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Hamiltonian chaos and statistical mechanics

Pierre Gaspard

Center for Nonlinear Phenomena and Complex Systems,
Université Libre de Bruxelles, Belgium

In systems of statistical mechanics, the chaotic dynamics is characterized by Lyapunov exponents which are of the order of the inverse of the intercollisional time between the particles. This time scale is the one of kinetics. Instead, the relaxation toward the thermodynamic equilibrium occurs on the longer time scale of hydrodynamics which is determined by transport properties such as diffusion, viscosity, or heat conductivity.

The connection between the chaotic and transport properties can be established thanks to the escape-rate theory or a newer theory which allows us to directly construct the hydrodynamic modes of diffusion and reaction-diffusion [1,2]. These hydrodynamic modes turn out to present fractal properties with a fractal dimension given in terms of the transport coefficients. The fractal character of the hydrodynamic modes results from the stretching and folding of nonequilibrium inhomogeneities induced by the chaotic dynamics. This mixing naturally leads to the entropy production expected from nonequilibrium thermodynamics [3].

The first lecture will be devoted to the relationship between chaos, transport properties, and entropy production. The second lecture will present the results of recent work on chaos in systems composed of many interacting particles [4].

[1] P. Gaspard, I. Claus, T. Gilbert, and J. R. Dorfman, ``Fractality of the Hydrodynamic Modes of Diffusion", Phys. Rev. Lett. 86 (2001) 1506.
[2] I. Claus and P. Gaspard, ``The fractality of the relaxation modes in deterministic reaction-diffusion systems", preprint cond-mat/0204264.
[3] J. R. Dorfman, P. Gaspard, and T. Gilbert, ``Entropy production of diffusion in spatially periodic deterministic systems", preprint nlin.CD/0203046.
[4] P. Gaspard and H. van Beijeren, ``When do tracer particles dominate the Lyapunov spectrum?", preprint nlin.CD/0112019.

next up previous
Next: Grabec Up: Abstracts Previous: Fujisaka