Dissipative Systems and Objective Description: Quantum Brownian Motion as an Example

A structure of generator of a quantum dynamical semigroup for the dynamics of a test particle interacting through collisions with the environment is considered, which has been obtained from a microphysical model. The related master-equation is shown to go over to a Fokker–Planck equation for the description of Brownian motion at quantum level in the long wavelength limit. The structure of this Fokker–Planck equation is expressed in this paper in terms of superoperators, giving explicit expressions for the coefficient of diffusion in momentum in correspondence with two cases of interest for the interaction potential. This Fokker–Planck equation gives an example of a physically motivated generator of quantum dynamical semigroup, which serves as a starting point for the theory of measurement continuous in time, allowing for the introduction of trajectories in quantum mechanics. This theory has in fact already been applied to the problem of Brownian motion referring to similar phenomenological structures obtained only on the basis of mathematical requirements. KEY WORDS: quantum theory; complete positivity; Fokker–Planck; quantum Brownian motion; dynamic structure factor