Journal
JOURNAL OF STATISTICAL PHYSICS
Volume 146, Issue 1, Pages 181-204Publisher
SPRINGER
DOI: 10.1007/s10955-011-0397-2
Keywords
Anomalous diffusion; Infinite horizon; Lorentz gas; Riemann hypothesis
Categories
Funding
- Royal Society
- Chaotic and Transport Properties of Higher-Dimensional Dynamical Systems
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The Lorentz gas is a billiard model involving a point particle diffusing deterministically in a periodic array of convex scatterers. In the two dimensional finite horizon case, in which all trajectories involve collisions with the scatterers, displacements scaled by the usual diffusive factor root t are normally distributed, as shown by Bunimovich and Sinai in 1981. In the infinite horizon case, motion is superdiffusive, however the normal distribution is recovered when scaling by root t ln t, with an explicit formula for its variance. Here we explore the infinite horizon case in arbitrary dimensions, giving explicit formulas for the mean square displacement, arguing that it differs from the variance of the limiting distribution, making connections with the Riemann Hypothesis in the small scatterer limit, and providing evidence for a critical dimension d=6 beyond which correlation decay exhibits fractional powers. The results are conditional on a number of conjectures, and are corroborated by numerical simulations in up to ten dimensions.
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