On The Weak-Coupling Limit for Bosons and Fermions

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 in spirit 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.