ON THE WEAK-COUPLING LIMIT FOR BOSONS AND FERMIONS

Abstract

In this paper we consider a large system of bosons or fermions. We start with an initial datum which is compatible with the Bose–Einstein, respectively Fermi–Dirac, statistics. We let the system of interacting particles evolve in a weak-coupling regime. We show that, in the limit, and up to the second order in the potential, the perturbative expansion expressing the value of the one-particle Wigner function at time t, agrees with the analogous expansion for the solution to the Uehling–Uhlenbeck equation. This paper follows the same spirit as the companion work, 2 where the authors investigated the weak-coupling limit for particles obeying the Maxwell–Boltzmann statistics: here, they proved a (much stronger) convergence result towards the solution of the Boltzmann equation.