On the generalized Boltzmann equation of a quantum gas II

Abstract

Synopsis The generalized Boltzmann equation for a homogeneous imperfect gas reported in the previous paper [Physica 27 (1961) 940] is found to be inadequate. For the case of Fermi-Dirac and Maxwell-Boltzmann statistics it is replaced by ∂ n ¯ p ∂ t = 2 π ∑ n ′ δ ( e n ′ − e n ) 〈 n | { A † | n ′ 〉 〈 n ′ A } p | n 〉 ( n p ′ − n p ) ¯ ¯ , where n, n′ stand for quantum numbers describing the free-particle system, A for transition matrix, and the double bar means that the product of occupation numbers npn1n2 is to be replaced by the product of average numbers npn1n2…; The suffix p means that contributions only of proper linked diagrams are to be retained. The derivation of this result and the conditions of validity are critically discussed with the aid of Feynman diagrams, representing contractions referring to the n−n matrix element. A proper diagram is defined as that which does not contain particle lines with common momenta in it. It is argued that a proper diagram represents a collision process which may or may not conserve the energy. The case of Bose-Einstein statistics is not discussed here.