Insights into the superdiffusive dynamics through collision statistics in periodic Lorentz gas and Sinai billiard [article]

Valery B. Kokshenev, Eduardo Vicentini
2004 arXiv   pre-print
We report on the stationary dynamics in classical Sinai billiard (SB) corresponding to the unit cell of the periodic Lorentz gas (LG) formed by square lattice of length L and dispersing circles of radius R placed in the center of unit cell. Dynamic correlation effects for classical particles, initially distributed by random way, are considered within the scope of deterministic and stochastic descriptions. A temporal analysis of elastic reflections from the SB square walls and circle obstacles
more » ... given for distinct geometries in terms of the wall-collision and the circle-collision distributions. Late-time steady dynamic regimes are explicit in the diffusion exponent z(R), which plays a role of the order-disorder crossover dynamical parameter. The ballistic (z_0=1) ordered motion in the square lattice (R=0) switches to the superdiffusion regime with z_1=1.5, which is geometry-independent when R<L√(2)/4. This observed universal dynamics is shown to arise from long-distance particle jumps along the diagonal and nondiagonal Bleher corridors in the LG with the infinite horizon geometry. In the corresponding SB, this universal regime is caused by the long-time wall-collision memory effects attributed to the bouncing-ball orbits. The crossover nonuniversal behavior with 1.5<z<2 is due to geometry with L√(2)/4≤ R<L/2, when only the nondiagonal corridors remain open. All the free-motion corridors are closed in LG with finite horizon (R≥ L/2) and the interplay between square and circle geometries results in the chaotic dynamics ensured by the normal Brownian diffusion (z_2=2) and by the normal Gaussian distribution of collisions.
arXiv:math-ph/0405033v1 fatcat:tyqvjamb5ndd3da7z2j4cl62jq