Langevin equation for the extended Rayleigh model with an asymmetric bath

TL;DR

This paper derives a non-linear Langevin equation for a piston in a two-gas system with asymmetric parameters, providing microscopic expressions and exact solutions for ideal gases, revealing fluctuation-induced drift and transient behaviors.

Contribution

It introduces a first-principles derivation of a non-linear Langevin equation for asymmetric gas-piston systems, including explicit kinetic coefficients and solutions.

Findings

Stationary solutions exhibit directional fluctuation-induced drift.

Exact solutions are obtained for ideal gases with quadratic repulsive potential.

Transient behavior matches previous hard-wall interaction results.

Abstract

In this paper a one-dimensional model of two infinite gases separated by a movable heavy piston is considered. The non-linear Langevin equation for the motion of the piston is derived from first principles for the case when the thermodynamic parameters and/or the molecular masses of gas particles on left and right sides of the piston are different. Microscopic expressions involving time correlation functions of the force between bath particles and the piston are obtained for all parameters appearing in the non-linear Langevin equation. It is demonstrated that the equation has stationary solutions corresponding to directional fluctuation-induced drift in the absence of systematic forces. In the case of ideal gases interacting with the piston via a quadratic repulsive potential, the model is exactly solvable and explicit expressions for the kinetic coefficients in the non-linear Langevin equation are derived. The transient solution of the non-linear Langevin equation is analyzed perturbatively and it is demonstrated that previously obtained results for systems with the hard-wall interaction are recovered.