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

Abstract

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].

Infinite horizon billiards: Transport at the border between Gauss and Lévy universality classes

“…Here η is a free parameter which cannot be determined uniquely by the aforementioned steps alone. We found that taking η = 1 produces good results for P (x, N ) and alternatively for P d (r, t), see Figs 4,5,9, Each plot consists out of approximately 10 6 points. The Lorentz gas displays strong similarity between repeated draws and the simulation data, meaning that the renewal condition is indeed fulfilled.…”

“…one neglects correlations between consecutive collisions. It was shown that for the cross-like configuration of the Lorentz gas and in the limit of large scattering centers, this condition is nullified as an effective trapping mechanism emerges [4]. In contrast, when the scattering centers are not too large, the Lévy walk with the obtained cumulative distribution function (CDF) of the waiting times works perfectly, as demonstrated below.…”

“…The infinite horizon Lorentz gas [1] is a paradigmatic model of deterministic classical transport, thoroughly studied by physicists [2][3][4][5][6][7][8] and mathematicians [9][10][11][12][13][14][15][16][17][18][19][20][21][22]. It consists of an infinite periodic lattice of convex obstacles, and point-like particles which undergo elastic collisions with them.…”

Lévy walks

“…The latter process was observed when tracking trajectories of a Hamiltonian particle moving over an egg-grate potential; see Klafter and Zumofen (1994). Adapted from Klafter, Shlesinger, and Zumofen, 1996. type of two-dimensional Lévy walks with the exponent γ ¼ 2 is relevant for the description of the diffusion in Sinai billiards with infinite horizon (Bouchaud and Georges, 1990), as shown recently by Cristadoro et al (2014).…”

Dispersion of particles in an infinite-horizon Lorentz gas

“…The Lorentz gas (LG) is a classical model of transport [1,2], in which a pointlike particle moves at a constant speed, while undergoing elastic collisions with fixed scatterers [1][2][3][4][5][6][7][8][9][10][11][12][13][14][15][16][17]. When the free paths in this model are unbounded, it is termed the infinite-horizon LG (see Ref.…”