of electrical potential) in solid-phase systems of different nature in the absence of any external effects (e.g., electric current) at a fixed temperature reveals the thermodynamically “nonequilibrium nature” of the measured noise. Effective thermodynamic “openness” of real solid-phase systems, disclosed in this way, may be due to local activation rearrangements of their nonequilibrium structure, redistribution of “structural energy” in solid-phase matrix fragments (for details, see [18]). Therefore, a number of questions arise in quantum mechanics in connection with the “classical instrument.” Specifically, these questions concern the extent to which (1) such effects may be significant in quantum-mechanical measurements, (2) the instrument used in these measurements may be regarded as meeting definition of “classic instrument,” and (3) an experimental study of the dynamics of a quantum subsystem and the “noises” of instruments may be helpful in “narrowing” the uncertainty intervals in description of quantum transitions via correlating the recorded scatter in the measured values with changes in the state of the “instrument.” The FNS methodology may provide answers to these questions. In author’s opinion, no reliable “quantum computer” can be developed without addressing this type of problems at the quantitative level. As regards the discussion around quantum informatics and development of quantum computers, a number of recent theoretical studies should be mentioned. Based on the general postulates of quantum theory, they led to a conclusion that the probability of observing the same quantum state in consecutive measurements on a quantum system significantly 231 exceeds that in the case of time-separated measurements, which contradicts the Schrödinger’s and Heisenberg’s postulate [64–66]. This finding may be regarded as a kind of the quantum “Zeno’s effect.” It can be suggested that, under controlled formation of the structure of a macroscopic item on the nanometer scale, it is possible to exert an effect on deceleration of local relaxation rearrangements of the functional fragments in the structure of such an item whose state changes upon the interaction with a quantummechanical subsystem, i.e., on realization of Zeno’s paradox formulated on the basis of the concept of “now” ([67], pp. 309–310), on “stopping motion.” Clearly, only experimental studies to be followed by FNS analysis of signals recorded in a macrosystem (possibly, by Voss and Clarke’s techniques) will allow tackling these problems. It should be reminded that, in his wonderful lectures on physics, delivered at the California Institute of Technology nearly fifty years, R. Feynman was very severe upon his contemporary philosophers: “These philosophers are always with us, struggling in the periphery to try to tell us something, but they never really understand the subtleties and the depth of the problem” ([68], p. 284). This opinion is fairly widespread now, in which connection it should be noted that this great physicist’s statement did not refer to the philosophical tradition of penetration into being or to those philosophy classics who were the creators of this tradition. It should rather be seen as a criterion to be satisfied by new philosophical constructions which are expected to bring philosophy “on track” from “roadside” so as it could be involved in the general process of real perception. Successful application of the FNS methodology to analyzing the dynamics of various processes and complex structures [18, 21–23, 37, 44, 47–58] suggests that such a “process has begun.” As demonstrated by examples of identifying highly individual specific features of natural, in particular, biomedical signals, the FNS phenomenology “in every detail can be subjected to the inexorable test of experiment” (see the first epigraph to this paper). The FNS phenomenology was applied to analyzing numerous temporal and spatial signals in order to derive information about evolution of a variety of open systems, features of emerging structures, and dynamics of the connections between subsystems of complex multifactor systems. This information is essential both RUSSIAN JOURNAL OF GENERAL CHEMISTRY Vol. 81 No. 1 2011 232 TIMASHEV for solving practical problems and for subsequent theoretical description of the phenomena examined. In the latter case, the derived FNS parameter sets may fulfill the function of the “first principles” and point to the “causes” from which “all other things come to be known,” as figuratively described by Aristotle (see an epigraph to this paper). Essentially, the discussed “phenomenology of complexity,” a method of perceiving the essential points of a phenomenon based on analysis of the primary digital information derived from a study of this phenomenon, appears as “metaphysics” in interpretation given by Stagiritis in the treatise on the “first philosophy” almost twenty four centuries ago. However, through the centuries that passed since then, the term “metaphysics” has changed, and after Kant it is associated with the transcendental essence existing as a necessary element in the “thing in itself” image, forming (reflecting) a unique, though essential, element in the methodology of contemplation (perception) of the essence in Husserlian terms. 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