Dynamical transition on the periodic Lorentz gas: Stochastic and deterministic approaches (original) (raw)

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.

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].

Properties of some chaotic billiards with time-dependent boundaries

Journal of Physics A: Mathematical and General, 2000

A dispersing billiard (Lorentz gas) and focusing billiards (in the form of a stadium) with time-dependent boundaries are considered. The problem of particle acceleration in such billiards is studied. For the Lorentz gas two cases of the time dependence are investigated: stochastic perturbations of the boundary and its periodic oscillations. Two types of focusing billiards with periodically forced boundaries are explored: a stadium with strong chaotic properties and a nearrectangle stadium. It is shown that in all cases billiard particles can reach unbounded velocities. Average velocities of the particle ensemble as functions of time and the number of collisions are obtained.

Numerical study of aD-dimensional periodic Lorentz gas with universal properties

Journal of statistical physics, 1985

We give the results of a numerical study of the motion of a point particle in a ddimensional array of spherical scatterers (Sinai's billiard without horizon). We find a simple universal law for the Lyapounov exponent (as a function of d) and a stretched exponential decay for the velocity autocorrelation as a function of the number of collisions. The diffusion seems to be anomalous in this problem. Ergodicity is used to predict the shape of the probability distribution of long free paths. Physical interpretations or clues are proposed.

Measuring logarithmic corrections to normal diffusion in infinite-horizon billiards

2014

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 meansquared 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.

Directed transport in classical and quantum chaotic billiards

Journal of Physics A: Mathematical and Theoretical, 2008

We construct an autonomous chaotic Hamiltonian ratchet as a channel billiard subdivided by equidistant walls attached perpendicularly to one side of the channel, leaving an opening on the opposite side. A static homogeneous magnetic field penetrating the billiard breaks time-reversal invariance and renders the classical motion partially chaotic. We show that the classical dynamics exhibits directed transport, owing to the asymmetric distribution of regular regions in phase space. The billiard is quantized by a numerical method based on a finite-element algorithm combined with the Landau gauge and the Bloch formalism for periodic potentials. We discuss features of the billiard eigenstates such as node lines and vortices in the probability flow. Evidence for directed quantum transport, inherited from the corresponding features of the classical dynamics, is presented in terms of level-velocity statistics.

Deterministic Diffusion in a Gravitational Billiard

Progress of Theoretical Physics Supplement, 2003

Numerical study of the motion of a classical particle in a homogeneous gravitational field bouncing off elastically from a piecewise linear periodic boundary shows that it is characterized by diffusion constant, but nonlinear scaling of the mean-square displacement is also observed. It is shown that periodic by modulus trajectories cannot have a segment with vertical flight.

Characterizing a transition from limited to unlimited diffusion in energy for a time-dependent billiard

arXiv (Cornell University), 2022

We explore Fermi acceleration in a driven oval billiard which shows unlimited to limited diffusion in energy when passing from the free to the dissipative case. We provide evidence for a secondorder phase transition taking place while detuning the corresponding restitution coefficient from one responsible for the degree of dissipation. A corresponding order parameter is suggested, and its susceptibility is shown to diverge at the transition point. We also discuss the underlying symmetry breaking and the elementary excitation of the controlled diffusion process.