Temperature of a granular material "fluidized" by external vibrations

The scaling for the temperature of a granular material "fluidized" by external vibrations is determined in the Limit where the dissipation of energy in a collision due to inelasticity, or between successive collisions due to viscous drag, is small compared to the energy of the particles. An asymptotic scheme is used, where the dissipation of energy is neglected in the leading approximation, and the Boltzmann equation for the system is identical to that for a gas at equilibrium in a gravitational field. The density variation in the "fluidized" material is given by the Boltzmann distribution, and the velocity distribution is given by the Maxwell-Boltzmann distribution. However, the "temperature" of the material is not specified by thermodynamic considerations, but is determined by a balance between the source of energy due to the vibrating surface and the dissipation of energy. This balance indicates that the dependence of temperature on the amplitude of the vibrating surface is sensitively dependent on the mechanism of dissipation (inelastic collisions or viscous drag), and also on whether the amplitude function for the velocity of the vibrating surface is symmetric or asymmetric about zero velocity. However, the temperature turns out to have the same functional dependence on the properties of the system in two and three dimensions.