Autocorrelation Function and Dynamical Transition on the Periodic Lorentz Gas: Stochastic and Deterministic Approaches

The Periodic Lorentz Gas is a system where a point mass moves freely between a periodical arrangement of scatterer disks, often modeled by Sinai billiards.
The main interest in its study is the diffusion coefficient, which shows the connection between the billiard geometry and its statistical and dynamical properties.
It is known that the geometric property that mostly influences the dynamics is the horizon, which defines whether a trajectory along corridors without collisions is possible (infinite horizon or H$\infty$) or not (without horizon or H0).
The dynamics is hyperbolic and normal diffusion is observed on H0 billiards, and it is non-hyperbolic and anomalously superdiffusive on H$\infty$.
The stochastic approach to diffusion in billiards has recently gained a lot of interest. It is based on random walks of particles between traps formed by adjacent scatterers, so the diffusion coefficient is determined in function of escape time and lattice spacing between traps.
Another method to calculate the diffusion coefficient is the integration of the velocity autocorrelation function $C(t)$. This technique is interesting since the $C(t)$ decay provides information on the billiards's dynamics as it is known for decaying exponentially on H0 and algebraically on H$\infty$ billiards. Our interest is on the transition of geometries between H$\infty$ and H0, exploring the variety of configurations of a double square lattice by finely tuning the disks radii.
In our case, we propose an escape-rate formalism, where the particles perform a continuous-time step between traps but maintain correlations if they travel along corridors on H$\infty$. We have introduced and demonstrated that the survival probability $\phi(t)$ provides information on the dynamics as well as the diffusion coefficient.
By numerical simulation, $\phi(t)$, $C(t)$ and the diffusion coefficient were determined. We observed similar behavior in the decay of $\phi(t)$ and $C(t)$, which suggests that both quantities are originated from the same dynamics within the billiard. Nevertheless, the asymptotic behavior of $\phi(t)$ shows a smooth curve which is advantageous over $C(t)$ since the latter oscillates strongly at long times. The divergence and convergence of the diffusion coefficient on H$\infty$ and H0 configurations, respectively, are also observed. On H$\infty$ geometries with narrow corridors, the normal diffusion regime is still dominant when the phase space of the corridors are very small. However, $\phi(t)$ is sensible enough to the dynamics to show the algebraic tail and transition time between exponential and algebraic decay even on this limit.