Quantum Fokker–Planck Dynamics

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 non-commutative analysis. In closing, we then construct and examine the asymptotic behaviour of the corresponding Markov semigroups. We also present a non-commutative 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.