On the nonequilibrium relation between potential and stationary distribution for driven diffusion

TL;DR

This paper explores the complex, long-range relationship between potential functions and stationary distributions in nonequilibrium driven diffusion processes, revealing that local potential changes have global effects and vice versa.

Contribution

It provides a variational framework linking stationary distributions and potentials in nonequilibrium diffusion, addressing both direct and inverse problems.

Findings

Local potential perturbations cause global density changes

Rearranging the potential is necessary for local density modifications

Variational characterization connects potentials and stationary densities

Abstract

We investigate the relation between an applied potential and the corresponding stationary state occupation for nonequilibrium and overdamped diffusion processes. This relation typically becomes long ranged resulting in global changes for the relative density when the potential is locally perturbed, and inversely, we find that the potential needs to be wholly rearranged for the purpose of creating a locally changed density. The direct question, determining the density as a function of the potential, comes under the response theory out of equilibrium. The inverse problem of determining the potential that produces a given stationary distribution naturally arises in the study of dynamical fluctuations. This link to the fluctuation theory results in a variational characterization of the stationary density upon a given potential and vice versa.