Brownian motion in the presence of a temperature gradient

TL;DR

This paper derives the Fokker-Planck equation for Brownian motion in a temperature gradient using non-equilibrium thermodynamics, linking it to temperature field equations and previous statistical methods.

Contribution

It introduces a thermodynamic approach to derive the Fokker-Planck equation in temperature gradients, connecting microscopic and macroscopic descriptions.

Findings

Derived the Fokker-Planck equation for Brownian particles in a temperature gradient.

Established the coupling between temperature field and particle dynamics.

Reproduced equations previously obtained by statistical mechanical methods.

Abstract

By considering an ensemble of Brownian particles suspended in a heat bath as a thermodynamic system with an internal degree of freedom it is possible to obtain the Fokker-Planck equation for Brownian motion in a temperature gradient, by applying the scheme of non-equilibrium thermodynamics. We recover explicitely the equations derived in particular by Zubarev and Bashkirov using statistical mechanical and kinetic methods. In addition when the temperature gradient does not have an externally imposed magnitude we obtain the differential equation for the temperature field, which is coupled to the Fokker-Planck equation.