Mathematical, Thermodynamical, and Experimental Necessity for Coarse Graining Empirical Densities and Currents in Continuous Space

TL;DR

This paper explores the importance of spatial coarse graining in analyzing fluctuations and correlations of empirical densities and currents in Markovian diffusion, revealing deep connections with thermodynamics and time-reversal symmetry.

Contribution

It provides a comprehensive theoretical framework demonstrating the necessity of coarse graining for accurate inference of dissipation and fluctuation analysis in continuous space.

Findings

Coarse graining uncovers features of currents breaking detailed balance.

Optimal coarse graining improves dissipation inference.

Without coarse graining, fluctuations diverge in higher dimensions.

Abstract

We present general results on fluctuations and spatial correlations of the coarse-grained empirical density and current of Markovian diffusion in equilibrium or non-equilibrium steady states on all time scales. We unravel a deep connection between current fluctuations and generalized time-reversal symmetry, providing new insight into time-averaged observables. We highlight the essential role of coarse graining in space from mathematical, thermodynamical, and experimental points of view. Spatial coarse graining is required to uncover salient features of currents that break detailed balance, and a thermodynamically "optimal" coarse graining ensures the most precise inference of dissipation. Defined without coarse graining, the fluctuations of empirical density and current are proven to diverge on all time scales in dimensions higher than one, which has far-reaching consequences for the central-limit regime in continuous space. We apply the results to examples of irreversible diffusion. Our findings provide new intuition about time-averaged observables and allow for a more efficient analysis of single-molecule experiments.