Cauchy universality and random billiards (original) (raw)

Deterministic diffusion in flower-shaped billiards

Physical Review E, 2002

We propose a flower shape billiard in order to study the irregular parameter dependence of chaotic normal diffusion. Our model is an open system consisting of periodically distributed obstacles of flower shape, and it is strongly chaotic for almost all parameter values. We compute the parameter dependent diffusion coefficient of this model from computer simulations and analyze its functional form by different schemes all generalizing the simple random walk approximation of Machta and Zwanzig. The improved methods we use are based either on heuristic higher-order corrections to the simple random walk model, on lattice gas simulation methods, or they start from a suitable Green-Kubo formula for diffusion. We show that dynamical correlations, or memory effects, are of crucial importance to reproduce the precise parameter dependence of the diffusion coefficent.

Anomalous diffusion in infinite horizon billiards

Physical Review E, 2003

We consider the long time dependence for the moments of displacement |r| q of infinite horizon billiards, given a bounded initial distribution of particles. For a variety of billiard models we find |r| q ∼ t γq (up to factors of log t). The time exponent, γq, is piecewise linear and equal to q/2 for q < 2 and q − 1 for q > 2. We discuss the lack of dependence of this result on the initial distribution of particles and resolve apparent discrepancies between this time dependence and a prior result. The lack of dependence on initial distribution follows from a remarkable scaling result that we obtain for the time evolution of the distribution function of the angle of a particle's velocity vector.

Transport in polygonal billiards

Physica D: Nonlinear Phenomena, 2004

We present in this work a numerical study of the dynamics of ensembles of point particles within a polygonal billiard chain. This billiard is a system with no exponential instability. Our numerical results suggest that some members of the family exhibit normal diffusive behavior while others present anomalous diffusion. Our conclusions are drawn from the numerical evaluation of the mean square displacement, the velocity autocorrelation function and its spectral analysis. Furthermore we analyze the properties of the incoherent scattering function. The super Burnett coefficient seems to be ill defined in all systems. The multifractal analysis of the spectrum of the velocity autocorrelation functions is also reported. Finally, we study the heat conduction in our polygonal chain.

Irregular diffusion in the bouncing ball billiard

Physica D-nonlinear Phenomena, 2004

We call a system bouncing ball billiard if it consists of a particle that is subject to a constant vertical force and bounces inelastically on a one-dimensional vibrating periodically corrugated floor. Here we choose circular scatterers that are very shallow, hence this billiard is a deterministic diffusive version of the well-known bouncing ball problem on a flat vibrating plate. Computer simulations show that the diffusion coefficient of this system is a highly irregular function of the vibration frequency exhibiting pronounced maxima whenever there are resonances between the vibration frequency and the average time of flight of a particle. In addition, there exist irregularities on finer scales that are due to higher-order dynamical correlations pointing towards a fractal structure of this curve. We analyze the diffusive dynamics by classifying the attracting sets and by working out a simple random walk approximation for diffusion, which is systematically refined by using a Green–Kubo formula.

Machta-Zwanzig regime of anomalous diffusion in infinite-horizon billiards

Physical review. E, Statistical, nonlinear, and soft matter physics, 2014

We study diffusion on a periodic billiard table with an infinite horizon in the limit of narrow corridors. An effective trapping mechanism emerges according to which the process can be modeled by a Lévy walk combining exponentially distributed trapping times with free propagation along paths whose precise probabilities we compute. This description yields an approximation of the mean squared displacement of infinite-horizon billiards in terms of two transport coefficients, which generalizes to this anomalous regime the Machta-Zwanzig approximation of normal diffusion in finite-horizon billiards [J. Machta and R. Zwanzig, Phys. Rev. Lett. 50, 1959 (1983)PRLTAO0031-900710.1103/PhysRevLett.50.1959].

Stochastic perturbations of convex billiards

Nonlinearity, 2015

We consider a strictly convex billiard table with C 2 boundary, with the dynamics subjected to random perturbations. Each time the billiard ball hits the boundary its reflection angle has a random perturbation. The perturbation distribution corresponds to a situation where either the scale of the surface irregularities is smaller than but comparable to the diameter of the reflected object, or the billiard ball is not perfectly rigid. We prove that for a large class of such perturbations the resulting Markov chain is uniformly ergodic, although this is not true in general.