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Nonequilibrium Statistcal Mechanics and Thermodynamics

30 Jul 2008 10:06

In equilibrium, we can use functions of states --- free energies, thermodynamic potentials --- to determine the most probable state. In fact, we can even determine the probability of arbitrary states. Out of equilibrium, it would seem that the natural generalization would be to use a functional of a sequence of states, of a trajectory, to determine the probability of trajectories. In the case of small, linear deviations from equilibrium, the Onsager-Machlup (or Onsager-Rayleigh) "action" gives us such a functional. What works far from equilibrium? In equilibrium, one can link the thermodynamic potentials to functions which specify the rate of decay of large deviations --- is this still true out of equilibrium? (Eyink, below, says yes, but I want to be sure. Keizer, for instance, also proposes an action, but I'm not sure it's the same as Eyink's, and, if they differ, which is right?)

Here's my argument for the ubiquity of effective actions. Markov processes have Gibbs distributions over sequences of states, and Gibbs distributions, just by definition, arise from an effective action. Many nonequilibrium systems can be described by Markov processes (say, deterministic trajectory plus noise). But I'd go further and argue that every nonequilbrium system can be represented as a Markov process --- that if you haven't found one, you're not looking hard enough. (That argument's in a separate paper.) So it should always be possible to find an effective action. But this doesn't establish that there should be a common form for these actions across different systems, which is what Eyink and Keizer claim. (The papers by Woo make similar claims, with special reference to hydrodynamics and spatially-extended systems; this is very exciting, and I need to re-read all of them carefully.)

Are there universal criteria for the stability of non-equilibrium steady states, or must be actually investigate entire paths? Landauer argued for the latter, convincingly to my mind, but I need to learn more here.

Approach to equilibrium doesn't interest me so much as sustained non-equilibrium situations, but like everybody else I suppose they're strongly connected. Fluctuation-dissipation results are accordingly interesting, especially ones which do not assume nearness to equilibrium. Having just read the paper by Carberry et al., below, I am seized with the desire to read up on the Evans-Searles fluctuation theorem, which now seems incredibly cool.

I should try to explain the new ideas about the role of smooth dynamical systems in the statistical mechanics here, but anyone who's geeky enough to be interested really ought to read Ruelle's review article rather than listen to me, and, after that, Dorfman's book.

See also Pattern Formation; Self-organization; Self-organized Critcality; Statistical Mechanics; Foundatons of Statisticcal Mechanics; Stochastic Processes; Interacting Particle Systems; Large Deviations

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