Classical dynamics on graphs - art. no. 066215

We consider the classical evolution of a particle on a graph by using a tim e-continuous Frobenius-Perron operator that generalizes previous propositio ns. In this way, the relaxation rates as well as the chaotic proper ties ca n be defined for the time-continuous classical dynamics on graphs. These pr operties are given as the zeros of some periodic-orbit zeta functions. We c onsider in detail the case of infinite periodic graphs where the particle u ndergoes a diffusion process. The infinite spatial extension is taken into account by Fourier transforms that decompose the observables and probabilit y densities into sectors corresponding to different values of the wave numb er. The hydrodynamic modes of diffusion are studied by an eigenvalue proble m of a Frobenius-Perron operator corresponding to a given sector. The diffu sion coefficient is obtained from the hydrodynamic modes of diffusion and h as the Green-Kubo form. Moreover, we study finite but large open graphs tha t 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 t he lifetime of a system that undergoes a diffusion process before it escape s.