Global stability and $H$-theorem in lattice models with non-conservative interactions

In kinetic theory, a system is usually described by its one-particle distribution function $f(\mathbf{r},\mathbf{v},t)$, such that $f(\mathbf{r},\mathbf{v},t)d\mathbf{r} d\mathbf{v}$ is the fraction of particles with positions and velocities in the intervals $(\mathbf{r}, \mathbf{r}+d\mathbf{r})$ and $(\mathbf{v}, \mathbf{v}+d\mathbf{v})$, respectively. Therein, global stability and the possible existence of an associated Lyapunov function or $H$-theorem are open problems when non-conservative interactions are present, as in granular fluids. Here, we address this issue in the framework of a lattice model for granular-like 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 non-increasing in the long time limit. Moreover, for two specific energy injection mechanisms, we are able to demonstrate that the proposed $H$-functional is non-increasing for all times. Also, we put forward a proof that clearly illustrates why the "classical" Boltzmann functional $H_{B}[f]=\int\! d\mathbf{r} \, d\mathbf{v} f(\mathbf{r},\mathbf{v},t) \ln f(\mathbf{r},\mathbf{v},t)$ is inadequate for systems with non-conservative interactions. Not only is this done for the simplified kinetic description that holds in the lattice models analysed here but also for a general kinetic equation, like Boltzmann's or Enskog's.