On spatially homogeneous solutions of a modified Boltzmann equation for Fermi-Dirac particles

摘要

The paper considers a modified spatially homogeneous Boltzmann equation for Fermi Dirac particles (BFD). We prove that for the BFD equation there are only two classes of equilibria: the first ones are Fermi Dirac distributions, the second ones are characteristic functions of the Euclidean balls, and they can be simply classified in terms of temperatures: T > 2/5 T-F and T = 2/5 T-F, where T-F denotes the Fermi temperature. In general we show that the L-infinity-bound 0 less than or equal to f less than or equal to 1/epsilon derived from the equation for solutions implies the temperature inequality T greater than or equal to 2/5 T-F and if T > 2/5 T-F, then f trend towards Fermi Dirac distributions; if T = 2/5 T-F, then f are the second equilibria. In order to study the long-time behavior, we also prove the conservation of energy and the entropy identity, and establish the moment production estimates for hard potentials. [References: 22]