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 tau(-1)(c), according to an inelastic collision rule. External driving, occurring with a rate t(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 < x*, where C(x,t) has a time-dependent form, whereas in the exterior 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.