Quantum FokkerPlanck Dynamics
Abstract
The FokkerPlanck 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 FokkerPlanck dynamics, as a means to describing quantum FokkerPlanck dynamics. Given that relevant models relate to the description of large systems, the quantization of the FokkerPlanck equation should be done in a manner that respects this fact, and is therefore carried out within the setting of noncommutative 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 CsiszarKullback 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.
 Publication:

Annales Henri Poincaré
 Pub Date:
 May 2022
 DOI:
 10.1007/s00023021011340
 arXiv:
 arXiv:2106.05718
 Bibcode:
 2022AnHP...23.1659L
 Keywords:

 Mathematics  Operator Algebras;
 Mathematical Physics;
 Quantum Physics;
 Primary: 46L55;
 47D07;
 Secondary 46L51;
 46N50;
 46L57
 EPrint:
 The final version submitted to AHP. A brief account of applied quantization as well as the comprehensive description of closability of quantum Laplacian is added