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

Valery B. Kokshenev·Eduardo Vicentini
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Abstract

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 is 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 $R<L\sqrt{2}/4$. 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 $L\sqrt{2}/4\leq R<L/2$, when only the nondiagonal corridors remain open. All the free-motion corridors are closed in LG with finite horizon ($R\geq L/2$) 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.

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Valery B. Kokshenev, Eduardo Vicentini. 2004-05-10. Insights into the superdiffusive dynamics through collision statistics in periodic Lorentz gas and Sinai billiard. https://arxiv.org/abs/math-ph/0405033

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