Measuring logarithmic corrections to normal diffusion in infinite-horizon billiards

TL;DR

This paper numerically investigates the super-diffusive behavior in a two-dimensional infinite-horizon billiard model, revealing challenges in observing the expected logarithmic corrections due to dominant linear growth in finite simulation times.

Contribution

It provides the first detailed numerical analysis of the logarithmic corrections to normal diffusion in infinite-horizon billiards, highlighting the dominance of linear growth over asymptotic logarithmic effects.

Findings

Logarithmic corrections are overshadowed by linear growth in accessible simulation times.

Numerical results align with analytical variance predictions for the limiting distribution.

Challenges in observing asymptotic behavior in finite-time simulations are demonstrated.

Abstract

We perform numerical measurements of the moments of the position of a tracer particle in a two-dimensional periodic billiard model (Lorentz gas) with infinite corridors. This model is known to exhibit a weak form of super-diffusion, in the sense that there is a logarithmic correction to the linear growth in time of the mean-squared displacement. We show numerically that this expected asymptotic behavior is easily overwhelmed by the subleading linear growth throughout the time-range accessible to numerical simulations. We compare our simulations to the known analytical results for the variance of the anomalously-rescaled limiting normal distributions.