The large-time and vanishing-noise limits for entropy production in nondegenerate diffusions

TL;DR

This paper studies the long-time and small-noise limits of entropy production in non-equilibrium diffusions, revealing a universal large deviation principle and symmetry properties through advanced analytical techniques.

Contribution

It introduces a unified analysis of entropy production functionals for nondegenerate diffusions, establishing a law of large numbers and large deviation principles independent of boundary term choices.

Findings

Law of large numbers for entropy production

Gallavotti--Cohen symmetry in large deviations

Eigenvalue analysis via semiclassical methods

Abstract

We investigate the behaviour of a family of entropy production functionals associated to stochastic differential equations of the form $\mathrm{d} X_s = -\nabla V(X_s) \, \mathrm{d} s + b(X_s) \, \mathrm{d} s + \sqrt{2\epsilon} \, \mathrm{d} W_s $, where $b$ is a globally Lipschitz nonconservative vector field keeping the system out of equilibrium, with emphasis on the large-time limit and then the vanishing-noise limit. Different members of the family correspond to different choices of boundary terms. Our analysis yields a law of large numbers and a local large deviation principle which does not depend on the choice of boundary terms and which exhibits a Gallavotti--Cohen symmetry. We use techniques from the theory of semigroups and from semiclassical analysis to reduce the description of the asymptotic behaviour of the functional to the study of the leading eigenvalue of a quadratic approximation of a deformation of the infinitesimal generator near critical points of $V$.