Journal of Statistical Physics, Vol. 105, Nos. 1/2, October 2001, p. 353-388.
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 > 2/5T_F and T=2/5T_F, where T_F denotes the Fermi temperature. In general we show that the L^(oo)-bound 0 \< f \< 1/E derived from the equation for solutions implies the temperature inequality T >/ 2/5T_F, and if T > 2/5T_F, then f trend towards Fermi–Dirac distributions; if T=2/5T_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.