High-energy tail of the velocity distribution of driven inelastic Maxwell gases

A model of homogeneously driven dissipative system, consisting of a collection of N particles that are characterized by only their velocities, is considered. Adopting a discrete time dynamics, at each time step, a pair of velocities is randomly selected. They undergo inelastic collision with probability p. With probability (1 - p), energy of the system is changed by changing the velocities of both the particles independently according to v -> -r(w)v+eta, where eta is a Gaussian noise drawn independently for each particle as well as at each time steps. For the case r(w) = -1, although the energy of the system seems to saturate (indicating a steady state) after time steps of O(N), it grows linearly with time after time steps of O(N-2), indicating the absence of a eventual steady state. For -1 < r(w) <= 1, the system reaches a steady state, where the average energy per particle and the correlation of velocities are obtained exactly. In the thermodynamic limit of large N, an exact equation is obtained for the moment generating function. In the limit of nearly elastic collisions and weak energy injection, the velocity distribution is shown to be a Gaussian. Otherwise, for vertical bar r(w)vertical bar < 1, the high-energy tail of the velocity distribution is Gaussian, with a different variance, while for r(w) = +1 the velocity distribution has an exponential tail. Copyright (C) EPLA, 2013