Dynamical Evolution in Noncommutative Discrete Phase Space and the Derivation of Classical Kinetic Equations

Details Dynamical Evolution in Noncommutative Discrete Phase Space and the Derivation of Classical Kinetic Equations Dynamical Evolution in Noncommutative Discrete Phase Space and the Derivation of Classical Kinetic Equations

1999-12-20

arxiv.org/abs/math-ph/9912016v1

Physics

Mathematical Physics

LaTeX2e, 40 pages, 1 Postscript figure, uses package epsfig

Scientific paper

10.1088/0305-4470/33/30/301

By considering a lattice model of extended phase space, and using techniques of noncommutative differential geometry, we are led to: (a) the conception of vector fields as generators of motion and transition probability distributions on the lattice; (b) the emergence of the time direction on the basis of the encoding of probabilities in the lattice structure; (c) the general prescription for the observables' evolution in analogy with classical dynamics. We show that, in the limit of a continuous description, these results lead to the time evolution of observables in terms of (the adjoint of) generalized Fokker-Planck equations having: (1) a diffusion coefficient given by the limit of the correlation matrix of the lattice coordinates with respect to the probability distribution associated with the generator of motion; (2) a drift term given by the microscopic average of the dynamical equations in the present context. These results are applied to 1D and 2D problems. Specifically, we derive: (I) The equations of diffusion, Smoluchowski and Fokker-Planck in velocity space, thus indicating the way random walk models are incorporated in the present context; (II) Kramers' equation, by further assuming that, motion is deterministic in coordinate space

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