Diffusion in nonuniform temperature and its geometric analog

TL;DR

This paper introduces a Langevin equation for systems in environments with nonuniform temperature, ensuring local Maxwellian steady states, local equipartition, and consistent thermodynamics, including a generalized First Law.

Contribution

It presents a novel Langevin framework that accurately models nonuniform temperature effects while maintaining thermodynamic consistency and local equilibrium properties.

Findings

The proposed Langevin equation admits a local Maxwellian steady state.

It demonstrates local equipartition in nonuniform temperature environments.

The model aligns with thermodynamic laws, including a generalized First Law.

Abstract

We propose a Langevin equation for systems in an environment with nonuniform temperature. At odds with an older proposal, ours admits a locally Maxwellian steady state, local equipartition holds and for detailed-balanced (reversible) systems statistical and physical entropies coincide. We describe its thermodynamics, which entails a generalized version of the First Law and Clausius's characterization of reversibility. Finally, we show that a Brownian particle constrained into a smooth curve behaves according to our equation, as if experiencing nonuniform temperature.