Rectification of thermal fluctuations in ideal gases

Abstract

We calculate the systematic average speed of the adiabatic piston and a thermal Brownian motor, introduced by C. Van

On the higher order corrections to the Fokker–Planck equation

“…Nevertheless, it is still possible to get for f (V, t) a differential equation of finite order analyzing dependence of coefficients a n on the small parameter λ and neglecting terms of higher order in λ. The Kramers-Moyal expansion in the form (4) is not appropriate for such a perturbation analysis because the dependence on λ is implicit in (4). Assuming that transition rates have certain scaling properties with respect to λ, van Kampen modified the Kramers-Moyal expansion transforming it into the form of the expansion in powers of λ.…”

“…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].…”

“…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].…”

“…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.…”

“…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.…”