Weak mixing and anomalous kinetics along filamented surfaces. (English) Zbl 1080.37584
Summary: We consider chaotic properties of a particle in a square billiard with a horizontal bar in the middle. Such a system can model field-line windings of the merged surfaces. The system has weak-mixing properties with zero Lyapunov exponent and entropy, and it can be also interesting as an example of a system with intermediate chaotic properties, between the integrability and strong mixing. We show that the transport is anomalous and that its properties can be linked to the ergodic properties of continued fractions. The distribution of Poincaré recurrences, distribution of the displacements, and the moments of the truncated distribution of the displacements are obtained. Connections between different exponents are found. It is shown that the distribution function of displacements and its truncated moments as a function of time exhibit log-periodic oscillations (modulations) with a universal period \(T_{\log} = \pi^2/12 \ln 2\). We note that similar results are valid for a family of billiard, particularly for billiards with square-in-square geometry.
MSC:
| 37J99 | Dynamical aspects of finite-dimensional Hamiltonian and Lagrangian systems |
| 37A25 | Ergodicity, mixing, rates of mixing |
| 37D50 | Hyperbolic systems with singularities (billiards, etc.) (MSC2010) |
| 70H99 | Hamiltonian and Lagrangian mechanics |
| 70K55 | Transition to stochasticity (chaotic behavior) for nonlinear problems in mechanics |
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