Large-deviation principles, stochastic effective actions, path entropies, and the structure and meaning of thermodynamic descriptions

TL;DR

This paper explores how large-deviation principles and stochastic effective actions can be used to understand the structure and meaning of thermodynamic descriptions, especially for non-equilibrium stochastic processes, through variational and entropy methods.

Contribution

It introduces a unified framework combining large-deviation theory, path-entropy, and variational methods to construct thermodynamic descriptions for stochastic systems beyond equilibrium.

Findings

Constructed entropy functions from large-deviations scaling.

Defined stochastic effective actions for non-equilibrium paths.

Linked entropy maximization principles to different thermodynamic descriptions.

Abstract

The meaning of thermodynamic descriptions is found in large-deviations scaling of the fluctuations probabilities. The primary large-deviations rate function is the entropy, which is the basis for both fluctuation theorems and for characterizing the thermodynamic interactions of systems. Freidlin-Wentzell theory provides a general formulation of large-deviations scaling for non-equilibrium stochastic processes, through a 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 and Peliti, appropriate to integer counting statistics, is widely used in reaction-diffusion theory. Using these tools together with a path-entropy method due to Jaynes, we show 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 of among all these levels of description may be constructed explicitly and understood in terms of information conditions. The example systems considered introduce 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.