Download Ruelle.pdf

Ann. Henri Poincaré 4, Suppl. 1 (2003) S489 – S495
c Birkhäuser Verlag, Basel, 2003
1424-0637/03/01S489-7
DOI 10.1007/s00023-003-0938-6
Annales Henri Poincaré
Is There a Unified Theory
of Nonequilibrium Statistical Mechanics?
David Ruelle
Abstract. This is a discussion of a certain number of questions that occur naturally in
the perspective of creating a unified theory of nonequilibrium statistical mechanics.
Introduction
As to the question in the title of this talk, the answer is currently: NO! There
is a fairly satisfactory theory of nonequilibrium close to equilibrium, with the
Onsager reciprocity relations, and the Green-Kubo formula, and many important
results have been obtained recently far from equilibrium, but not a unified theory.
In fact, nonequilibrium covers a variety of different physical phenomena, such as
hysteresis and decay of metastable states, and one does not really expect a unified
description for all those. But in situations like the study of nonequilibrium steady
states, one might hope for a unification of the different idealizations currently
studied (small system interacting with stochastic reservoirs, small system with
isokinetic thermostat, etc.). This unification has up to now not been attained, and
one may thus be concerned that perhaps some of the idealizations used are not
doing justice to the physics of the problem.
But let us not be too pessimistic. We do know a certain number of things
about the physics of nonequilibrium, and we also have some ideas of where to look
in order to progress. This is what I would like to discuss in this talk, and also why
one would like to have a more conceptual and unified theory.
One thing that characterizes nonequilibrium is (strictly positive) entropy production. It may be surprizing that one can speak of entropy production even though
entropy itself is not very well defined far from equilibrium. Indeed there have been
speculations that, because biological phenomena are far from equilibrium, they
could defeat the second law. In fact if one considers any process starting and ending close to equilibrium (however far one may go in between), the entropy at the
end is bigger than at the beginning (if this were not the case on could build a
device that would convert heat into mechanical work, an the evidence is against
the existence of such a device). So, we can speak of entropy production, and compute it in different idealizations of nonequilibrium far from equilibrium. This does
not mean that we have a good understanding of all the situations of interest. In
particular we shall want to come back to the discussion of biological phenomena
S490
D. Ruelle
Ann. Henri Poincaré
which are currently treated by ad hoc methods that remain unsatisfactory from a
fundamental point of view.
One mathematical description of entropy production is as en increase of the
volume of phase space “occupied by the system”. The idea is that microscopic
states that are indistinguishable macroscopically occupy a volume of phase space
that increases with time. Let us give only the briefest discussion of this somewhat
messy subject, where there is still disagreement between experts. The problem
is that Hamiltonian evolutions preserve phase space volume. One way to have
nevertheless an apparent volume increase is that the (Hamiltonian) time evolution
spreads an initial blob of phase space thinly so that it becomes dense in a larger
volume of phase space. But this leads one to introduce “coarse graining” which
is hard to handle in a mathematically satisfactory manner. One is then led to
studying infinite Hamiltonian systems, which are technically difficult and for which
the concept of volume no longer makes natural sense. For such infinite Hamiltonian
systems the entropy production arises from the fact that there are correlations in
the description of the system which, in the course of time, become spread out over
larger and larger regions of space, to be finally lost “at infinity”.
A mathematically convenient way to obtain nontrivial entropy production
is to use the non-Hamiltonian dynamics corresponding to one of the “Gaussian
thermostats” introduced by Hoover and Evans, for instance the isokinetic (IK)
thermostat. Here one subjects a finite Hamiltonian system (“small system”) to
an non-Hamiltonian force, and and adds a deterministic term corresponding to a
“thermostat”1 . The small system gives entropy to the thermostat, so that there is
volume contraction in the (compact) phase space of the small system. The average
rate of phase space contraction of the phase space volume of the small system is
in fact the average entropy production. It may be computed for a nonequilibrium
steady state represented by a singular measure invariant under time evolution.
Gaussian thermostats were introduced for the purpose of computer simulations,
but turn out to be very well suited to theoretical investigations. Not everyone is
happy with their physical interpretation, but they are mathematically very convenient, and allow an in depth discussion of otherwise inaccessible questions. Among
the results obtained is the important Fluctuation Theorem of Gallavotti and Cohen
[4], other developments are reviewed in [9].
While this is not the place to give an extensive bibliography we must also
mention here the rigorous results of Eckmann and coworkers [3] on a classical
small system (anharmonic chain) coupled to infinite reservoirs. The difficulty of
the mathematical problems encountered in nonequilibrium is illustrated by the
fact that no proof of Fourier’s law has been obtained at this time. Quantum spin
systems spin systems coupled to infinite reservoirs are relatively accessible mathematically, and have been studied by Ruelle [11], and by Jakšić and Pillet [6].
1 for the casual reader, let us insist that Gaussian thermostats have nothing to do with Gaussian stochastic forces, the Gaussian thermostat is strictly deterministic, as we shall see below in
the example of the IK thermostat.
Vol. 4, 2003
Is There a Unified Theory of Nonequilibrium Statistical Mechanics?
S491
Let us now look at the dynamical processes that can lead to entropy production. One has of course tried to study heat conduction through a chain of harmonic
oscillators, and the results obtained are pathological: Fourier’s law is violated. Why
is simple dynamics inadequate for nonequilibrium? Using the formalism of Gaussian thermostats where entropy production is the rate of volume contraction, or
equivalently minus the sum of all Lyapunov exponents, one sees that nontrivial
entropy production entails some non zero Lyapunov exponents, which means some
kind of dynamical chaos. Completely integrable systems, like harmonic chains, are
thus excluded. Chaotic dynamics appears thus to be linked with irreversible behavior. This remark applies without regard to the spatial extension of the system.
For systems with spatial extension, irreversibility is typically linked with diffusive
behavior. This is perhaps best seen in the parabolic character of the macroscopic
evolution equations (for material diffusion, heat propagation, of hydrodynamics).
At the microscopic level, diffusive behavior appears in the study of quantum spin
systems where the dynamics is deterministic. But microscopic diffusive behavior
also arises in classical systems as a consequence of chaos, because the dynamically
unstable time evolution approximates a stochastic motion.
We are loosely using the terms “nonequilibrium statistical mechanics” and
“irreversibility” to describe the microscopic behavior underlying the well known
macroscopic irreversible behavior described by nonequilibrium thermodynamics.
The microscopic dynamics may be chaotic, diffusive, or stochastic, and we have
seen the physical relations between these different types of dynamics. But at the
mathematical level we certainly do not have a unified theory.
In spite of this lack of unity one can observe a common concern for the analysis of fluctuations (as in the Gallavotti-Cohen Fluctuation Theorem), and also
of linear response of a system to small changes in the forces applied to it. The
two questions are connected. In fact, close to equilibrium we have a FluctuationDissipation Theorem, which relates the linear response (= dissipation) to the spontaneous fluctuations in the system. Technically, the fluctuations are expressed in
terms of time (auto-)correlation functions. The Fourier transform of a time autocorrelation function is a so-called power spectrum. The linear response is given by a
response function, which has a Fourier transform known as susceptibility. We shall
concern ourselves below with the problem of extending the fluctuation-dissipation
theorem to situations far from equilibrium. At this point let us remark that the
physical diversity of nonequilibrium systems expresses itself at the level of fluctuations. For instance, under the Gallavotti-Cohen chaotic hypothesis a small system
with Gaussian thermostat has in principle exponentially decreasing time correlation functions. But an extended system with diffusion in d dimensions has time
correlations decaying like t−d/2 .
S492
D. Ruelle
Ann. Henri Poincaré
Nonlocal fluctuations in stochastic models
The matrix method of Derrida permits a rigorous study of a one-dimensional lattice system (chain) where particles jump independently from any occupied site to
one of the two neighboring sites provided the latter is empty. Using this method,
Derrida, Lebowitz and Speer [2] have studied the continuous limit of a chain for
which the densities of occupied sites at the two ends are maintained at different
values. As could be expected one finds a linear density profile along the chain,
interpolating between the the values imposed at the endpoints. Interestingly one
can compute exactly the probability of density fluctuations, and the expression obtained turns out to be nonlocal, i.e., it is not an integral over the chain of a local
function of the density and its derivatives. This is quite different from the behavior of fluctuations in equilibrium statistical mechanics (and contradicts some older
ideas). Jona-Lasinio and coworkers [1] have obtained more general exact formulas
of the same type, but their approach is less rigorous. It is interesting that nonlocality is observed even though the dynamics is stochastic and thus not quite realistic.
For a more realistic (Hamiltonian) system, one does expect nonlocal fluctuations
because entropy production corresponds to loss of information at infinity, as we
have indicated earlier. Procaccia, who made an early study of the sort of model
discussed here [7], has convinced me that the nonlocality observed in the stochastic
models is of a somewhat more trivial nature as that which would correspond to
loss of information at infinity.
Interlude: biology
Nonequilibrium situations are, in many cases of physical interest, locally close to
equilibrium, i.e., close to equilibrium near each point of space and time. This is
believed to be usually the case in transport phenomena like diffusion, heat propagation, and hydrodynamics. It is however not the case in chemical reactions, and
in particular in most phenomena associated with life. Where biology is concerned
one is thus confronted with problems of nonequilibrium far from equilibrium, an
example being that of motor proteins, which are currently the object of intense investigation. Let me mention in this respect a recent book by Howard [5] which tries
to dispel a certain number of misconceptions. The author insists that the proteins
that play a role in biological motors have the stiffness of “hard plastic”, not jello,
and that friction is intense: inertia is unimportant, motion is overdamped, nonoscillatory. To study the physics of motor proteins, one uses the pheneomenological laws
of friction and other ad hoc assumptions. These are physically reasonable, but can
one not do better? If one had a fundamental theory of nonequilibrium statistical
mechanics that one could apply, this would be esthetically more satisfactory. But
more importantly, it would ensure a correct approach to the problem, and avoid
possibly serious inconsistencies.
Vol. 4, 2003
Is There a Unified Theory of Nonequilibrium Statistical Mechanics?
S493
Chaotic models with Gaussian thermostats
I want here to discuss the question of linear response in the framework of chaotic
models with a Gaussian thermostat. This is not a new topic, and the reason to
come back to it is that there has been some concern about the correctness of results
announced in [8]. Indeed the proof given there is careless, but the critics of the
earlier version (D. Dolgopyat and M. Jiang) accept the revised argument given
in [10].
Consider the Hamiltonian equations
ξ(q)
d p
=
dt q
p/m
where ξ(q) = −gradV (q). A nonequilibrium time evolution is obtained if we replace
ξ(q) by a non gradient term, in particular if ξ is a vector field which is locally but
not globally a gradient. But then the energy is not conserved, so that the time
evolution is no longer constrained to a compact submanifold of phase space. A
nice way to constrain the system to a compact manifold is due to Gauss. In the
special example where the kinetic energy is maintained at a fixed value (this is the
IK thermostat), the Hamiltonian evolution is replaced by
ξ(q) − αp
d p
=
dt q
p/m
where α = α(p, q) = p.ξ/p.p. For our purposes we may just as well consider a
general smooth time evolution on a compact manifold and, for simplicity, we shall
take the time variable to be discrete (in Z instead of R). Of the original physics we
retain that the entropy production is the rate of volume contraction (which volume
element is used will not be important). We shall impose that the time evolution
is chaotic in the strong form of the “chaotic hypothesis” of Gallavotti and Cohen,
i.e., assume that the time evolution is uniformly hyperbolic (in the sense of Smale’s
Axiom A). Finally, as “physical measure” describing a nonequilibrium steady state
we choose an SRB measure on an attractor for the Axiom A diffeomorphism f
describing the time evolution. If we start with a measure absolutely continuous
with respect to Lebesgue in the basin of the attractor, the SRB measure ρf is
its large time limit. The problem of linear response is to study how ρf changes,
to first order, when f is varied. We write δf ◦ f −1 = X, so that X is the vector
field associated with an infinitesimal change of f . On the attractor which supports
ρf , one can write X = X s + X u the decomposition of X into components in the
stable (contracting) and unstable (expanding) directions respectively. With these
definitions we have now the following theorem [8],[10]:
Let r ≥ 3, and K ⊂ M be a mixing Axiom A attractor with SRB state ρf
for the Cr diffeomorphism f : M → M . Then the map f → ρf is Cr−2 from the
Cr diffeos of M to the space Cr−1 (M )∗ of distributions on M . To first order we
S494
D. Ruelle
have
δρf (Φ) =
∞
Ann. Henri Poincaré
ρf grad(Φ ◦ f n ), X
n=0
=
∞
ρf [(gradΦ) ◦ f n , (T f n )X s − (Φ ◦ f n )divu X u ]
n=0
u
where div is the divergence on unstable manifolds computed with repect to the
natural measure on those manifolds.
The natural measure on unstable manifolds is that induced by the SRB measure. Since X u is in general only Hölder continuous, divu X u is in principle only
a distribution (as remarked by Dolgopyat). It turns out however that (because of
absolute continuity of the stable foliation), divu X u is in fact a Hölder continuous
function on K. Also, one needs to take the divergence with respect to the natural measure on unstable manifolds, otherwise an extra term appears as noted by
Jiang.
The above formulas are important in view of physical applications. Let us
introduce instead of X = δf ◦ f −1 a time dependent perturbation Xt = δt f ◦ f −1
(where f remains time independent). The perturbed state ρf + δt ρ is now given
(at time t = 0) by
∞
δ0 ρ(Φ) =
ρf grad(Φ ◦ f n ), X−n n=0
=
∞
s
u
ρf [(gradΦ) ◦ f n , (T f n )X−n
− (Φ ◦ f n )divu X−n
]
n=0
and similarly for continuous time. (We do not try to be precise here: there should
be conditions on which functions t → Xt are allowed). Note that δ0 ρ(Φ) is the
response at time 0 to the perturbations δt f , and only the pertubations for t ≤ 0
play a role (causality). The two terms in the right-hand side both converge exponentially in n. The function t → ρf [(Φ ◦ f t )divu X u ] is a time correlation function, and the fact that one term in the response function is a time correlation
function may be viewed as a form of the fluctuation dissipation theorem (what remains of the fluctuation-dissipation theorem far from equilibrium). But the term
ρf (gradΦ) ◦ f n , (T f n )X s is of a different sort. On one hand, since the attractor
contains the unstable directions, it is not too astonishing that the term containing X u be expressible in terms of spontaneous fluctuations of the system. On the
other hand the term containing X s corresponds to shaking the system into directions that it cannot spontaneously explore, this term is thus of a different nature.
[Near equilibrium, i.e., for Hamiltonian dynamics, the measure ρf is smooth in
stable as well as unstable directions, which explains the more complete form of
the fluctuation-dissipation theorem in that case].
A word of caution to finish. The above considerations are based on the chaotic
hypothesis of uniform hyperbolicity of the dynamics. This will virtually never
Vol. 4, 2003
Is There a Unified Theory of Nonequilibrium Statistical Mechanics?
S495
be satisfied in practice, and we are still far from a more general understanding
of dynamics which would permit a serious weakening of the chaotic hypothesis.
Furthermore, Gaussian thermostats are just one idealization of nonequilibrium
statistical mechanics, and it remains to be seen to what extent the results indicated
above extend to other idealizations.
References
[1] L. Bertini et al. “Fluctuations in stationary nonequilibrium states of irreversible processes.” Phys. Rev. Letters 87, 040601 (2001).
[2] B. Derrida, J.L. Lebowitz and E.R. Speer. “Free energy functional for nonequilibrium systems: an exactly solvable case.” Phys. Rev. Letters 87, 150601
(2001).
[3] J.-P. Eckmann, C.-A. Pillet and L. Rey-Bellet. “Entropy production in nonlinear, thermally driven Hamiltonian systems.” J. Statist. Phys. 95, 305–331
(1999).
[4] G. Gallavotti and E.G.D. Cohen. “Dynamical ensembles in stationary states.”
J. Statist. Phys. 80, 931–970 (1995).
[5] J. Howard.Mechanics of motor proteins and the cytoskeleton. Sinauer, Sunderland MA, 2001.
[6] V. Jakšić and C.-A. Pillet. “On entropy production in quantum statistical
mechanics.” Commun. Math. Phys. 217, 285–293 (2001).
[7] I. Procaccia et al. “Statistical mechanics of stationary states.” Phys. Rev. A
I: 19, 1290 (1979), II: 19, 1307 (1979), III: 19,1324(1979), IV: 20, 2533 (1979),
V: 22, 2809 (1980), VI: 22, 714 (1980).
[8] D. Ruelle. “Differentiation of SRB states.” Commun. Math. Phys. 187, 227–
241 (1997).
[9] D. Ruelle. “Smooth dynamics and new theoretical ideas in nonequilibrium
statistical mechanics.” J. Statist. Phys. 95, 393–468 (1999).
[10] D. Ruelle. “Differentiation of SRB states: correction and complements.” Commun. Math. Phys. 234, 185–190 (2003).
[11] D. Ruelle. “Entropy production in quantum spin systems.” Commun. Math.
Phys. 224, 3–16 (2001).
David Ruelle
IHES
F-91440 Bures sur Yvette
France