Langevin equation for the extended Rayleigh model with an asymmetric bath

Abstract

In this paper a one-dimensional model of two infinite gases separated by a movable heavy piston is considered. The nonlinear 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 the 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 nonlinear 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 nonlinear Langevin equation are derived. The transient solution of the nonlinear Langevin equation is analyzed perturbatively and it is demonstrated that previously obtained results for systems with the hard-wall interaction are recovered.

On the higher order corrections to the Fokker–Planck equation

“…These formulas are valid not only for the original Rayleigh model, but also for asymmetric models [2][3][4] when properties of the bath on the left and on the right of the particle are different. Of course, the explicit form of the coefficients α In what follows we restrict ourselves to the symmetric problem when the transition rate is given by Eq.…”

“…The systematic average velocity of the particle can be calculated as a perturbative solution of the Langevin equation with nonlinear corrections to the damping force [3]. Alternatively, one can use the corresponding nonlinear Fokker-Planck equation or an equivalent set of equations for the moments [4].…”

“…There has been renewed discussion recently of the Rayleigh model of nonlinear Brownian motion, stimulated primarily by findings of new qualitative effects governed by nonlinear stochastic processes such as rectification of fluctuations [1][2][3][4], stochastic resonance [5], etc. In the Rayleigh model the heavy Brownian particle of mass M is immersed in an ideal gas of molecules of mass m ≪ M and interacts with them through instantaneous elastic collisions.…”

“…If the temperatures on the left and right sides of the Rayleigh particle are different, the particle undergoes directional movement even when the pressure on both sides is the same [2][3][4]. The systematic average velocity of the particle can be calculated as a perturbative solution of the Langevin equation with nonlinear corrections to the damping force [3].…”

“…The Rayleigh model may be generalized in many ways to study effects of asymmetry of the surrounding bath [2][3][4], possibility of non-canonical distributions [10], finiterange interaction [11], strong non-equilibrium fluctuations [12], etc. To analyze these generalizations it would be of help to compare their predictions with the results of the original model.…”