Non-commutative Geometry and Kinetic Theory of Open Systems

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.

22 pages, LaTeX

References in corpus (1)

• Differential Calculi on Commutative Algebras

Cited by in corpus (7)

• Noncommutative Geometry of Finite Groups
• Differential Calculi and Linear Connections
• Noncommutative Geometries and Gravity
• Some aspects of noncommutative geometry and physics
• Noncommutative geometry and stochastic processes
• Soliton equations and the zero curvature condition in noncommutative geometry
• Dynamical Evolution in Noncommutative Discrete Phase Space and the Derivation of Classical Kinetic Equations