Driven inelastic Maxwell gases.

We consider the inelastic Maxwell model, which consists of a collection of particles that are characterized by only their velocities and evolving through binary collisions and external driving. At any instant, a particle is equally likely to collide with any of the remaining particles. The system evolves in continuous time with mutual collisions and driving taken to be point processes with rates τ(c)(-1) and τ(w)(-1), respectively. The mutual collisions conserve momentum and are inelastic, with a coefficient of restitution r. The velocity change of a particle with velocity v, due to driving, is taken to be Δv=-(1+r(w))v+η, where r(w)∈[-1,1] and η is Gaussian white noise. For r(w)∈(0,1], this driving mechanism mimics the collision with a randomly moving wall, where r(w) is the coefficient of restitution. Another special limit of this driving is the so-called Ornstein-Uhlenbeck process given by dv/dt=-Γv+η. We show that while the equations for the n-particle velocity distribution functions (n=1,2,...) do not close, the joint evolution equations of the variance and the two-particle velocity correlation functions close. With the exact formula for the variance we find that, for r(w)≠-1, the system goes to a steady state. Also we obtain the exact tail of the velocity distribution in the steady state. On the other hand, for r(w)=-1, the system does not have a steady state. Similarly, the system goes to a steady state for the Ornstein-Uhlenbeck driving with Γ≠0, whereas for the purely diffusive driving (Γ=0), the system does not have a steady state.