Driven inelastic Maxwell gas in one dimension

Abstract

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,” vi. Nearest neighbors on the lattice interact, with a rate τ−1c, according to an inelastic collision rule. External driving, occurring with a rate τ−1w, 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)=⟨vi(0)vi+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.