Generally covariant state-dependent diffusion

TL;DR

This paper develops a gauge-invariant, coordinate-covariant second-order Langevin framework for state-dependent diffusion, ensuring well-defined stochastic dynamics and equilibrium properties, applicable to diffusion in varying temperature heat baths.

Contribution

It introduces a novel gauge-invariant Langevin equation with state-dependent diffusion, addressing ambiguities in previous models and ensuring consistent equilibrium behavior.

Findings

Derives a second-order Langevin equation respecting gauge invariance.

Shows the resulting Fokker-Planck equation admits equilibrium steady states.

Proposes a model for diffusion in heat baths with spatially varying temperature.

Abstract

Statistical invariance of Wiener increments under SO(n) rotations provides a notion of gauge transformation of state-dependent Brownian motion. We show that the stochastic dynamics of non gauge-invariant systems is not unambiguously defined. They typically do not relax to equilibrium steady states even in the absence of extenal forces. Assuming both coordinate covariance and gauge invariance, we derive a second-order Langevin equation with state-dependent diffusion matrix and vanishing environmental forces. It differs from previous proposals but nevertheless entails the Einstein relation, a Maxwellian conditional steady state for the velocities, and the equipartition theorem. The over-damping limit leads to a stochastic differential equation in state space that cannot be interpreted as a pure differential (Ito, Stratonovich or else). At odds with the latter interpretations, the corresponding Fokker-Planck equation admits an equilibrium steady state; a detailed comparison with other theories of state-dependent diffusion is carried out. We propose this as a theory of diffusion in a heat bath with varying temperature. Besides equilibrium, a crucial experimental signature is the non-uniform steady spatial distribution.