Charged Brownian Particles: Kramers and Smoluchowski Equations and the Hydrothermodynamical Picture

TL;DR

This paper develops a comprehensive thermohydrodynamical framework for charged Brownian particles influenced by external fields and temperature gradients, deriving transport equations and exploring their implications for various physical systems.

Contribution

It introduces a novel approach to derive coupled transport equations for charged Brownian gases, including reactive and nonuniform effects, extending previous models with a complete thermodynamic picture.

Findings

Derived a Smoluchowski-reactive equation for particle density.

Established a generalized Ohm's law for momentum density.

Formulated a Maxwell-Cattaneo type equation for energy density.

Abstract

We consider a charged Brownian gas under the influence of external and non uniform electric, magnetic and mechanical fields, immersed in a non uniform bath temperature. With the collision time as an expansion parameter, we study the solution to the associated Kramers equation, including a linear reactive term. To first order we obtain the asymptotic (overdamped) regime, governed by transport equations, namely: for the particle's density, a Smoluchowski-reactive like equation; for the particle's momentum density, a generalized Ohm's like equation; and for the particle's energy density, a Maxwell-Cattaneo like equation. Defining a nonequilibrium temperature as the mean kinetic energy density, and introducing Boltzmann's entropy density via the one particle distribution function, we present a complete thermohydrodynamical picture for a charged Brownian gas. We probe the validity of the local equilibrium approximation, Onsager relations, variational principles associated to the entropy production, and apply our results to: carrier transport in semiconductors, hot carriers and Brownian motors. Finally, we outline a method to incorporate non linear reactive kinetics and a mean field approach to interacting Brownian particles.