Vita, J Thermodynam Cat 2012, 3:4
http://dx.doi.org/10.4172/2157-7544.1000e108
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On Information Thermodynamics and Scale Invariance in Fluid Dynamics
A. Di Vita*
DICAT, University of Genova, Genova, Italy
Entropy S attains a maximum in an isolated system (i.e., with
given internal energy and volume) at thermodynamic equilibrium.
According to Gibbs, the macrostate which is most likely to be observed
includes the largest number of microstates; the latter are equiprobable
because of Liouville’s theorem of mechanics.
As for steady states far from equilibrium, the amount of entropy
produced per unit time by all irreversible processes (“entropy
production”) is a minimum against simultaneous perturbations of
thermodynamical luxes and forces (Minimum Entropy Production,
MEP) in non-isolated, discontinuous systems where Onsager
symmetry holds. (A system is discontinuous if we may split it in many
mutually interacting regions, the entropy density being uniform in
each region). If either forces or luxes are ixed, then MEP is replaced
by the least dissipation principle, i.e. maximisation of the diference
between entropy production and Rayleigh’s dissipation function [1].
Unfortunately, Onsager symmetry relies on very stringent assumptions
[2] which hold e.g. in solid-state physics [3]. In continuous systems
(where entropy density is a continuously varying function of space)
Onsager holds for particular problems only, like e.g. particle difusion
coupled with chemical reactions whose chemical ainity is much
lower than thermal energy [4]. MEP predicts self-organisation in
simple networks of autocatalyic chemical reactions [5], and has been
also applied to radiative transport [6]. But MEP provides no correct
predictions in simple problems of heat transport, Ohmic heating [7]
and luid dynamics [8].
Alternative approaches are extensively investigated. Kirchhof
has shown that Ohmic heating power gets minimized in steady state
electric conductors at constant resistivity [9]. his result reduces to
Steenbeck’s principle for electric arcs [10], and has been postulated
for electron-positron plasmas in [11]. Moreover, it has been shown
[12] that viscous power is minimised in viscous, steady, Newtonian
luids [13] at low Reynolds’ number [14]. Chandrasekhar has proven
constrained minimisation of Rayleigh’s number in many problems
involving Bénard convection cells [15].
Some researchers have also postulated variational principles in
order to get closure conditions in the description of steady turbulence.
Busse invokes maximisation of the amount of momentum transported
by convection in turbulent Couette low [16]. Malkus maximizes the
total rate of energy dissipated per unit mass in steady-state, viscous,
incompressible turbulent shear lows with ixed averaged velocity
[17]. Paltridge’s maximisation of the averaged amount of entropy
produced per unit time by energy exchange between equatorial and
polar regions of Earth’s atmosphere [18] is generalised in [19,20] and
extended to further problems in turbulence in [20,21]. In contrast
with MEP, all these approaches maximise the entropy produced per
unit time, and are collectively known as MEPP (Maximum Entropy
Production Principle). Examples of MEPP may be found also outside
luid dynamics, like the “orthogonality principle” postulated in [22] for
problems of plasticity, as well as the applications to crystal growth [23].
Unfortunately, no generally accepted proof of MEPP exists to date.
Attempts to derive MEPP are unconvincing since they oten require
J Thermodynam Cat
ISSN: 2157-7544 JTC, an open access journal
introduction of additional hypotheses, which by themselves are less
evident than MEPP itself. For example, the orthogonality principle
of [22] has its statistical substantiation only if the deviation from
equilibrium is small [23]. MEPP holds for selected problems only. For
example, some transport properties in luids which satisfy MEP satisfy
maximisation criteria [24]. Kirchof’s result deies universal MEPP
validity, and it is possible to reconcile Steenbeck’s principle with MEPP
for linear laws only [25].
An approach to MEPP for selected problems involving the amount
of entropy produced per unit time by irreversible processes across the
boundaries of the system is discussed in [26] and includes the results of
[15], [17] and [21]; it has been applied to the transition to turbulence
in viscous, turbulent, incompressible Couette low [27]. In contrast, a
popular approach (‘MaxEnt’ = Maximum Entropy) is to link MEPP
with the maximisation of entropy (not of entropy production) within
the so called ‘information thermodynamics’ developed by Jaynes
[28,29]. We are going to discuss a particular application of MaxEnt to
luid dynamics below.
Maximisation of suitably deined “entropy” for steady, farfrom-equilibrium states is nothing new. In many problems where
turbulence leads to spontaneous formation and self-sustainment
of discrete, coherent structures like vortices, ilaments and the like
[30], the distribution of such structures at a given time plays the role
of Gibbs’ microstate at thermodynamic equilibrium. his formal
analogy allowed researchers to deine an “entropy”, whose maximum
corresponds to the most probable coniguration of the system as a
whole. Constraints are e.g. given by conservation of total energy, etc.
For example, steady tokamak plasmas are described as a constrained
maximum of the entropy built on a system of toroidal ilaments of
electric current, the constraint being given by a ixed value of total
electric current [31]. A similar approach to 2D magnetohydrodynamic
steady turbulence leads to a correct dimensional analysis of transport
coeicient in turbulent plasmas [32,33]. It is even possible to introduce
the equivalent of absolute temperature T, which may become
negative in case of self-organisation. However, this “temperature” is
model-dependent; this feature is in common e.g. with the ‘Extended
Irreversible hermodynamics’ [34].
MaxEnt looks for the description of the system which requires a
minimum amount of information given the constraints provided by
experiments and/or general symmetry considerations like e.g. energy
*Corresponding author: Di Vita, DICAT, University of Genova, Genova, Italy,
E-mail: Andrea.DiVita@aen.ansaldo.it
Received July 21, 2012; Accepted July 24, 2012; Published July 26, 2012
Citation: Vita AD (2012) On Information Thermodynamics and Scale Invariance in
Fluid Dynamics. J Thermodynam Cat 3:e108. doi:10.4172/2157-7544.1000e108
Copyright: © 2012 Vita AD. This is an open-access article distributed under the
terms of the Creative Commons Attribution License, which permits unrestricted
use, distribution, and reproduction in any medium, provided the original author and
source are credited.
Volume 3 • Issue 4 • 1000e108
Citation: Vita AD (2012) On Information Thermodynamics and Scale Invariance in Fluid Dynamics. J Thermodynam Cat 3:e108. doi:10.4172/21577544.1000e108
Page 2 of 3
conservation. he aim is to obtain the best estimates that can be made
on the basis of the information available, independently of any physical
argument. Dewar [35,36] derived MEPP from Jaynes’ results; his
proof is criticised in Sec. 2.3.4 of [23] and in [37], and its validity is
limited to the contributions of some irreversible processes only to the
entropy production [38]. Niven links MaxEnt and a control volume
approach [39]; Niven’s work is criticised in [26]. Dewar’s argument
[38] is similar to Gibbs’ treatment of equilibrium: the diference is that
the “entropy” is not deined by a probability measure on phase space,
but on path space. A “path” is an allowable trajectory for the system
which evolves across the phase space in time, or, in other words, a
microstate of the evolution of the system. A macrostate is a set of timedependent values of few macroscopic quantities. Many microstates
correspond to one macrostate, just like in Gibbs’ thermodynamics. he
most likely evolution of the system maximizes an “entropy”, which is
built starting from the paths. In MaxEnt, a macroscopic steady system
far from equilibrium is equally likely to follow any of its paths–see
equation “qi=s−1” page 021113-2 of [39]. At equilibrium, microstates
are equiprobable because of Liouville equation; unfortunately, no
Maxent equivalent of Liouville equation is stated explicitly. his agrees
well with Jaynes’ philosophy, but casts doubt on the physical relevance
of MaxEnt.
As an example, we discuss an application of MaxEnt to a 2D
incompressible low behind a circular cylinder with Reynolds number
100 [40]. Meaningful physics is invariant with respect to scaling
transformations, like those involved when changing the system of
units from SI to CGS [41]. Constraints like energy conservation are
automatically scale-invariant: regardless of the actual value of the total
energy ETOT, relationship E=ETOT is scale-invariant because if E → λE
(with λ ≠ 0) then also ETOT → λETOT for dimensional reasons. Physics
does not depend on λ; indeed, equation (3.17) of [40], which plays the
role of energy conservation, is scale-invariant. Now, let us suppose we
ind a coniguration which maximizes an “entropy” H while using a
system of units where λ=1. he condition H=max. implies dH/dλ=0 for
λ=1. Together, “entropy” maximization and scale invariance require:
H=max. and dH /dλ=0 for arbitrary λ
(1)
Conditions (1) are automatically satisied at thermodynamical
equilibrium, where H=S depends on E through Boltzmann’s
exponential exp(-E/kBT), kB Boltzmann’s constant; if E → λE then kBT
→ λ kBT for dimensional reasons. In contrast, the treatment of [40]
violates (1). In fact, equations (3.8), (3.14) and (3.15) of [40] show
that H depends just on a subset the coordinates of the system, namely
on those responsible for the occurrence of coherent structures in the
luid. For the sake of clarity, let us suppose H to depend just on 2
quantities a1, a2; generalization is straightforward. We follow § 5 and
equation (3.14) of [40], recall that both a1 and a2 have the dimension
of a velocity, take a2=α a12 where the constant α has the dimension
of 1/velocity, and eliminate therefore the dependence of H on a2.
p(a ,a )
According to (3.8), we write H = − da1da 2 p(a1 ,a 2 )ln 1 2 where
q(a1 ,a 2 )
∫
p and q are suitable quantities with dimension of 1/velocity2. Of course
dH/dλ=0, but what about H = max.? Taking a2=αa12 requires that we
write p(a1 ,a 2 ) = w(a1 ) ⋅∂ a 2 − α ⋅ a12 = 0 and q(a1,a 2 ) = z(a1) ⋅∂ a 2 − α ⋅ a12 = 0 ,
where both functions w and z have dimensions 1/velocity.
(
∫
(
)
)
Accordingly, H = H * , H * ≡ − da1 p(a1 ) ln p(a1 ) where we recall that
2α
a1
J Thermodynam Cat
ISSN: 2157-7544 JTC, an open access journal
q(a1 )
∂ ( f(x) ) =
∂ ( x − x0 )
df dx
if f(x = x0) = 0. he condition H = max. is therefore
x = x0
equivalent to H* = max.; but dH* /dλ ≠ 0 because H* has the dimension
of 1/velocity, and (1) is violated. MaxEnt makes the separation between
coordinates of coherent structures and other coordinates to violate
scale-invariance even if H is scale-invariant.
References
1. Gyarmati I (1970) Non equilibrium Thermodynamics. Springer Berlin.
2. Casimir HBG (1945) On Onsager’s Principle of Microscopic Reversibility. Rev
Mod Phys 17: 343-350.
3. Ichiyanagi M (1994) Variational principles of irreversible processes. Phys Rep
243: 125-182.
4. Kondraputi D, Prigogine I (1998) Modern Thermodynamics. Wiley.
5. Nieto-Villar JM, García-Fernández JM, Rieumont-Briones J (1995) The rate of
entropy production as an evolution criterion in chemical systems: I. Oscillating
reactions. Phys Scr 52: 30.
6. Kroell WJ (1967) Quantitative Spectroscopy and Radiative Transfer. 715-723.
7. Jaynes ET (1980) The Minimum Entropy Production Principle. Ann Rev Phys
Chem 31: 579-601.
8. Bertola V, Cafaro E (2008) A critical analysis of the minimum entropy
production theorem and its application to heat and luid low. Intl J A Heat and
Mass Transfer 51: 1907–1912.
9. Hermann F (1986) Eur J Phys 7: 130.
10. Steenbeck M (1940) Wissenschaftlichen Veroeffentlichungen aus den
Siemens. Werke 1: 59
11. Bhattacharyya R (2003) Phys Lett A 315: 120–125.
12. Lamb H (1906) Hydrodynamics Cambridge.
13. Astarita GJ (1977) of Non-Newtonian Fluid Mechanics 2: 343-351.
14. Takaki R (2007) Variational principle to derive the Stokes equations. Fluid
Dynamics Research 39: 590–594.
15. Chandrasekhar S (1961) Hydrodynamic and Hydromagnetic Stability. Oxford
University Press New York.
16. Busse FHJ (1970) Fluid Mech 41: 219-240.
17. Malkus WVR (1956) Outline of a theory of turbulent shear low. J Fluid Mech
1: 521-539.
18. Paltridge GW (1979) Climate and thermodynamic systems of maximum
dissipation. Nature 279: 630-631.
19. Kleidon A, Phil Trans R Soc B (2010) 365: 1303-1315.
20. Ozawa H, Ohmura A, Lorenz RD, Pujol T (2003) The second law of
thermodynamics and the global climate system: A review of the maximum
entropy production principle. Rev Geophys 41:1018.
21. Ozawa H, Shimokawa S, Sakuma H (2001) Thermodynamics of luid
turbulence: A uniied approach to the maximum transport properties. Phys Rev
E 64: 026303.
22. Ziegler H (1983) Chemical reactions and the principle of maximal rate of
entropy production. ZAMP 34: 832-844.
23. Martyushev LM, Seleznev VD (2006) Maximum entropy production principle in
physics, chemistry and biology. Phys Rep 426: 1-45.
24. Lewalle J (1987) Phys Lett A 122: 338-340.
25. Christen T (2009) Modeling Electric Discharges with Entropy Production Rate
Principles. Entropy 11: 1042-1054.
26. Di Vita A (2010) Phys Rev E 81: 041137.
27. Monokrousos (2011) A PRL 106:134502.
Volume 3 • Issue 4 • 1000e108
Citation: Vita AD (2012) On Information Thermodynamics and Scale Invariance in Fluid Dynamics. J Thermodynam Cat 3:e108. doi:10.4172/21577544.1000e108
Page 3 of 3
36. Dewar RC (2005) Maximum entropy production and the luctuation theorem. J
Phys A 38: L371.
28. Jaynes ET (1957) Phys Rev 106: 620.
29. Jaynes ET (1957) Phys Rev 108: 171.
30. Brown MR (1997) J Plasma Physics 57: 203-229.
37. Grinstein G, Linsker R (2007) Comments on a derivation and application of the
‘maximum entropy production’ principle. J Phys A: Math Theor 40: 9717.
31. Minardi E, Lampis G (1990) Maximum entropy Tokamak conigurations.
Plasma Phys Contr Fus 32: 819.
38. Bruers S (2007) A discussion on maximum entropy production and information
theory.
32. Taylor JB (1997) Plasma Phys Contr Fus 39 A1.
39. Niven RK (2009) Steady state of a dissipative low-controlled system and the
maximum entropy production principle. Phys Rev E 80: 021113.
33. Taylor JB (1991) Phys Fluids 17: 1492.
34. Jou D, Casas-Vàzquez J, Lebon G
thermodynamics. Rep Prog Phys 51: 1105.
(1988)
Extended
irreversible
35. Dewar R (2003) Information theory explanation of the luctuation theorem,
maximum entropy production and self-organized criticality in non-equilibrium
stationary states. J Phys A 36: 631.
40. Noack BR, Niven RK (2012) Maximum-entropy closure for a Galerkin model of
an incompressible periodic wake. J Fluid Mech 700: 187-213.
41. Qian JJ (1996) Phys A: Math Gen 29: 1305-1309.
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Volume 3 • Issue 4 • 1000e108