Homogeneous steady state of a confined granular gas

Abstract

The nonequilibrium statistical mechanics and kinetic theory for a model of a confined quasi-two-dimensional gas of inelastic hard spheres is presented. The dynamics of the particles includes an effective mechanism to transfer the energy injected in the vertical direction to the horizontal degrees of freedom. The Enskog approximation is formulated and used as the basis to investigate the temperature and the distribution function of the steady state eventually reached by the system. An exact scaling of the distribution function of the system having implications on the form of its moments is pointed out. The theoretical predictions are compared with numerical results obtained by a particle simulation method, and a good agreement is found.

Kinetic equation and nonequilibrium entropy for a quasi-two-dimensional gas

“…Recently, the interest in the slit geometry with two parallel, hard plates of a single component system of hard spheres, with a separation between the two plates smaller than two particle diameters, has largely increased, due to a series of experimental observations in systems of macroscopic particles [9][10][11][12]. Although it is quite sure that the inelasticity of collisions, inherent to the macroscopic character of the spheres, and the subsequently needed energy injection to reach and maintain a steady state, play a crucial role in many of the observed features [13][14][15], it is clear that before considering the consequences of these factors, the idealized system of elastic hard spheres must be addressed. The central issue is whether it is possible to formulate a macroscopic, hydrodynamic-like theory to describe the two-dimensional dynamics of the system when observed from above or from below and, if the answer is affirmative, which is the form of the equations and the expressions of the coefficients appearing in them.…”

Homogeneous hydrodynamics of a collisional model of confined granular gases

“…From the Liouville equation, the Born-Bogoliubov-Green-Kirkwood-Ivon (BBGKY) hierarchy of equations for the reduced distribution functions is easily derived [14]. The first equation of the hierarchy reads…”

“…The dynamics consists of free streaming until a given pair of particles i, j are at contact. At this moment, the velocities of the two particles change instantaneously according to the inelastic collision rule [13,14] …”

“…Very recently, an alternative approach to the description of the two-dimensional dynamics of a confined granular gas has been proposed [13,14]. The idea is to substitute the real three-dimensional collision rule by an effective two-dimensional one, trying to capture the mechanism for which the particles convert their kinetic energy associated to the vertical component of the velocity into kinetic energy in the horizontal plane.…”

“…The main features of its homogeneous steady state were analyzed in ref. [14]. Here, the dynamics of the system towards the steady state, assuming that it remains always homogeneous, will be addressed.…”