Coarse Graining Empirical Densities and Currents in Continuous-Space Steady States

TL;DR

This paper develops a framework for analyzing the fluctuations of empirical densities and currents in steady-state diffusion processes, emphasizing the role of spatial coarse graining and symmetries to infer physical properties like dissipation.

Contribution

It introduces a novel approach using generalized time-reversal symmetry to understand coarse-grained fluctuations and provides insights into inferring dissipation bounds from empirical data.

Findings

Coarse graining scale variation helps infer dissipation bounds.

Emerging symmetries in empirical statistics are identified.

Stochastic calculus offers a practical alternative to traditional methods.

Abstract

We present the conceptual and technical background required to describe and understand the correlations and fluctuations of the empirical density and current of steady-state diffusion processes on all time scales -- observables central to statistical mechanics and thermodynamics on the level of individual trajectories. We focus on the important and non-trivial effect of a spatial coarse graining. Making use of a generalized time-reversal symmetry we provide deeper insight about the physical meaning of fluctuations of the coarse-grained empirical density and current, and explain why a systematic variation of the coarse-graining scale offers an efficient method to infer bounds on a system's dissipation. Moreover, we discuss emerging symmetries in the statistics of the empirical density and current, and the statistics in the central-limit regime. More broadly our work promotes the application of stochastic calculus as a powerful direct alternative to Feynman-Kac theory and path-integral methods.