The Quantum Mean Field State as a Limit of Canonical States: Maxwell-Boltzmann Statistics

Abstract

This is a report on joint work with M.Pulvirenti and A.Teta¹. Stimulated by a recent paper² devoted to the mathematics of the quantum mean-field description of a one-component plasma (existence, uniqueness and classical limit of solutions to the Schrodinger-Poisson problem), we proposed to understand what kind of many-body system is that theory relevant to, more precisely to obtain it as a limit of N-particle quantum theories with suitably scaled interactions. The result, which I shall present below in a slightly more general form, is that the canonical k-particle reduced density matrices at fixed temperature for N particles obeying Maxwell- Boltzmann statistics, with charges of the order N ^−1/2, and in a fixed confining potential (e.g. in a finite box) converge, as N → ∞, to the tensor product of k one-particle density matrices, which are solutions of the self-consistency equations considered in ref. 2. Another possible choice would be to scale temperature like N ^1/3 and both mass and charge like N ^−1/3, what would correspond to a hot dense plasma. Though this convergence relates the Hartree theory of ref. 2 to interacting quantum Coulomb systems, not much insight is gained in this way concerning the range of applicability of the former. It turned out however that the techniques used in deriving it are simple enough to provide an alternate more transparent treatment and a better understanding of the Hartree theory itself (in particular, an easier control of the classical limit), and to apply hopefully to more subtle cases of mean field limits, e.g. for quantum statistics and not purely repulsive forces. This is why I shall include also a short outline of the proof.