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Deterministic chaos and diffusion in maps and billiard

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Mathematics for the Fluid Earth

A fundamental problem of statistical mechanics and dynamical systems theory is to understand transport processes such as diffusion on the basis of deterministic chaos. In my talk I will discuss this issue for deterministic random walks in one and two dimensions generated by simple dynamical systems. For a class of piecewise linear maps lifted onto the whole real line the parameter-dependent diffusion coefficient can be calculated exactly analytically. It turns out that the response of these systems to parameter variations is non-trivial by displaying both linear and fractal parameter dependencies in the diffusion coefficient. Computer simulations predict analogous results for Hamiltonian particle billiards like the periodic Lorentz gas. These results are supported by systematic approximations based on a Taylor-Green-Kubo formula. [1] R.Klages, N.Korabel, Understanding deterministic diffusion by correlated random walks, J.Phys.A: Math. Gen.35, 4823 (2002) [2] R.Klages, Microscopic Chaos, Fractals and Transport in Nonequilibrium Statistical Mechanics (World Scientific, Singapore, 2007)

This talk is part of the Isaac Newton Institute Seminar Series series.

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