On the derivation of a quantum Boltzmann equation from the periodic Von-Neumann equation

Abstract

We present the semi-conductor Boltzmann equation, which is time-reversible, and indicate that it can be formally derived by considering the large time and small perturbing potential limit in the Von-Neumann equation (time-reversible). We then rigorously compute the corresponding asymptotics in the case of the Von-Neumann equation on the Torus. We show that the limiting equation we obtain does not coincide with the physically realistic model. The former is indeed an equation of Boltzmann type, yet with memory in time, so that it appears to be time-reversible. We comment on this point, and further describe the properties of the limiting equation.