New horizons in multidimensional diffusion: The Lorentz gas and the Riemann Hypothesis

Abstract

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 $\sqrt{t}$ 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 $\sqrt{t\ln t}$, 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.

Now 23 pages, including a new section 2 proposing that the mean square displacement differs from the variance of the limiting distribution for this model; there are also changes to notation, figures and references

References in corpus (5)

• Physics of the Riemann Hypothesis
• Open circular billiards and the Riemann hypothesis
• Diffusive properties of persistent walks on cubic lattices with application to periodic Lorentz gases
• Continuity properties of transport coefficients in simple maps
• Normal diffusion in crystal structures and higher-dimensional billiard models with gaps

Cited by in corpus (17)

• Free path lengths in quasicrystals
• Normal and Anomalous Diffusion in Soft Lorentz Gases
• Displacement Autocorrelation Functions for Strong Anomalous Diffusion: A Scaling Form, Universal Behavior, and Corrections to Scaling
• Measuring logarithmic corrections to normal diffusion in infinite-horizon billiards
• Recurrence of particles in static and time varying oval billiards
• The shape of shortest paths in random spatial networks
• Dispersion of particles in an infinite-horizon Lorentz gas
• Infinite horizon billiards: Transport at the border between Gauss and Lévy universality classes
• Power-law distributions for the free path length in Lorentz gases
• Tail asymptotics of free path lengths for the periodic Lorentz process. On Dettmann's geometric conjectures
• Time irreversible billiards with piecewise-straight trajectories
• Horizons and free path distributions in quasiperiodic Lorentz gases
• Random walks and Lorentz processes
• Efficient algorithms for general periodic Lorentz gases in two and three dimensions
• Periodic compression of an adiabatic gas: Intermittency enhanced Fermi acceleration
• Diffusion in the Inverted Triangular Soft Lorentz Gas
• Escape Dynamics of Many Hard Disks