Driven inelastic Maxwell gas in one dimension

A lattice version of the driven inelastic Maxwell gas is studied in one dimension with periodic boundary conditions. Each site i of the lattice is assigned with a scalar "velocity," v_{i}. Nearest neighbors on the lattice interact, with a rate τ_{c}^{-1}, according to an inelastic collision rule. External driving, occurring with a rate τ_{w}^{-1}, sustains a steady state in the system. A set of closed coupled equations for the evolution of the variance and the two-point correlation is found. Steady-state values of the variance, as well as spatial correlation functions, are calculated. It is shown exactly that the correlation function decays exponentially with distance, and the correlation length for a large system is determined. Furthermore, the spatiotemporal correlation C(x,t)=〈v_{i}(0)v_{i+x}(t)〉 can also be obtained. We find that there is an interior region -x^{*}x^{*}, the correlation function remains the same as the initial form. C(x,t) exhibits second-order discontinuity at the transition points x=±x^{*}, and these transition points move away from the x=0 with a constant speed.