Mathematical Theory of Nonequilibrium Steady States: On the Frontier of Probability and Dynamical Systems

276 Pages·2004·2.699 MB·English

by  Da-Quan Jiang, Min Qian, Min-Ping Qian (auth.)

Introduction
This book addresses the issue of a systematic mathematical exposition of the
conceptualproblemsofnonequilibriumstatisticalphysics,especiallythosere-
lated to the second law of thermodynamics, which in the alternative form for
open systems means positivity of entropy production and related topics. As
pointed out by Ya. G. Sinai [461, pp. 207] in 1994, the problem concerning
the irreversibility of nonequilibrium systems was not yet considered mathe-
matically with appropriate generality. The following passage from the review
article by D. Ruelle [430] in 1999 describes well enough the present status of
nonequilibrium statistical physics:
“Statistical mechanics, which was created at the end of the 19th century by
such people as Maxwell, Boltzmann and Gibbs, consists of two rather differ-
ent parts: equilibrium and nonequilibrium statistical mechanics. The success
of equilibrium statistical mechanics has been spectacular. It has been developed
to a high degree of mathematical sophistication 1, and applied with success to
subtle physical problems like the study of critical phenomena. ... By contrast,
theprogressofnonequilibriumstatisticalmechanicshasbeenmuchslower.We
still depend on the insights of Boltzmann for our basic understanding of irre-
versibility,andthisunderstandingremainsratherqualitative.Furtherprogress
has been mostly on dissipative phenomena close to equilibrium: Onsager reci-
procity relations, Green-Kubo formula, and related results. ...”
Atypicalmacroscopicsysteminoureverydaylifeconsistsofanenormous
number(orderof1023)ofmicroscopicelements,suchasmoleculesandatoms.
The fundamental problem in nonequilibrium statistical physics is to explain
theprevalentirreversiblephenomenaofthesemacroscopicsystemsontheba-
sis of reversible microscopic evolution, and to give quantitative predictions,
especially for dissipative systems. The solution to this problem begin with
Boltzmann’s derivation, in 1872, of what are now known as the Boltzmann
1 A highly condensed book on this respect which should be further digested by
mathematical physicists is Ruelle’s Thermodynamic Formalism. — note by the
authors of this book.
D.-Q.Jiang,M.Qian,andM.-P.Qian:LNM1833,pp.1–10,2004.
(cid:1)c Springer-VerlagBerlinHeidelberg2004
2 Introduction
transportequationandH-theorem[36,159,273].TheBoltzmannequationde-
termines the evolution of the velocity distribution function of molecules in a
dilute gas. In terms of the distribution function, a function of time, H(t) can
be defined. It decreases monotonically in time and reaches a constant value
when the velocity distribution function reaches the Maxwell-Boltzmann equi-
libriumdistribution.Inthisequilibriumsituation,theH-functionturnsoutto
be the minus thermodynamic entropy for an ideal gas, up to the Boltzmann
constant. However, there were objections to Boltzmann’s derivation based
upon the time reversal invariance of Newton’s equations of motion, called
Loschmidt’sparadox,orthroughthePoincar´erecurrencetheorem,calledZer-
melo’s paradox. The problem of deriving macroscopic irreversibility from mi-
croscopic reversibility, or more generally, the “arrow of time” problem is far
from being clear even physically, not to say mathematically and philosophi-
cally [71,222,358,369].
Now two correlated mathematical approaches are adopted to deal with
problems in nonequilibrium statistical physics; one is to model physical sys-
tems by stochastic processes and the other by deterministic or random dy-
namical systems. Chapters 1–6 of this book are devoted to the mathematics
in the modelling using stochastic processes, including Markov chains with
discrete or continuous time parameter and diffusion processes in Euclidean
spaces or on Riemannian manifolds; While Chapters 7–9 are devoted to that
related with dynamical systems, including deterministic hyperbolic dynam-
ical systems and random hyperbolic dynamical systems. The main subjects
that we are interested in are different from those related to H-theorem, and
we only investigate the nonequilibrium steady state, which is the chief point
emphasized by I. Prigogine [188,344] (see next section for more details).
According to classical mechanics, the microscopic evolution of a macro-
scopic system is characterized by a set of equations of motion of the micro-
scopic elements making up of the system. This can be taken as the origin of
the dynamical systems approach, which will be described in Sect. 0.2 of this
introduction.
Although the true mechanics of the microscopic world is quantum, in this
book we do not touch upon quantum mechanics, but only list in the biblio-
graphy some works on the irreversibility and entropy production of quantum
systems(see[245–247,290,291,432]).Webelievethatlogicallythelinkingbe-
tween the microscopic and macroscopic worlds does not depend much upon
themechanicsgoverningtheformer,andourmathematicalresultsaboutclas-
sical systems can be extended to quantum systems.
0.1 Approach of Stochastic Processes
The idea of the stochastic-process approach can be traced back to Einstein’s
celebrated work on Brownian motion in 1905 [115], and even back to the
0.1 Approach of Stochastic Processes 3
derivationoftheBoltzmannequation.Inordertodescribeamacroscopicsys-
temconsistingofanenormousnumberofmicroscopicelements,theextremely
complexanduntraceablemotionofthemicroscopicelementshastobestatis-
tically projected onto a much smaller number of macroscopic or mesoscopic
variables. In other words, to obtain the macroscopic or mesoscopic descrip-
tion of the system starting from the fundamental microscopic equations of
motion, spatial, temporal, or spatiotemporal coarse graining has to be intro-
duced; then a stochastic process is obtained and the stochastic ingredients
of the description come from the incompleteness of information. (See Kubo,
etc. [273] for detailed discussions.) For example, the Brownian motion is the
projection of microscopic motion of a pollen particle together with all the
molecules of the surrounding liquid onto the dimension of the motion of the
pollenparticleonly.Thereisnoabsolutevacuuminnature,soanysystemhas
to interact with its environment. Therefore, a complete description is impos-
sible except for idealization. The stochastic ingredients can also represent the
spontaneous fluctuations of the macroscopic variables due to the thermal ag-
itation of the microscopic elements [310,348–350], which is different from the
noiseperturbationsonthesystembytheenvironment.Nowstochasticmodels
arewidelyusedinphysics,chemistry,biologyandevenineconomics.However,
thederivationofmacroscopicirreversibilitybasedoncoarsegrainingmaylead
tothedelusionthatthevalidityofthesecondlawofthermodynamicsdepends
on the techniques of physicists who carry through the experiments or obser-
vations, and that irreversibility is due to the incompleteness of information.
Such a delusion is strongly opposed by I. Prigogine and his school [369].
The researches on irreversible systems close to equilibrium via stochastic
processes, especially Gaussian processes arose from the fluctuation problem
of Brownian motion. Since Einstein [115] put forward the physical model of
Brownianmotion,thetheoryofBrownianmotionwasfurtherdevelopedbyP.
Langevin, M. Smoluchowski, G.E. Uhlenbeck, L.S. Ornstein [479] and many
others;Wiener,Itoˆ,etc.didmanymathematicalresearches,andatthebegin-
ning of 1950’s gave birth to the theory of diffusion processes and stochastic
differential equations [241,259,409,467]. In the meantime, Onsager [348,349]
exploited stochastic processes, especially Gaussian processes to discuss sys-
tematically irreversible thermodynamics for systems close to equilibrium and
in the linear response regime; In 1953, he and Machlup [310,350] proposed
the Onsager-Machlup principle, which is actually a functional formula about
the probability density of a stochastic process. Such kind of formulas can
also be generalized to systems in the nonlinear response regime [193–196].
See [28,29,52,86–89,147,174,309] for some recent researches related to the
Onsager-Machlup theory.
The researches on irreversible systems far from equilibrium began with
the works by Haken [210,211] about laser and Prigogine, etc. [188,344] about
oscillationsofchemicalreactions.Butwhataresystemsfarfromequilibrium?
Nicolis and Prigogine [344] argued that they arise from equilibrium via bi-
furcations. However, only for deterministic dynamical systems does bifurca-
4 Introduction
tion have clear meaning. Therefore, Prigogine’s explanation of phenomena
far from equilibrium is descriptive. Nicolis and Prigogine [344] also regarded
that a nonequilibrium system is a stationary open system with positive en-
tropy production rate, which means exchange of substances and energy with
its environment. Then they put forward the concept of dissipative structure
to denote the macroscopic periodic phenomena originating from the cooper-
ation of the subsystems in the nonequilibrium situation. They called ordered
such periodic phenomena, including spatial and temporal ones, and named
their existence “self-organization”. If one only considers the temporally or-
dered phenomena, then one can use Markov chains and diffusion processes as
mathematical tools to model nonequilibrium states.
To discuss ordered phenomena, the first step is to distinguish equilibrium
andsteadynonequilibriumstates.Inphysics,theconditionofdetailedbalance
was already known by Boltzmann; the reversibility of a Markov chain was in-
troduced by Kolmogorov; it is no accident that the mathematical essence of
these two concepts actually turns out to be the same. As is well known, ther-
modynamic equilibrium is in general maintained through detailed balance; so
what characteristic should a nonequilibrium steady state have? The answer
is contained in the definition of irreversibility and a theorem revealing the
appearance of circulations, which is the chief point in [257,391,400,405,406].
This result can be most clearly expressed by the circulation decomposition
theorem in the case of Markov chains. The trajectories of an ergodic recur-
rentMarkovchaincompletecyclesincessantly.Thecirculationdecomposition
theoremtellsthattheprobabilityfluxbetweeneachtwostatescanbedecom-
posed into two parts: the part of detailed balance and that of circulation. If
onlythepartofdetailedbalanceappears,thentheMarkovchainisreversible;
otherwise, there is a net circulation on at least one cycle and the chain is ir-
reversible. We can also introduce the concept of entropy production rate for
Markov chains. The entropy production rate of a stationary Markov chain
can be expressed in terms of the circulations on its cycles. A Markov chain
with net circulations has positive entropy production rate, and vice versa. So
Markov chains with net circulations can be taken as models of systems far
from equilibrium, and the appearance of net circulations can be regarded as
an ordered phenomenon.
The above results can be applied to explain some biochemical phenomena
closelyrelatedtothestudyofpolymers.Biochemicalreactionsareapparently
irreversiblewithrespecttotime.Onecanevensaythatirreversibilityisoneof
the chief characteristics of life activities. Although it is not common in text-
booksofphysicsandchemistrytostatethatnonequilibriumsteadystatesare
maintained via circulation balance, the appearance of cycles in biochemistry
iscertainlyanestablishedfact.Hill’stheory[223–226]offreeenergytransduc-
tion in living organisms supplies a convincing example. Since 1966, T.L. Hill,
etc. constructed a general mesoscopic model for the combination and trans-
formation of biochemical polymers in vivid metabolic systems. Their results
can be applied to explain the mechanism of muscle contraction and active
0.1 Approach of Stochastic Processes 5
transport, such as the Na and K ions actively transferring and penetrating
throughorganicmembranesintheHodgkin-Huxleymodel[223,227].Hill’sba-
sic method is diagrammatic. Then appeared Schnakenberg’s works [439,440]
which are close to those of Hill, but the emphasis is on general principles and
thedefinitionofentropyproduction.OnecantakeastationaryMarkovchain
as the mathematical model of Hill’s theory on cycle fluxes. Each state of the
Markovchaincorrespondstoamesoscopicstateofpolymers.Andonewillsee
thatHill’scyclefluxesareequivalenttothecirculationratesinthecirculation
decomposition theorem of Markov chains [257,400,405,406].
In irreversible processes of free energy transduction, there must exist con-
comitant dissipation of free energy. Hill’s model relevant to biochemical phe-
nomena is a completely analyzed example of mesoscopic dissipative systems
withorderedphenomena.Theentropyproductioninthesedissipativesystems
is just the dissipation of free energy. Given a stationary Markov chain mod-
ellingthecombinationandtransformationofbiochemicalpolymers,writethe
stationary distribution as
e−Fi/kBT
π = (cid:1)
i e−Fj/kBT
j
with i in the state space, then under the condition of detailed balance, F is
i
just the free energy of the system in state i; the transition from state i to j
results in the free energy dissipation F −F . But in the irreversible case, this
i j
kindoftransitionscanresultintheemissionofenergy,whichmaycorrespond
tothephenomenonofbiologicalfluorescence.WereferthereadertoH.Qian’s
work[377,378,380]fordetaileddiscussiononfreeenergydifferenceassociated
with equilibrium fluctuations and nonequilibrium deviations.
According to the original equilibrium assumption in the Hodgkin-Huxley
model, the power spectrum of electronic current should be Lorentz-typed,
but H.M. Fishman [56,138] observed biased peaky power spectrum in the
experiment on the axon of squid. Hill, etc. pointed out that their circulation
modelcanbeusedtoappropriatelyexplainsuchphenomena.Hereweremark
that stochastic resonance marked by the indispensable biased peaky power
spectrum [25,26,96,142,166,255,339,384,389,390,394,395], and molecular
motors marked by the unidirectional circulation (current) (see [12,256,373,
378,408]andreferencestherein),arebothirreversiblephenomenaofstochastic
systems.
Chapters 1 and 2 of this book are mainly dedicated to providing a firm
mathematical foundation for Hill’s theory in the situations of Markov chains
with discrete or continuous time parameter. For a stationary Markov chain
with discrete time parameter, in Chapter 1 we obtain the formulas for cir-
culations on cycles via introducing its so-called “derived chain”, prove the
circulation decomposition theorem, define its entropy production rate, all in
the measure-theoretic sense. We get the entropy production formula in terms
of the circulations, and prove that the chain is reversible if and only if its en-
tropyproductionratevanishes,orifftherearenonetcirculations.InChapter
6 Introduction
2,parallelresultsareobtainedforastationaryMarkovchainwithcontinuous
time parameter via its embedded chain. We also prove the so-called fluctua-
tion theorem for Markov chains (see Lebowitz and Spohn [286]): The sample
entropyproductionrateshavealargedeviationpropertyandthelargedevia-
tion rate function has a symmetry of Gallavotti-Cohen type. See Sect. 0.2 for
more detailed discussion about the fluctuation theorem.
The existing theory on the existence and uniqueness of diffusion processes
assolutionstostochasticdifferentialequationsusuallyimposeveryrestrictive
conditions on the diffusion and drift coefficients, however, most of the inter-
esting applications of stochastic differential equations could not meet these
requirements. So, in order to get the results in Chapters 1 and 2 in the case
of diffusion processes, in Chapter 3, we construct general minimal diffusion
processes in Euclidean space Rd by the approach of semigroups and partial
differential equations, and prove the weak Foguel Alternatives [281]. In case
the minimal diffusion process has an invariant probability measure, we give
by a heuristic argument, a rigorous definition of the entropy production rate
andprovethatthediffusionprocessisreversibleifandonlyifitsinfinitesimal
generator is self-adjoint, or iff its entropy production rate vanishes, etc.
In Chapter 4, we first give a measure-theoretic definition of the entropy
production rate (as the expectation of the logarithm of a Radon-Nikodym
derivative)ofastationarydiffusionprocessandderivetheentropyproduction
formula, which is heuristically obtained in Chapter 3, from the Cameron-
Martin-Girsanov formula. (Here, in this respect, we would like to mention
the papers [313–316] by C. Maes and his collaborators, where interacting
particle systems are also discussed.) Then we give a probabilistic definition
of the “flux” (current velocity) of a diffusion process as was considered by
Nelson[337][189,Chap.6].Lastly,weprovetheEinsteinrelationforreversible
diffusionprocesses,andtheGreen-KuboformulaforgeneralreversibleMarkov
processes.
DuetothetrivialtopologyofRd,therearenodiscretecirculationsassoci-
ated to diffusion processes on Rd. In Chapter 5, we consider an arbitrary dif-
fusionprocess(driftedBrownianmotion)onacompactRiemannianmanifold
M. We define its entropy production rate measure-theoretically and give the
entropyproductionformula.Weprovethetheoremssimilartothoseobtained
in Chapter 3. Furthermore, the entropy production rate can be decomposed
into two parts—in addition to the first part analogous to that of a diffusion
process on Rd, some discrete circulations (or say, rotation numbers) intrinsic
to the topology of M appear! The first part is called the hidden circulation
and can be explained as the circulation of a lifted process on M ×S1 around
the circle S1. The entropy production rate can be expressed as a linear sum
of its rotation numbers around elements of the fundamental group of M and
the hidden circulation, very similar to the discrete case of Markov chains. We
also prove that the diffusion process is reversible if and only if the hidden
circulation and the rotation numbers all vanish.
0.2 The Dynamical Systems Approach 7
InChapter6westudyaspecialsystemofN-coupledoscillatorswithwhite
noise.Wegetapositiverecurrentdiffusionprocessbywindingthesolutionof
thissystemonacylinderalongaspecialdirection.Bythisway,weprovethat
thelimitsofthefrequenciesoftheoscillatorsexist;moreover,theyareidentical
and independent of the initial values, no matter how large the white noise is.
Thismeansthatthesystemhasthepropertyofbeingfrequency-locked.Thus
we can define the rotation number of the system as the common limit of
the frequencies of the oscillators. The winded process can be regarded as an
example of diffusion processes on non-compact Riemannian manifolds, and
the rotation number of the system of N-coupled oscillators can be regarded
as the counterpart of those in Chapter 5 for diffusion processes on compact
Riemannian manifolds.
We point out now that the entropy production rate of each stochastic
process considered in Chapters 1–5 is consistently and measure-theoretically
defined as the specific relative entropy of the probability distribution of the
process on the path space with respect to that of its time reversal, although
the entropy production formulas are different in various concrete cases. For
the system modelled by the stochastic process, the specific relative entropy
describesthedifferencebetweentheforwardevolutionandthebackwardone,
therefore, the entropy production rate characterizes the macroscopic irre-
versibility of the system.
0.2 The Dynamical Systems Approach
Now we enter into the dynamical-system approach to nonequilibrium statis-
tical physics. In 1973 Ruelle made a suggestion that it might be possible
to develop a general theory for nonequilibrium stationary systems by apply-
ing the theory of smooth dynamical systems, which was written down later
in [420,424]:
“If one is optimistic, one may hope that the asymptotic measures will play
for dissipative systems the sort of role which the Gibbs ensembles played for
statistical mechanics. Even if that is the case, the difficulties encountered in
statistical mechanics in going from Gibbs ensembles to a theory of phase
transitions may serve as a warning that we are, for dissipative systems, not
yet close to a real theory of turbulence.” [420]
ThisproposalisveryambitiousandsuggeststhatSRBmeasuresinthetheory
of smooth dynamical systems should be the ensembles that describe steady
states of macroscopic systems, whether in equilibrium or not. In the last
decade, there appeared many attempts to connect the chaotic microscopic
dynamics of particle systems to the macroscopic properties of systems in
nonequilibriumsteadystates,viathetheoryofdynamicalsystems.J.R.Dorf-
man [101], P. Gaspard, T. Gilbert [102–104,170,171,175–178,184,218], G.
Nicolis [179,268], D.J. Evans, G.P. Morriss [123,332], W. Breymann, T. T´el,
8 Introduction
J. Vollmer [47,321,322,485–489], etc. try to relate the transport properties
of irreversible processes to the characteristic quantities of chaos such as the
Lyapunov exponents, the Kolmogorov-Sinai entropy, the escape rate, and the
fractal dimensions, seeking to incorporate Irreversible Thermodynamics into
the framework of dynamical systems theory.
In 1995, Gallavotti and Cohen [150,162,163] developed Ruelle’s idea and
proposedthechaotichypothesisthatforthepurposeofstudyingmacroscopic
properties, the time evolution of a many-particle system in a stationary state
can be regarded as a transitive Anosov system. Under this hyperbolicity as-
sumption, they obtained the fluctuation theorem [149,163], which is the first
oneamongthephysicalpioneeringworksonnonequilibriumstatisticalphysics
by the approach of dynamical systems. It says that the probability distribu-
tions of the phase space contraction averaged over large time spans have a
large deviation property, and the large deviation rate function has a symme-
try. (The phase space contraction rate has been identified with the entropy
production rate [163,425].) From then on, Gallavotti [150,162,163] and Ru-
elle [425,427,430], etc. use smooth dynamical systems or smooth random
dynamicalsystemstomodelchaoticsystemsinstatisticalphysics,whetherin
equilibriumornot.Ruelle[430]reviewedvariousapplicationsofthetheoryof
smooth dynamical systems to conceptual problems of nonequilibrium statis-
tical mechanics. Their emphasis is on understanding nonequilibrium steady
states themselves, which are described by SRB measures. Their idea is using
SRBstatestomakeinterestingphysicalpredictions,andthestronghyperbol-
icity condition is assumed to prove the Gallavotti-Cohen fluctuation theorem
and derive a general linear response formula. For systems near equilibrium,
they recover, in particular, the Onsager reciprocity relations and the Green-
Kubo formula.
Historically, early in 1993, Evans, Cohen and Morriss [121] found in com-
puter simulations that the natural invariant measure of a stationary nonequi-
librium system has a symmetry, and by a general formula, gave the proba-
bility ratio of observing trajectories that satisfy or violate the second law of
thermodynamics. This might be the origin leading to the work by Gallavotti
and Cohen [149,163], which contained the first mathematical presentation
of the fluctuation theorem. Many papers then appeared in its wake. Evans
andSearles[14,125–127,444–446]consideredtransient,ratherthanstationary,
nonequilibriumsystemsandemployedaknownequilibriumstate(suchasthe
Liouville measure) as the initial distribution to derive a transient fluctuation
theorem. Gallavotti [157] and Evans, et al. [15,446] proposed a local version
of the fluctuation theorem. Kurchan [276] pointed out that the fluctuation
theorem also holds for certain diffusion processes. Lebowitz and Spohn [286]
extendedKurchan’sresultstoquitegeneralMarkovprocesses,andMaes[311]
thought of the fluctuation theorem as a property of space-time Gibbs mea-
sures. Searles and Evans [443] derived informally the transient fluctuation
theorem for non-stationary stochastic systems.
0.2 The Dynamical Systems Approach 9
For systems close to equilibrium, the fluctuation theorem yields the well-
known Green-Kubo formula and the Onsager reciprocity relations [150,151,
286,311,430], i.e. the symmetry of the transport coefficients matrix which re-
late thermodynamic “forces” and “fluxes”. Surprisingly, the fluctuation theo-
remisalsovalidforsystemsinthenonlinearresponseregimefarfromequilib-
rium. In this sense, it can be thought of as an extension, to arbitrarily strong
externalfields,ofthefluctuation-dissipationtheorem,whichholdsforsystems
in the linear response regime close to equilibrium.
As for entropy production rate, Andrey [7] and Ruelle [425] gave the def-
inition respectively for deterministic dynamical systems with continuous and
discrete time parameter from the physical point of view. Ruelle [425] calcu-
lated the rate of change of the Gibbs entropy for a system with an initial
state described by an absolutely continuous measure on the phase space, and
defined the entropy production rate of the system in the steady state de-
scribed by an SRB measure as the limit of the minus changing rate. He [427]
also defined entropy production rate similarly for smooth random dynamical
systems.FromRuelle’sdefinition,onecannotseedirectlytherelationshipbe-
tween the entropy production rate and the macroscopic irreversibility of the
dissipative system, as is the case for stochastic processes. Naturally, one may
ask whether Ruelle’s definition has any measure-theoretic basis, and whether
the entropy production rate thus defined and the one defined for stochastic
processes are essentially in the similar spirit. As we will see, the answer is
positiveinthecaseofdeterministicorrandomhyperbolicdynamicalsystems.
As is well known, one can study the dynamical behaviors of an Axiom
A system by studying those of its symbolic representation (subshift of finite
type), which is obtained via Markov partition (coarse graining) (cf. [43,457]).
In Chapter 7 we introduce the concept of specific information gain (or say,
specific relative entropy) for subshifts of finite type and Axiom A systems.
For a basic set ∆ of a C2 Axiom A diffeomorphism (M,f), let µ+ and µ−
be respectively the generalized SRB measures for f and f−1 on ∆. We com-
pare the entropy production rate e (f,µ ) defined by Ruelle to the specific
p +
information gain h(µ+,µ−) of µ+ with respect to µ−. In the special case of
Anosovdiffeomorphisms,thetwoquantitiesh(µ+,µ−)andep(f,µ+)coincide,
and moreover, ep(f,µ+) = 0 if and only if µ+ = µ−, or if and only if µ+ is
absolutely continuous with respect to the Lebesgue measure on M. In the
general case, Ruelle’s definition needs to be modified due to “diffusion”; from
the measure-theoretic point of view, h(µ+,µ−) can be adopted as a modified
definitionoftheentropyproductionrateoff| inthestationarystateµ .As
∆ +
h(µ+,µ−)describesthedifferencebetweentheforwardevolution(M,f|∆,µ+)
and the backward one (M,f−1|∆,µ−), one can say that the entropy produc-
tion rate characterizes the degree of macroscopic irreversibility of the system.
In this chapter, we also give a short and strict proof of the Gallavotti-Cohen
fluctuation theorem after presenting the level-2 large deviation property of
the Axiom A diffeomorphism (M,f).
10 Introduction
In Chapter 8 we prove that for an attractor ∆ of a C2 Axiom A diffeo-
morphism (M,f), Lebesgue-almost every point x in the basin of attraction
Ws(∆) of the attractor ∆ is positively regular, and the Lyapunov exponents
of (f,Tf) at the point x are the same as those of (f,Tf) with respect to
theSRBmeasureon∆.Similarresultholdstruefornonuniformlycompletely
hyperbolic attractors with SRB measures. This “large ergodic property” of
Lyapunov exponents w.r.t. SRB measures justifies the choice of initial points
close to attractors uniformly with respect to Lebesgue measures which facili-
tatestheapproximatecomputationofLyapunovexponents(andthenentropy
productionrates)inphysicalapplications.Ingeneral,thehyperbolicattractor
hasafractalstructureandtheSRBmeasureonitissingular,sotheLebesgue
measure is a much more useful reference measure for sampling than the SRB
measure.
Chapter 9 is devoted to the measure-theoretic exposition of the entropy
productionrateofsmoothrandomdynamicalsystemsdefinedbyRuelle[427]
from the physical point of view. We introduce the concept of specific infor-
mation gain (or say, specific relative entropy) for random subshifts of finite
typeandrandomhyperbolicdynamicalsystemswhicharegeneratedbysmall
diffeomorphism-typeperturbationsofanAxiomAbasicset.Letµ+andµ−be
respectively the generalized SRB measures for such a random hyperbolic dy-
namicalsystemG anditstimereversalG−1.Wecomparetheentropyproduc-
tionrateep(G,µ+)definedbyRuelletothespecificinformationgainh(µ+,µ−)
of µ+ with respect to µ−. Then a generalization of the results in Chapter 7
is obtained. A random version of the result in Chapter 8 also holds true for
random hyperbolic systems arising from small perturbations of an Axiom A
attractor.
Althoughtheentropyproductionratesofthestochasticprocesses,andthe
deterministic or random hyperbolic dynamical systems have different expres-
sionsasgivenbySchnakenberg[439],theQians[385,401–403],Gallavottiand
Ruelle[163,425,427],etal.,theyallcanbemeasure-theoreticallyexpressedas
thespecificrelativeentropybetweentheforwardandthebackwardevolution,
as is shown in this book. The entropy production rate of a stationary system
vanishes if and only if the system is reversible and in equilibrium.