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

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 distribution s, 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/eps ilon derived from the equation for solutions implies the temperature inequa lity T greater than or equal to 2/5 T-F and if T > 2/5 T-F, then f trend to wards Fermi Dirac distributions; if T = 2/5 T-F, then f are the second equi libria. In order to study the long-time behavior, we also prove the conserv ation of energy and the entropy identity, and establish the moment producti on estimates for hard potentials.