Level 2 large deviation functionals for systems with and without detailed balance

TL;DR

This paper derives and generalizes large deviation functionals for stochastic systems, including non-equilibrium cases, highlighting the role of empirical flux and entropy production, with illustrative examples.

Contribution

It rederives the Donsker-Varadhan functional using path integrals and extends it to systems without detailed balance, incorporating empirical flux.

Findings

Generalized large deviation functionals for non-equilibrium systems

Emphasized importance of empirical probability flux

Connected large deviation functionals with entropy production

Abstract

Large deviation functions are an essential tool in the statistics of rare events. Often they can be obtained by contraction from a so-called level 2 large deviation {\em functional} characterizing the empirical density of the underlying stochastic process. For Langevin systems obeying detailed balance, the explicit form of this functional has been known ever since the mathematical work of Donsker and Varadhan. We rederive the Donsker-Varadhan result by using stochastic path-integrals and then generalize it to situations without detailed balance including non-equilibrium steady states. The proper incorporation of the empirical probability flux turns out to be crucial. We elucidate the relation between the large deviation functional and different notions of entropy production in stochastic thermodynamics and discuss some aspects of the ensuing contractions. Finally, we illustrate our findings with examples.