Towards nonlinear quantum Fokker-Planck equations

TL;DR

This paper develops a new semiclassical Klein-Kramers equation for the Wigner function, improving the description of quantum Brownian motion and proposing extensions to nonlinear quantum Fokker-Planck equations using Fisher information.

Contribution

It introduces a novel semiclassical Klein-Kramers equation and extends it to nonlinear quantum Fokker-Planck equations based on Fisher information, advancing quantum stochastic process modeling.

Findings

Derived a new semiclassical Klein-Kramers equation for the Wigner function.

Formulated a semiclassical Smoluchowski equation generalizing classical Brownian motion.

Discussed extensions to nonlinear quantum Fokker-Planck equations using Fisher information.

Abstract

It is demonstrated how the equilibrium semiclassical approach of Coffey et al. can be improved to describe more correctly the evolution. As a result a new semiclassical Klein-Kramers equation for the Wigner function is derived, which remains quantum for a free quantum Brownian particle as well. It is transformed to a semiclassical Smoluchowski equation, which leads to our semiclassical generalization of the classical Einstein law of Brownian motion derived before. A possibility is discussed how to extend these semiclassical equations to nonlinear quantum Fokker-Planck equations based on the Fisher information.