Some considerations on the derivation of the nonlinear quantum Boltzmann equation II: The low density regime

In this paper we analyse the asymptotic dynamics of a system of N identical quantum particles in a low-density regime. Our approach follows the strategy introduced by the authors in a previous work,((2)) to treat the simpler weak coupling regime. The time evolution of the Wigner transform of the one-particle reduced density matrix is represented by means of a perturbative series. The expansion is obtained upon iterating the Duhamel formula, in the spirit of the paper by Lanford.((32)) For short times and small interaction potential, we rigorously prove that a subseries of the complete perturbative series, converges to the solution of the nonlinear Boltzmann equation that is physically relevant in the context. An important point is that we completely identify the cross-section entering the limiting Boltzmann equation, as being the Born series expansion of quantum scattering. As in ref. 2, our convergence result is only partial, in that we merely characterize the asymptotic behaviour of a subseries of the complete original perturbative expansion. We only have plausibility arguments in the direction of proving that the terms we neglect, when going from the original series to its associated subseries, are indeed vanishing in the limit. The present study holds in any dimension d >= 3.