Non-commutative geometry and kinetic theory of open systems

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 of at most second order. For open systems interacting with a bath at canonical equilibrium they have the 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 that certain conditions, geometric in character, are fulfilled. To this end, symplectic geometry on M is developed in this context, and an outline of the required tensor analysis and differential geometry is given. Certain questions as regards the possible mathematical interpretation of this structure are also discussed.