Nonequilibrium Statistcal Mechanics and Thermodynamics

24 Jul 2023 22:17

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 of trajectories. What works far from equilibrium? In equilibrium, one can link the thermodynamic potentials to functions which specify the rate of decay of large deviations, and this is still true out of equilibrium (see, e.g., Touchette's great review paper), but this is more of a mathematical result than a "physical" one.

Here's an 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 e.g., Keizer and Woo (separately) claim.

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. The Evans-Searles fluctuation theorem, which is well-supported by experiments (see e.g. the Carberry et al. paper) is extremely interesting.

I should try to explain some 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.

*: Pedantically, a Gibbs distribution over a collection of variables \( X_t \), index by \( \mathcal{T} \), collectively \( X \), will take the form \( \mathbb{P}(X=x) \propto e^{-u(x)} \), and the "potential" \( u(x) = \sum_{A \in 2^{\mathcal{T}}}{u_A(x_A)} \), where \( u_A \) is a function only of the values of the random variables at the indices \( A \). (Obvious measure-theoretic complications for continuous index sets are obvious.) The sets \( A \) which actually contribute to the potential are called "cliques". In the particular case of Markov processes in discrete time, the cliques are pairs of successive indices \( \left\{ t, t+1 \right\} \), so we can think of \( u(x) \) as being basically a discrete-time effective action. For continuous-time Markov processes, see under Path Integrals and Feynman Diagrams for Classical Stochastic Processes.