Stochastic Dynamics of Extended Objects in Driven Systems: I. Higher-Dimensional Currents in the Continuous Setting

TL;DR

This paper extends the analysis of stochastic currents in driven Langevin systems to higher-dimensional objects, introducing new observables called higher-dimensional currents and expressing their mean fluxes via supersymmetric Fokker-Planck solutions.

Contribution

It generalizes the concept of stochastic currents to higher-dimensional objects on manifolds, providing a topologically protected measure of nonequilibrium behavior.

Findings

Introduction of higher-dimensional currents as intersection counts

Expression of mean fluxes using supersymmetric Fokker-Planck solutions

Generalization of conventional current formulas to higher dimensions

Abstract

The probability distributions, as well as the mean values of stochastic currents and fluxes, associated with a driven Langevin process, provide a good and topologically protected measure of how far a stochastic system is driven out of equilibrium. By viewing a Langevin process on a compact oriented manifold of arbitrary dimension m as a theory of a random vector field associated with the environment, we are able to consider stochastic motion of higher-dimensional objects, which allow new observables, called higher-dimensional currents, to be introduced. These higher dimensional currents arise by counting intersections of a k-dimensional trajectory, produced by a evolving (k-1)-dimensional cycle, with a reference cross section, represented by a cycle of complimentary dimension (m - k). We further express the mean fluxes in terms of the solutions of the Supersymmetric Fokker-Planck (SFP), thus generalizing the corresponding well-known expressions for the conventional currents.