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Register a new userChaotic and Regular Motion in Dissipative Gravitational Billiards
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We consider the motion of a particle subjected to the constant gravitational field and scattered inelasticaly by hard boundaries which possess the shape of parabola, wedge, and hyperbola. The billiard itself performs oscillations. The linear dependence of the restitution coefficient on the particle velocity is assumed. We demonstrate that this dynamical system can be either regular or chaotic, which depends on the billiard shape and the oscillation frequency. The trajectory calculations are compared with the experimental data; a good agreement has been achieved. Moreover, the properties of the system has been studied by means of the Lyapunov exponents and the Kaplan-Yorke dimension. Chaotic and nonuniform patterns visible in the experimental data are interpreted as a result of large embedding dimension.
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The dynamics in three-dimensional billiards leads, using a Poincare section, to a four-dimensional map which is challenging to visualize. By means of the recently introduced 3D phase-space slices an intuitive representation of the organization of the mixed phase space with regular and chaotic dynamics is obtained. Of particular interest for applications are constraints to classical transport between different regions of phase space which manifest in the statistics of Poincare recurrence times. For a 3D paraboloid billiard we observe a slow power-law decay caused by long-trapped trajectories which we analyze in phase space and in frequency space. Consistent with previous results for 4D maps we find that: (i) Trapping takes place close to regular structures outside the Arnold web. (ii) Trapping is not due to a generalized island-around-island hierarchy. (iii) The dynamics of sticky orbits is governed by resonance channels which extend far into the chaotic sea. We find clear signatures of partial transport barriers. Moreover, we visualize the geometry of stochastic layers in resonance channels explored by sticky orbits.
We apply a molecular dynamics scheme to analyze classically chaotic properties of a two-dimensional circular billiard system containing two Coulomb-interacting electrons. As such, the system resembles a prototype model for a semiconductor quantum dot. The interaction strength is varied from the noninteracting limit with zero potential energy up to the strongly interacting regime where the relative kinetic energy approaches zero. At weak interactions the bouncing maps show jumps between quasi-regular orbits. In the strong-interaction limit we find an analytic expression for the bouncing map. Its validity in the general case is assessed by comparison with our numerical data. To obtain a more quantitative view on the dynamics as the interaction strength is varied, we compute and analyze the escape rates of the system. Apart from very weak or strong interactions, the escape rates show consistently exponential behavior, thus suggesting strongly chaotic dynamics and a phase space without significant sticky regions within the considered time scales.
We consider classical dynamical properties of a particle in a constant gravitational force and making specular reflections with circular, elliptic or oval boundaries. The model and collision map are described and a detailed study of the energy regimes is made. The linear stability of fixed points is studied, yielding exact analytical expressions for parameter values at which a period-doubling bifurcation occurs. The dynamics is apparently ergodic at certain energies in all three models, in contrast to the regularity of the circular and elliptic billiard dynamics in the field-free case. This finding is confirmed using a sensitive test involving Lyapunov weighted dynamics. In the last part of the paper a time dependence is introduced in the billiard boundary, where it is shown that for the circular billiard the average velocity saturates for zero gravitational force but in the presence of gravitational it increases with a very slow growth rate, which may be explained using Arnold diffusion. For the oval billiard, where chaos is present in the static case, the particle has an unlimited velocity growth with an exponent of approximately 1/6.
A recent model of Ariel et al. [1] for explaining the observation of Levy walks in swarming bacteria suggests that self-propelled, elongated particles in a periodic array of regular vortices perform a super-diffusion that is consistent with Levy walks. The equations of motion, which are reversible in time but not volume preserving, demonstrate a new route to Levy walking in chaotic systems. Here, the dynamics of the model is studied both analytically and numerically. It is shown that the apparent super-diffusion is due to sticking of trajectories to elliptic islands, regions of quasi-periodic orbits reminiscent of those seen in conservative systems. However, for certain parameter values, these islands coexist with asymptotically stable periodic trajectories, causing dissipative behavior on very long time scales.
This article reports on a joint theoretical and experimental study of the Pauli quantum-mechanical stress tensor $T_{alpha beta}(x,y)$ for open two-dimensional chaotic billiards. In the case of a finite current flow through the system the interior wave function is expressed as $psi = u+iv$. With the assumption that $u$ and $v$ are Gaussian random fields we derive analytic expressions for the statistical distributions for the quantum stress tensor components $T_{alpha beta}$. The Gaussian random field model is tested for a Sinai billiard with two opposite leads by analyzing the scattering wave functions obtained numerically from the corresponding Schroedinger equation. Two-dimensional quantum billiards may be emulated from planar microwave analogues. Hence we report on microwave measurements for an open 2D cavity and how the quantum stress tensor analogue is extracted from the recorded electric field. The agreement with the theoretical predictions for the distributions for $T_{alpha beta}(x,y)$ is quite satisfactory for small net currents. However, a distinct difference between experiments and theory is observed at higher net flow, which could be explained using a Gaussian random field, where the net current was taken into account by an additional plane wave with a preferential direction and amplitude.
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