Transport properties, Lyapunov exponents, and entropy per unit time

Abstract

For dynamical systems of large spatial extension giving rise to transport phenomena, like the Lorentz gas, we establish a relationship between the transport coefficient and the difference between the positive Lyapunov exponent and the Kolmogorov-Sinai entropy per unit time, characterizing the fractal and chaotic repeller of trapped trajectories. Consequences for nonequilibrium statistical mechanics are discussed.

Deterministic chaos and the foundations of the kinetic theory of gases

“…Here we argue that it is possible to apply dynamical systems theory to such a case, and that this application shows a deep connection between transport coefficients and the properties of chaotic dynamical systems. This connection was first made by Gaspard and Nicolis [17]. This is an appropriate place to say that by a chaotic dynamical system we mean one that has a positive KS entropy, i.e., exponential separation of trajectories, and a bounded phase space so that folding takes place.…”

“…where the summation is over all the positive Lyapunov exponents, λ +,i (R) for trajectories on the repeller, and h KS (R) is the KS entropy for these trajectories [1,17,32,35]. Note that we have a system for which Pesin's theorem does not hold, but by a small amount of order L −2 .…”

“…Eq. (33) is due to Gaspard and Nicolis [17]. Gaspard and Dorfman have extended this method to apply to the other transport coefficients, such as the shear and bulk viscosities, and thermal conductivity of a fluid, as well as for chemical reaction rates [37].…”

“…In Section 3 we consider some recent applications of ideas from dynamical systems theory to the theory of transport phenomena, and illustrate the role of fractal structures that are responsible for diffusion and other transport processes. We consider there the application of escape-rate methods to transport theory, as developed by P. Gaspard and G. Nicolis [17], and the method of thermostatted dynamical systems, developed by D. Evans, W. Hoover, H. A. Posch, G. Morriss, and E. G. D. Cohen [18], as well as other authors. In Section 4 we use these ideas to present our current understanding of the dynamical origins of the Boltzmann equation, and to show how this equation, in turn, can be used to calculate quantities such as Lyapunov exponents which measure the chaotic nature of the relevant dynamics.…”

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“…The escape rate can also be interpreted as a macroscopic quantity resulting e. g. from a diffusion process described by a Fokker-Planck equation for the macroscopic density of particles. This connection between microscopic dynamics and macroscopic processes, known as the escape rate formalism [9,10,11,12,13], yields expressions of the transport coefficients, e. g. the diffusion coefficient, in terms of the dynamical quantities. The existence of this connection relies heavily on the hyperbolic properties of the system, i. e. (i) (almost) every point in phase space is assumed to be of saddle type, and (ii), for the open boundaries, the repeller is fractal.…”