Noise Rectification and Fluctuations of an Asymmetric Inelastic Piston

We consider a massive inelastic piston, whose opposite faces have different coefficients of restitution, moving under the action of an infinitely dilute gas of hard disks maintained at a fixed temperature. The dynamics of the piston is Markovian and obeys a continuous Master Equation: however, the asymmetry of restitution coefficients induces a violation of detailed balance and a net drift of the piston, as in a Brownian ratchet. Numerical investigations of such non-equilibrium stationary state show that the velocity fluctuations of the piston are symmetric around the mean value only in the limit of large piston mass, while they are strongly asymmetric in the opposite limit. Only taking into account such an asymmetry, i.e. including a third parameter in addition to the mean and the variance of the velocity distribution, it is possible to obtain a satisfactory analytical prediction for the ratchet drift velocity. Introduction. – Granular materials have been the subject of intense research in the last 20 years in Physics [1]. Most of the non-trivial phenomena that can be observed in a shaken box of sand are due to the inelasticity of collisions among grains [2]. Kinetic energy is dissipated into heat, introducing an intrinsic time irreversibility in the “microscopic” dynamics which can have consequences at a more macroscopic level: for instance species segregation [3], breakdown of energy equipartition [4], apparent Maxwell-demon-like properties such as heat currents against a temperature gradient [5] or mass current against a density gradient [6], ratchet-like net drift of a asymmetrically shaped tracers [7], and so on. It is well known, moreover, that the space asymmetry of an adiabatic piston results in a stationary macroscopic motion [8]. Here we consider, instead, a model of asymmetric granular piston with different coefficients of restitution which, with respect to a previously presented model of granular Brownian ratchet [7], has the advantages of being simpler to be studied analytically as well as realized in the laboratory, and at the same time displays unexpected peculiar properties: in particular we will show how its stationary state is characterized by asymmetric velocity fluctuations, and that this asymmetry becomes crucial when the piston mass is smaller than the mass of surrounding disks. Such a light piston limit, which could be regarded as purely academic in the case of a sect separating two molecular gases, is instead realistic in the case of granular gases, where the role “molecules” is played by large grains. Model. – The 2D model consists of a piston of mass M and height L surrounded by a dilute gas ofN hard disks of mass m and density ρ. The faces of the piston have two different values of inelasticity, characterized by the coefficients of restitution α1, on the left, and α2, on the right. The piston can only slide, without rotating, along the direction x, perpendicular to its faces. The thickness of the piston can be neglected for the purpose of our simulations and calculations. The particles-piston binary collisions are described by the rule: V = V ′ − (1 + αi) ǫ 1 + ǫ2 (V ′ − v x) vx = v ′ x + (1 + αi) 1 1 + ǫ2 (V ′ − v x) vy = v ′ y (1) where v and v are the post-collisional and pre-collisional disk velocities respectively, while V and V ′ are the corresponding velocities of the piston, while ǫ = m/M . Because of the constraint on the piston, its vertical velocity