Non-commutative Geometry and Kinetic Theory of Open Systems

TL;DR

This paper explores how non-commutative geometry can be used to derive generalized Fokker-Planck equations for open systems interacting with a thermal bath, linking geometric structures with kinetic theory.

Contribution

It introduces a non-commutative differential geometric framework to derive kinetic equations for open systems, extending classical Hamiltonian dynamics to include dissipative effects.

Findings

Derivation of generalized Fokker-Planck equations from non-commutative Hamiltonian structures

Development of symplectic geometry tailored for open systems

Discussion on mathematical interpretation of non-commutative differential structures

Abstract

The basic mathematical assumptions for autonomous linear kinetic equations for a classical system are formulated, leading to the conclusion that if they are differential equations on its phase space $M$, they are at most of the 2nd order. For open systems interacting with a bath at canonical equilibrium they have a particular form of an equation of a generalized Fokker-Planck type. We show that it is possible to obtain them as Liouville equations of Hamiltonian dynamics on $M$ with a particular non-commutative differential structure, provided certain geometric in character, conditions are fulfilled. To this end, symplectic geometry on $M$ is developped in this context, and an outline of the required tensor analysis and differential geometry is given. Certain questions for the possible mathematical interpretation of this structure are also discussed.