Global stability and H theorem in lattice models with nonconservative interactions

In kinetic theory, a system is usually described by its one-particle distribution function f(r, v, t), such that f (r, v, t) dr dv is the fraction of particles with positions and velocities in the intervals (r, r + dr) and (v, v + dv), respectively. Therein, global stability and the possible existence of an associated Lyapunov function or H theorem are open problems when nonconservative interactions are present, as in granular fluids. Here, we address this issue in the framework of a lattice model for granularlike velocity fields. For a quite general driving mechanism, including both boundary and bulk driving, we show that the steady state reached by the system in the long-time limit is globally stable. This is done by proving analytically that a certain H functional is nonincreasing in the long-time limit. Moreover, for a quite general energy injection mechanism, we are able to demonstrate that the proposed H functional is nonincreasing for all times. Also, we put forward a proof that clearly illustrates why the classical Boltzmann functional H-B[f] = integral dr dv f (r, v, t) ln f (r, v, t) is inadequate for systems with nonconservative interactions. This is done not only for the simplified kinetic description that holds in the lattice models analyzed here but also for a general kinetic equation, like Boltzmann's or Enskog's.