About an H-theorem for systems with non-conservative interactions

We exhibit some arguments in favour of an H-theorem for a generalization of the Boltzmann equation including non-conservative interactions and a linear Fokker-Planck-like thermostatting term. Such a non-linear equation describing the evolution of the single particle probability $P_i(t)$ of being in state $i$ at time $t$, is a suitable model for granular gases and is indicated here as Boltzmann-Fokker-Planck (BFP) equation. The conjectured H-functional, which appears to be non-increasing, is $H_C(t)=\sum_i P_i(t) \ln P_i(t)/\Pi_i$ with $\Pi_i = \lim_{t \to \infty} P_i(t)$, in analogy with the H-functional of Markov processes. The extension to continuous states is straightforward. A simple proof can be given for the elastic BFP equation. A semi-analytical proof is also offered for the BFP equation for so-called inelastic Maxwell molecules. Other evidence is obtained by solving particular BFP cases through numerical integration or through "particle schemes" such as the Direct Simulation Monte Carlo.