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

Abstract

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.

15 pages, 4 figures, submitted

References in corpus (4)

• Structure factors in granular experiments with homogeneous fluidization
• Entropy of continuous mixtures and the measure problem
• Velocity fluctuations in a one dimensional Inelastic Maxwell model
• Monotonic entropy growth for a nonlinear model of random exchanges

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