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Classical dynamics on graphs.

F Barra1, P Gaspard

  • 1Center for Nonlinear Phenomena and Complex Systems, Université Libre de Bruxelles, Campus Plaine Code Postal 231, B-1050 Brussels, Belgium.

Physical Review. E, Statistical, Nonlinear, and Soft Matter Physics
|June 21, 2001
PubMed
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This study introduces a time-continuous Frobenius-Perron operator for particle dynamics on graphs, defining relaxation rates and chaotic properties. It analyzes diffusion on infinite periodic graphs and finite open graphs, yielding a Green-Kubo form for the diffusion coefficient.

Area of Science:

  • Statistical Mechanics
  • Dynamical Systems Theory
  • Graph Theory

Background:

  • Classical particle evolution on graphs lacks a unified framework for continuous time.
  • Defining relaxation rates and chaotic properties for continuous-time graph dynamics is challenging.

Purpose of the Study:

  • To generalize classical particle dynamics on graphs using a time-continuous Frobenius-Perron operator.
  • To define and analyze relaxation rates and chaotic properties for these dynamics.
  • To investigate diffusion processes on infinite periodic and finite open graphs.

Main Methods:

  • Utilized a time-continuous Frobenius-Perron operator.
  • Defined chaotic properties via zeros of periodic-orbit zeta functions.
  • Employed Fourier transforms for infinite graphs to decompose into wave number sectors.

Related Experiment Videos

  • Analyzed diffusion via an eigenvalue problem of the Frobenius-Perron operator.

    • Main Results:

    • Established a method to define relaxation rates and chaotic properties for continuous-time graph dynamics.
    • Derived the diffusion coefficient for infinite periodic graphs in Green-Kubo form.
    • Demonstrated that particle lifetime on large open graphs matches diffusion escape time.

    Conclusions:

    • The time-continuous Frobenius-Perron operator provides a robust framework for studying particle dynamics on graphs.
    • The study offers insights into diffusion processes and chaotic behavior in both infinite and finite graph systems.
    • Results connect particle lifetime in finite systems to diffusion escape in the infinite limit.