Dynamical evolution in non-commutative discrete phase space and the derivation of classical kinetic equations

Citation
A. Dimakis et C. Tzanakis, Dynamical evolution in non-commutative discrete phase space and the derivation of classical kinetic equations, J PHYS A, 33(30), 2000, pp. 5267-5301
Citations number
50
Language
INGLESE
art.tipo
Article
Categorie Soggetti
Physics
Journal title
JOURNAL OF PHYSICS A-MATHEMATICAL AND GENERAL
ISSN journal
0305-4470 → ACNP
Volume
33
Issue
30
Year of publication
2000
Pages
5267 - 5301
Database
ISI
SICI code
0305-4470(20000804)33:30<5267:DEINDP>2.0.ZU;2-A
Abstract
By considering a lattice model of extended phase space, and using technique s of non-commutative differential geometry, we are led to: (a) the concept of vector fields as generators of motion and transition probability distrib utions 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 classica l dynamics. We show that, in the limit of a continuous description, these r esults lead to the time evolution of observables in terms of (the adjoint o f) generalized Fokker-Planck equations having: (1) a diffusion coefficient given by the limit of the correlation matrix of the lattice coordinates wit h 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 two -dimensional problems. Specifically we derive: (I) the equations of diffusi on, Smoluchowski and Fokker-Planck in velocity space. thus indicating the w ay random-walk models are incorporated in the present context; (II) Kramers ' equation, by further assuming that, motion is deterministic in coordinate space.