Aspects of diffusion in the stadium billiard

Period-doubling bifurcation in surface radio-frequency trap: Transition to chaos through Feigenbaum scenario

We have numerically investigated the dynamics of charged microparticles in a "five-wire" surface radio-frequency trap. The period-doubling bifurcation conditions have been shown to depend on the particle, the trap, and the alternating voltage parameters. For a comprehensive study of the dynamics chaotization through a cascade of period doubling, we have used Fourier analysis of a particle trajectory as well as the calculations of a non-trivial Lyapunov exponent map. We have demonstrated that the period-doubling bifurcation is consistent with a Feigenbaum scenario. A new approach to particle property determination can, thus, be based on observing a period-doubling bifurcation.

Spectral Form Factor and Dynamical Localization

Quantum dynamical localization occurs when quantum interference stops the diffusion of wave packets in momentum space. The expectation is that dynamical localization will occur when the typical transport time of the momentum diffusion is greater than the Heisenberg time. The transport time is typically computed from the corresponding classical dynamics. In this paper, we present an alternative approach based purely on the study of spectral fluctuations of the quantum system. The information about the transport times is encoded in the spectral form factor, which is the Fourier transform of the two-point spectral autocorrelation function. We compute large samples of the energy spectra (of the order of 106 levels) and spectral form factors of 22 stadium billiards with parameter values across the transition between the localized and extended eigenstate regimes. The transport time is obtained from the point when the spectral form factor transitions from the non-universal to the universal regime predicted by random matrix theory. We study the dependence of the transport time on the parameter value and show the level repulsion exponents, which are known to be a good measure of dynamical localization, depend linearly on the transport times obtained in this way.

Diffusion phenomena in a mixed phase space

We show that, in strongly chaotic dynamical systems, the average particle velocity can be calculated analytically by consideration of Brownian dynamics in a phase space, the method of images, and the use of the classical diffusion equation. The method is demonstrated on the simplified Fermi-Ulam accelerator model, which has a mixed phase space with chaotic seas, invariant tori, and Kolmogorov-Arnold-Moser islands. The calculated average velocities agree well with numerical simulations and with an earlier empirical theory.

Statistical properties of the localization measure of chaotic eigenstates and the spectral statistics in a mixed-type billiard

We study the quantum localization in the chaotic eigenstates of a billiard with mixed-type phase space [J. Phys. A: Math. Gen. 16, 3971 (1983)JPHAC50305-447010.1088/0305-4470/16/17/014; J. Phys. A: Math. Gen. 17, 1049 (1984)JPHAC50305-447010.1088/0305-4470/17/5/027], after separating the regular and chaotic eigenstates, in the regime of slightly distorted circle billiard where the classical transport time in the momentum space is still large enough, although the diffusion is not normal. This is a continuation of our recent papers [Phys. Rev. E 88, 052913 (2013)PLEEE81539-375510.1103/PhysRevE.88.052913; Phys. Rev. E 98, 022220 (2018)2470-004510.1103/PhysRevE.98.022220]. In quantum systems with discrete energy spectrum the Heisenberg time t_{H}=2πℏ/ΔE, where ΔE is the mean level spacing (inverse energy level density), is an important timescale. The classical transport timescale t_{T} (transport time) in relation to the Heisenberg timescale t_{H} (their ratio is the parameter α=t_{H}/t_{T}) determines the degree of localization of the chaotic eigenstates, whose measure A is based on the information entropy. We show that A is linearly related to normalized inverse participation ratio. The localization of chaotic eigenstates is reflected also in the fractional power-law repulsion between the nearest energy levels in the sense that the probability density (level spacing distribution) to find successive levels on a distance S goes like ∝S^{β} for small S, where 0≤β≤1, and β=1 corresponds to completely extended states. We show that the level repulsion exponent β is empirically a rational function of α, and the mean 〈A〉 (averaged over more than 1000 eigenstates) as a function of α is also well approximated by a rational function. In both cases there is some scattering of the empirical data around the mean curve, which is due to the fact that A actually has a distribution, typically with quite complex structure, but in the limit α→∞ well described by the beta distribution. The scattering is significantly stronger than (but similar as) in the stadium billiard [Nonlin. Phenom. Complex Syst. (Minsk) 21, 225 (2018)] and the kicked rotator [Phys. Rev. E 91, 042904 (2015)PLEEE81539-375510.1103/PhysRevE.91.042904]. Like in other systems, β goes from 0 to 1 when α goes from 0 to ∞. β is a function of 〈A〉, similar to the quantum kicked rotator and the stadium billiard.

Structure, size, and statistical properties of chaotic components in a mixed-type Hamiltonian system

We perform a detailed study of the chaotic component in mixed-type Hamiltonian systems on the example of a family of billiards [introduced by Robnik in J. Phys. A: Math. Gen. 16, 3971 (1983)JPHAC50305-447010.1088/0305-4470/16/17/014]. The phase space is divided into a grid of cells and a chaotic orbit is iterated a large number of times. The structure of the chaotic component is discerned from the cells visited by the chaotic orbit. The fractal dimension of the border of the chaotic component for various values of the billiard shape parameter is determined with the box-counting method. The cell-filling dynamics is compared to a model of uncorrelated motion, the so-called random model [Robnik et al. J. Phys. A: Math. Gen. 30, L803 (1997)JPHAC50305-447010.1088/0305-4470/30/23/003], and deviations attributed to sticky objects in the phase space are found. The statistics of the number of orbit visits to the cells is analyzed and found to be in agreement with the random model in the long run. The stickiness of the various structures in the phase space is quantified in terms of the cell recurrence times. The recurrence time distributions in a few selected cells as well as the mean and standard deviation of recurrence times for all cells are analyzed. The standard deviation of cell recurrence time is found to be a good quantifier of stickiness on a global scale. Three methods for determining the measure of the chaotic component are compared and the measure is calculated for various values of the billiard shape parameter. Lastly, the decay of correlations and the diffusion of momenta is analyzed.