We present some arguments in favor 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 Pi(t) of being in state i at time t is a suitable model for granular gases and is referred to here as the Boltzmann–Fokker–Planck (BFP) equation. The conjectured H-functional, which appears to be non-increasing, is HC(t) = ∑iPi(t)lnPi(t)/Πi with Πi = limt→∞Pi(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.
About an H-theorem for systems with non-conservative interactions
MARINI BETTOLO MARCONI, Umberto;
2013-01-01
Abstract
We present some arguments in favor 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 Pi(t) of being in state i at time t is a suitable model for granular gases and is referred to here as the Boltzmann–Fokker–Planck (BFP) equation. The conjectured H-functional, which appears to be non-increasing, is HC(t) = ∑iPi(t)lnPi(t)/Πi with Πi = limt→∞Pi(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.I documenti in IRIS sono protetti da copyright e tutti i diritti sono riservati, salvo diversa indicazione.
