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arXiv · hep-th/9508035

Non-commutative Geometry and Kinetic Theory of Open Systems

A. Dimakis·C. Tzanakis
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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.

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A. Dimakis, C. Tzanakis. 1995-08-08. Non-commutative Geometry and Kinetic Theory of Open Systems. https://doi.org/10.1088/0305-4470%2F29%2F3%2F012

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