Boltzmann equation and wave kinetic equations

The well-known nonlinear kinetic equations (in particular, the wave kinetic equation and the quantum Nordheim – Uehling – Uhlenbeck equations) are considered as a natural generalization of the classical spatially homogeneous Boltzmann equation. To this goal we introduce the generalized kinetic equation that depends on a function of four real variables $F(x_1, x_2; x_3, x_4)$. The function F is assumed to satisfy certain commutation relations. The general properties of this equation are studied. It is shown that the above mentioned kinetic equations correspond to different forms of the function (polynomial) $F$. Then the problem of discretization of the generalized kinetic equation is considered on the basis of ideas which are similar to those used for construction of discrete models of the Boltzmann equation. The main attention is paid to discrete models of the wave kinetic equation. It is shown that such models possses a monotone functional similar to Boltzmann $H$-function. The behaviour of solutions of the simplest Broadwell model for the wave kinetic equation is discussed in detail.