Inhomogeneous Boltzmann equations: distance, asymptotics and comparison of the classical and quantum cases

The notion of distance between a global Maxwellian function and an arbitrary solution $f$ (with the same total density $\rho$ at the fixed moment $t$) of Boltzmann equation is introduced. In this way we essentially generalize the important Kullback-Leibler distance, which was used before. Namely, we generalize it for the spatially inhomogeneous case. An extremal problem to find a solution of the Boltzmann equation, such that $\text{dist}\{M,f\}$ is minimal in the class of solutions with the fixed values of energy and of $n$ moments, is solved. The cases of the classical and quantum (for Fermi and Bose particles) Boltzmann equations are studied and compared. The asymptotics and stability of solutions of the Boltzmann equations are also considered.