Classical dynamics on graphs
Abstract
We consider the classical evolution of a particle on a graph by using a timecontinuous FrobeniusPerron operator that generalizes previous propositions. In this way, the relaxation rates as well as the chaotic properties can be defined for the timecontinuous classical dynamics on graphs. These properties are given as the zeros of some periodicorbit zeta functions. We consider in detail the case of infinite periodic graphs where the particle undergoes a diffusion process. The infinite spatial extension is taken into account by Fourier transforms that decompose the observables and probability densities into sectors corresponding to different values of the wave number. The hydrodynamic modes of diffusion are studied by an eigenvalue problem of a FrobeniusPerron operator corresponding to a given sector. The diffusion coefficient is obtained from the hydrodynamic modes of diffusion and has the GreenKubo form. Moreover, we study finite but large open graphs that converge to the infinite periodic graph when their size goes to infinity. The lifetime of the particle on the open graph is shown to correspond to the lifetime of a system that undergoes a diffusion process before it escapes.
 Publication:

Physical Review E
 Pub Date:
 June 2001
 DOI:
 10.1103/PhysRevE.63.066215
 arXiv:
 arXiv:nlin/0011045
 Bibcode:
 2001PhRvE..63f6215B
 Keywords:

 02.50.r;
 03.65.Sq;
 05.60.Cd;
 45.05.+x;
 Probability theory stochastic processes and statistics;
 Semiclassical theories and applications;
 Classical transport;
 General theory of classical mechanics of discrete systems;
 Nonlinear Sciences  Chaotic Dynamics;
 Condensed Matter  Statistical Mechanics;
 Nonlinear Sciences  Cellular Automata and Lattice Gases
 EPrint:
 42 pages and 8 figures