Non derivation of the quantum Boltzmann equation from the periodic von Neumann equation

We consider the quantum dynamics of an electron in a periodic box of large size L, for long time scales T, in d dimensions dimensions of space, d greater than or equal to 3. One obstacle occupying a volume 1 is present in the box. The coupling constant between the electron and the obstacle is lambda. The model is described by a scaled periodic von Neumann equation with a potential, a time-reversible equation. We investigate the asymptotic dynamics in the typical low-density regime T similar to L-d, L --> infinity. The coupling constant has to be rescaled and small, namely lambda similar to L-d+2 --> 0. More general regimes are in fact considered. Our analysis is easily adapted in the case of Dirichlet boundary conditions. Loosely speaking, the dynamics of an electron moving in a field of obstacles and in the low-density regime is, in general, asymptotically described by a time-irreversible Boltzmann equation. Large finite boxes are often used in the physical literature to formally justify this statement. However, the above asymptotics has only been proved true for randomly distributed obstacle, say. On the more, the physical derivations relying on taking large finite boxes are mathematically as well as physically questionable. Starting from these observations, we investigate here in a quantitative way the case of an electron moving in a large periodic (or Dirichlet) box, with a given deterministic obstacle. We prove here that both periodicity and the fact that the obstacle is deterministic, create strong phase coherence effects which dominate the asymptotic process. This implies that, (a) the limiting dynamics is not the Boltzmann equation, (b) it is time-reversible, (c) it is the same for any time scale T such that, roughly, T/L-2 --> infinity, and (d) the unusual rescaling of lambda is needed as well. However, the convergence proved here only holds as a term-by-term convergence of certain series. Our result relies on the analysis of certain Riemann sums with arithmetic constraints, and number theoretic considerations relating the asymptotic distribution of integer vectors on spheres of large radius happen to play a key role in this paper.