• DOI:10.1007/s00023-021-01134-0
• Corpus ID: 235390751

Quantum Fokker–Planck Dynamics

@article{Labuschagne2021QuantumFD,
  title={Quantum Fokker–Planck Dynamics},
  author={Louis E. Labuschagne and W. Adam Majewski},
  journal={Annales Henri Poincar{\'e}},
  year={2021}
}

• L. Labuschagne, W. Majewski
• Published 10 June 2021
• Mathematics
• Annales Henri Poincaré

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 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 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.