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and nonequilibrium statistical mechanics

Contributed by: G. Gallavotti (Roma 1), Nov. 6, 1998.

Abstract:   How to define entropy creation rate in a classical system in a stationary state, under external nonconservative forces balanced (in the average) by forces modeling a thermostat, is in general an open problem. Attempts at defining it at least in special cases seem to indicate that a general definition might be possible, and that a theory of ``non equilibrium ensembles'' with general non trivial (although simple) predictions may arise from it.

In nonequilibrium statistical mechanics the notion of entropy and of entropy creation are not well established. New definitions and proposals arise continuously. And recently there have been several new examples of such definitions.

A fundamental definition would be highly desirable. By fundamental I mean a definition that should hold for very general systems in stationary states: and it should not be restricted to (stationary) systems close to equilibrium. This means that it should be defined even in situations where the other fundamental thermodynamics quantity, i.e. the temperature, may itself be also in need of a proper definition. And furthermore it should be a notion accessible to experiments, on numerical simulations and possibly on real systems.

Recently some simple applications have been made which point to the possibility that such a definition might after all exist. They point in the direction of an adaptation of the very first definitions of Boltzmann and Gibbs and rely on the recent works on chaotic dynamics, both in the mathematical domain and in the physical domain.

The theory of the SRB distributions and Ruelle's proposal that they may constitute the foundations of a general theory of chaotic motions provides us with formal expressions of the probability distributions describing stationary states. This is a surprising achievement (relying on the basic work of Sinai on Markov partitions) and the hope is that such formal expressions can be used to derive relations between observable quantities whose values (much as it is already the case in equilibrium statistical mechanics) there is no hope to ever be able to compute via the solution of the equations of motion.

I have in mind general relations like Boltzmann's heat theorem:

        (dU + p dV) T = exact

which involves averaged {U,p} (computed, say, in the canonical ensemble), with dependence on the ``parameters'' V and T that we cannot hope ever to compute but which nevertheless is a very important, non trivial and useful relation. Are similar relations possible between dynamical averages in stationary nonequilibrium states?
After all -- a great part of equilibrium statistical mechanics is dedicated to obtaining similar (if less shiny) relations, from involved N--dimensional integrals (with N very large).

Of course such results are difficult: but they might be not impossible. In situations close to equilibrium there are, in fact, classical examples like the Onsager's reciprocal relations and Green--Kubo's transport coefficients expressions: these are parameterless relations essentially independent of the stationary state

  1. Ruelle, D.: Annals of the New York Academy of Sciences, 356, 408--416, 1978.
  2. Ruelle, D.: Annals of the New York Academy of Sciences, volume Nonlinear dynamics, ed. R.H.G. Helleman, 357, 1--9, 1980.
  3. Ruelle, D.: Journal of Statistical Physics, 85, 1--25, 1996.
  4. Ruelle, D.: New theoretical ideas in non-equilibrium statistical mechanics, Lecture notes at Rutgers University, fall 1998.
  5. Gallavotti, G.: Chaos, 8, 384--392, 1998.
  6. Bonetto, F., Chernov, N., Lebowitz, J.: mp_arc 98-269.
  7. Kurchan, J.: Journal of Physics, A, 31, 3719--3729, 1998.

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giovanni.gallavotti@roma1.infn.it (contributor),
aizenman@princeton.edu (editor)