A derivation of a new set of equations at the onset of the Bose–Einstein condensation

Abstract

In this paper, we derive the equations characterizing the boundary layer which describes the transition of the distribution function of a gas of weakly interacting bosons to the distribution function of the gas in the presence of a Bose–Einstein condensate. To this end, we first rederive the classical Uehling–Uhlenbeck equation very briefly, taking as a starting point the dynamics of a system of many weakly interacting quantum particles. The solutions of the Uehling–Uhlenbeck equation yield blow up in finite time. Near the blow-up time, the approximations used to derive the Uehling–Uhlenbeck equation break down. We derive the set of equations that describe the building of correlations and the onset of quantum interference effects for the many-particle Hamiltonian system under the assumption that the blow-up for the Uehling–Uhlenbeck equation takes place in a self-similar form.