Authors

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.