The advances in the theory of dynamical systems have brought a new life to Boltzmann's mechanical formulation of statistical mechanics, especially for systems near or far from equilibrium, and yielded new sets of microscopic dynamics formulas for macroscopic observables such as the transport coefficients.

The classical Boltzmann equation for evolution of 1-particle density is based on stosszahlansatz, neglect of particle correlations prior to, or after a 2-particle collision. It is a very good approximate description of dilute gas dynamics, but a difficult starting point for inclusion of systematic corrections. In the theory of deterministic diffusion developed in recent years, no correlations are neglected - they are all included in the exact cycle expansions for transport coefficients such as the diffusion constant.

The exact results are sometimes counterintuitive, and might help us decide whether a diffusive phenomenon whose microscopic dynamics is hard to observe directly, such as conductance fluctuations in a mesoscopic device, is due to impurities or to deterministic transport. For example, as some parameter (such as mean free flight time) is increased, the deterministic diffusion coefficient reveals a non-monotone, fractal dependence.

I will illustrate the theory with a ``baby'' example, computation of the difussion constant for deterministic diffusion in periodic Lorentz gas. For systems of a few degrees of freedom these results are on rigorous footing, but there are indications that they capture the essential dynamics of systems of many degrees of freedom as well. Honesty in advertising requires disclaimer; no such fractal behavior of the conductance has been detected experimentally so far.

May 17, 1999           NU CondMat seminar