2000/11/24 by Felipe Barra, F. Barra, Pierre Gaspard +1 · 1 citation
Mathematics · Physics and Astronomy · #Chaotic #Continuous spectrum #Diffusion #Diffusion process #Discrete mathematics #Eigenvalues and eigenvectors #Fourier transform #Graph #Mathematical analysis #Mathematics #Operator (biology) #Physics #Quantum chaos and dynamical systems #Quantum mechanics #Spectroscopy and Quantum Chemical Studies #Theoretical and Computational Physics #cond-mat.stat-mech #nlin.CD #nlin.CG
paper · pdf · doi:10.1103/physreve.63.066215
42 pages and 8 figures
arxiv created 2000/11/24 · openalex publication_date 2001/05/23 · arxiv updated 2009/11/30 · openalex created_date 2025/10/10 · openalex updated_date 2026/08/05
We consider the classical evolution of a particle on a graph by using a time-continuous Frobenius-Perron operator that generalizes previous propositions. In this way, the relaxation rates as well as the chaotic properties can be defined for the time-continuous classical dynamics on graphs. These properties are given as the zeros of some periodic-orbit 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 Frobenius-Perron operator corresponding to a given sector. The diffusion coefficient is obtained from the hydrodynamic modes of diffusion and has the Green-Kubo 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.