From the text: We consider the quantum dynamics of an electron in a periodic box of large size \(L\), for long time scales \(T\), in a \(d\)-dimensional space with \(d\geq 3\). One obstacle occupying a volume of order \(O(1)\) 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. The paper investigates the asymptotic dynamics in the typical low-density regime \(T\sim L^d\), \(L\to\infty\). The coupling constant has to be rescaled and small, namely \(\lambda\sim L^{-d+2}\to 0\). More general regimes are in fact also considered. The analysis is easily adapted in the case of Dirichlet boundary conditions.
The motivation of this work is based on the following observations. First, the dynamics of an electron moving in a field of obstacles and in a low-density regime is ‘in general’ asymptotically described by a time-irreversible Boltzmann equation. Large finite boxes, which is where the limit in \(L\) originates, are often used in the physical literature to formally justify this statement. Secondly, many rigorous mathematical works establish precise convergence results of the von Neumann equation towards a (linear) Boltzmann equation in various situations where the obstacles are randomly distributed. In particular, the initially time-reversible model is proved to be asymptotically described by a time-irreversible equation, but the convergence has only been proved true in expectation value. Hence the result typically fails for some ‘zero-measure set’ of configurations of the obstacles. This is due to the fact that the limiting equation is time-irreversible. However, these works do not give any quantitative information about the specific configurations of obstacles where the convergence actually fails. Moreover, the physical derivations relying on taking large finite boxes are mathematically as well as physically questionable.
Starting from these observations, the present paper is at variance with a model which is deterministic. We exhibit an effective setting where the formal convergence towards a Boltzmann equation actually fails. We prove 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, contrary to what is expected in more general geometries or distributions of obstacles, (b) it is time-reversible, (c) it is the same for any time scale \(T\) such that \(T/L^2\to\infty\), 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 main 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.