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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 $v\rightarrow -r_w v +η$, where $η$ 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 \leq 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 $|r_w| < 1$, 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.