Abstract
In kinetic theory, a system is usually described by its one-particle distribution function , such that is the fraction of particles with positions and velocities in the intervals and , respectively. Therein, global stability and the possible existence of an associated Lyapunov function or 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 functional is nonincreasing in the long-time limit. Moreover, for a quite general energy injection mechanism, we are able to demonstrate that the proposed functional is nonincreasing for all times. Also, we put forward a proof that clearly illustrates why the “classical” Boltzmann functional 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.
- Received 20 February 2017
DOI:https://doi.org/10.1103/PhysRevE.95.052121
©2017 American Physical Society