Dynamical evolution in noncommutative discrete phase space and the derivation of classical kinetic equations
Abstract
By considering a lattice model of extended phase space, and using techniques of noncommutative differential geometry, we are led to: (a) the concept 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 evolution of the observables 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 FokkerPlanck 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 one and twodimensional problems. Specifically, we derive: (I) the equations of diffusion, Smoluchowski and FokkerPlanck in velocity space, thus indicating the way randomwalk models are incorporated in the present context; (II) Kramers' equation, by further assuming that, motion is deterministic in coordinate space.
 Publication:

Journal of Physics A Mathematical General
 Pub Date:
 August 2000
 DOI:
 10.1088/03054470/33/30/301
 arXiv:
 arXiv:mathph/9912016
 Bibcode:
 2000JPhA...33.5267D
 Keywords:

 Mathematical Physics
 EPrint:
 LaTeX2e, 40 pages, 1 Postscript figure, uses package epsfig