Lyapunov exponents and anomalous diffusion of a Lorentz gas with infinite horizon using approximate zeta functions

Abstract

We compute the Lyapunov exponent, generalized Lyapunov exponents and the diffusion constant for a Lorentz gas on a square lattice, thus having infinite horizon. Approximate zeta functions, written in terms of probabilities rather than periodic orbits, a re used in order to avoid the convergence problems of cycle expansions. The emphasis is on the relation between the analytic structure of the zeta function, where a branch cut plays an important role, and the asymptotic dynamics of the system. We find a diverging diffusion constant $D(t)\sim \log t$ and a phase transition for the generalized Lyapunov exponents.

14 pages LaTeX, figs 2-3 on .uu file, fig 1 available from author

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• On the decay of correlations in Sinai billiards with infinite horizon
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• Measuring logarithmic corrections to normal diffusion in infinite-horizon billiards
• Periodic orbit theory of strongly anomalous transport
• The role of singularities in chaotic spectroscopy
• Error of semiclassical eigenvalues in the semiclassical limit - An asymptotic analysis of the Sinai billiard
• On the effect of pruning on the singularity structure of zeta functions
• Slow relaxation in weakly open vertex-splitting rational polygons
• Boltzmann-type approach to transport in weakly interacting one-dimensional fermionic systems