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A model of homogeneously driven dissipative system, consisting of acollection of $N$ particles that are characterized by only their velocities, isconsidered. Adopting a discrete time dynamics, at each time step, a pair ofvelocities is randomly selected. They undergo inelastic collision withprobability $p$. With probability $(1-p)$, energy of the system is changed bychanging the velocities of both the particles independently according to$v\rightarrow -r_w v +\eta$, where $\eta$ is a Gaussian noise drawnindependently 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 asteady state) after time steps of $O(N)$, it grows linearly with time aftertime 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 energyper particle and the correlation of velocities are obtained exactly. In thethermodynamic limit of large $N$, an exact equation is obtained for the momentgenerating function. In the limit of nearly elastic collisions and weak energyinjection, 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 anexponential tail.

High-Energy Tail of the Velocity Distribution of Driven Inelastic Maxwell Gases