Diffusive behaviour of the Wigner function and decoherence

Abstract

In this chapter we express the classical Wigner function by a degenerate diffusion process [164] solving a randomly perturbed Newton equation. We discuss first some estimates on the solution of these equations and the long time behaviour of the diffusion process. We discuss a relation between the classical Wigner function and the quantum one. We approximate the large time behaviour of the density matrix by the classical Wigner function. In this way we obtain some estimates on the decoherence in space expressed by the correlation functions of the diffusion process.