Stationary motion of the adiabatic piston

Abstract

We consider a one-dimensional system consisting of two infinite ideal fluids, with equal pressures but different temperatures T1 and T2, separated by an adiabatic movable piston whose mass M is much larger than the mass m of the fluid particules. This is the infinite version of the controversial adiabatic piston problem. The stationary non-equilibrium solution of the Boltzmann equation for the velocity distribution of the piston is expressed in powers of the small parameter ǫ = m/M , and explicitly given up to order ǫ 2 . In particular it implies that although the pressures are equal on both sides of the piston, the temperature difference induces a non-zero average velocity of the piston in the direction of the higher temperature region. It thus shows that the asymmetry of the fluctuations induces a macroscopic motion despite the absence of any macroscopic force. This same conclusion was previously obtained for the non-physical situation where M = m.

On the motion of a body in thermal equilibrium immersed in a perfect gas

“…A similar model with elastic collisions between fluid particles and body has been introduced in connection with the so called piston problem (see [7,8] and references quoted therein). An investigation on the approach to equilibrium in such a model has been done in [3,4].…”

Multiplicative Langevin equation to reproduce long-time properties of nonequilibrium Brownian motion

“…Using the Kramers-Moyal expansion and a perturbation expansion in powers of ε for (7), we have [22,36] ∂P ∂t…”

“…Here, γ 0 and γ 1 are given by (15) and (16), respectively. T = √ T L T R corresponds to the kinetic temperature of the piston, i.e., [21,22]. Ignoring the contribution of O(ε 2 ) by noting that both γ 0 and γ 1 are of O(ε), (19) corresponds to a Langevin equation…”

“…By rescaling the variables according to (22), (23) and (24) as t → τ , X → X , and V → V, (A.1) is rewritten in dimensionless form as ∂P(X , V, τ ) ∂τ = −V ∂P ∂X Substituting these evaluations into (A.2), we obtain (26), that is…”

“…To demonstrate, we adopt a standard model, called the Rayleigh piston, which is used to derive the equilibrium Langevin equation from the microscopic dynamics of a large number of degrees of freedom [16,17]. Its nonequilibrium version is presented in textbooks and has been studied during the past few decades [18][19][20][21][22][23][24][25][26][27][28][29][30][31][32][33]. We numerically identify a multiplicative Langevin equation (13) that reproduces the cumulants of long-time displacement of this nonequilibrium Brownian motion at least up to the third order.…”