Global stability and H theorem in lattice models with nonconservative interactions.

Abstract

In kinetic theory, a system is usually described by its one-particle distribution function f(r,v,t), such that f(r,v,t)drdv 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]=∫drdvf(r,v,t)lnf(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.