Paper #: 11-02-009
The meaning of thermodynamic descriptions is found in large-deviations scaling [1, 2] of the probabilities for fluctuations of averaged quantities. The central function expressing large-deviations scaling is the entropy, which is the basis for both fluctuation theorems and for characterizing the thermodynamic interactions of systems. Freidlin-Wentzell theory  provides a quite general formulation of large-deviations scaling for non-equilibrium stochastic processes, through a remarkable representation in terms of a Hamiltonian dynamical system. A number of related methods now exist to construct the Freidlin-Wentzell Hamiltonian for many kinds of stochastic processes; one method due to Doi [4, 5] and Peliti [6, 7], appropriate to integer counting statistics, is widely used in reaction-diffusion theory.
Using these tools together with a path-entropy method due to Jaynes , this review shows how to construct entropy functions that both express large-deviations scaling of fluctuations, and describe system-environment interactions, for discrete stochastic processes either at or away from equilibrium. A collection of variational methods familiar within quantum field theory, but less commonly applied to the Doi-Peliti construction, is used to define a “stochastic effective action”, which is the large-deviations rate function for arbitrary non-equilibrium paths.
We show how common principles of entropy maximization, applied to different ensembles of states or of histories, lead to different entropy functions and different sets of thermodynamic state variables. Yet the relations among all these levels of description may be constructed explicitly and understood in terms of information conditions. Although the example systems considered are limited, they are meant to provide a self-contained introduction to methods that may be used to systematically construct descriptions with all the features familiar from equilibrium thermodynamics, for a much wider range of systems describable by stochastic processes.