Quantum Fokker-Planck Dynamics

TL;DR

This paper develops a quantum version of the Fokker-Planck equation using non-commutative analysis, enabling the study of quantum stochastic dynamics and their asymptotic properties.

Contribution

It introduces a non-commutative quantization of the Fokker-Planck equation within von Neumann algebras, including a generalized Laplace operator, potential terms, and entropy-based asymptotic analysis.

Findings

Constructed a quantum Fokker-Planck model within non-commutative analysis.

Established a noncommutative Csiszar-Kullback inequality related to entropy.

Analyzed the asymptotic behavior of quantum Markov semigroups.

Abstract

The Fokker-Planck equation is a partial differential equation which is a key ingredient in many models in physics. This paper aims to obtain a quantum counterpart of Fokker-Planck dynamics, as a means to describing quantum Fokker-Planck dynamics. Given that relevant models relate to the description of large systems, the quantization of the Fokker-Planck equation should be done in a manner that respects this fact, and is therefore carried out within the setting of non-commutative analysis based on general von Neumann algebras. Within this framework we present a quantization of the generalized Laplace operator, and then go on to incorporate a potential term conditioned to noncommutative analysis. In closing we then construct and examine the asymptotic behaviour of the corresponding Markov semigroups. We also present a noncommutative Csiszar-Kullback inequality formulated in terms of a notion of relative entropy, and show that for more general systems, good behaviour with respect to this notion of entropy ensures similar asymptotic behaviour of the relevant dynamics.