State-dependent diffusion: thermodynamic consistency and its path integral formulation

TL;DR

This paper develops a thermodynamically consistent Langevin framework for particles with position-dependent friction, clarifies the role of different stochastic calculus conventions, and introduces a path integral approach for analyzing such systems.

Contribution

It formulates a general Langevin equation with multiplicative noise that ensures thermodynamic consistency and provides a path integral formulation for arbitrary noise interpretations.

Findings

The drift term depends on the noise interpretation convention.

The framework ensures the Boltzmann distribution at equilibrium.

Path integral representations facilitate analysis of correlations.

Abstract

The friction coefficient of a particle can depend on its position as it does when the particle is near a wall. We formulate the dynamics of particles with such state-dependent friction coefficients in terms of a general Langevin equation with multiplicative noise, whose evaluation requires the introduction of specific rules. Two common conventions, the Ito and the Stratonovich, provide alternative rules for evaluation of the noise, but other conventions are possible. We show the requirement that a particle's distribution function approach the Boltzmann distribution at long times dictates that a drift term must be added to the Langevin equation. This drift term is proportional to the derivative of the diffusion coefficient times a factor that depends on the convention used to define the multiplicative noise. We explore the consequences of this result in a number examples with spatially varying diffusion coefficients. We also derive path integral representations for arbitrary interpretation of the noise, and use it in a perturbative study of correlations in a simple system.