We develop novel methods to compute auto-correlation functions, or power spectral densities, for chaotic dynamical systems generated by an inverse method whose starting point is an invariant distribution and a two-form. In general, the inverse method makes some aspects of chaotic dynamics calculable by methods familiar in quantum field theory. This approach has the numerical advantage of being amenable to Monte-Carlo parallel computation. We demonstrate the approach on a specific example, and show how auto-correlation functions can be computed without any direct numerical simulation, by Pade approximants of a short time expansion.
We present an experimental and numerical study of missing-level statistics of chaotic three-dimensional microwave cavities. The nearest-neighbor spacing distribution, the spectral rigidity, and the power spectrum of level fluctuations were investigated. We show that the theoretical approach to a problem of incomplete spectra does not work well when the incompleteness of the spectra is caused by unresolved resonances. In such a case the fraction of missing levels can be evaluated by calculations based on random matrix theory.
Large deviations in chaotic dynamics have potentially significant and dramatic consequences. We study large deviations of series of finite lengths $N$ generated by chaotic maps. The distributions generally display an exponential decay with $N$, associated with large-deviation (rate) functions. We obtain the exact rate functions analytically for the doubling, tent, and logistic maps. For the latter two, the solution is given as a power series whose coefficients can be systematically calculated to any order. We also obtain the rate function for the cat map numerically, uncovering strong evidence…
We present a semiclassical approach to n-point spectral correlation functions of quantum systems whose classical dynamics is chaotic, for arbitrary n. The basic ingredients are sets of periodic orbits that have nearly the same action and therefore provide constructive interference. We calculate explicitly the first correlation functions, to leading orders in their energy arguments, for both unitary and orthogonal symmetry classes. The results agree with corresponding predictions from random matrix theory, thereby giving solid support to the conjecture of universality.
We propose a novel measure of chaotic scattering amplitudes. It takes the form of a log-normal distribution function for the ratios $r_n={δ_n}/{δ_{n+1}}$ of (consecutive) spacings $δ_n$ between two (consecutive) peaks of the scattering amplitude. We show that the same measure applies to the quantum mechanical scattering on a leaky torus as well as to the decay of highly excited string states into two tachyons. Quite remarkably the $r_n$ obey the same distribution that governs the non-trivial zeros of Riemann zeta function.
Typical eigenstates of quantum systems, whose classical limit is chaotic, are well approximated as random states. Corresponding eigenvalue spectra is modeled through appropriate ensemble of random matrix theory. However, a small subset of states violate this principle and display eigenstate localization, a counter-intuitive feature known to arise due to purely quantum or semiclassical effects. In the spectrum of chaotic systems, the localized and random states interact with one another and modifies the spectral statistics. In this work, a $3 \times 3$ random matrix model is used to obtain exac…
Starting from a semiclassical approach recently developed for spectral correlation functions of quantum systems whose classical dynamics is chaotic, we focus on the case of broken time-reversal symmetry, the so-called unitary class. We obtain to all orders in perturbation theory the non-oscillatory parts of all correlation functions, showing that the off-diagonal contributions to these correlation functions cancel and the conjectured universailty holds. The innovation that allows this calculation to be performed is the introduction of an auxiliary matrix model which is governed by the same dia…
Dynamical localization of chaotic eigenstates in the mixed-type systems: spectral statistics in a billiard system after separation of regular and chaotic eigenstates
Interpreting the oscillations of massive and intermediate mass stars remains a challenging task. In fast rotators, the oscillation spectrum of p-modes is a superposition of sub-spectra which correspond to different types of modes, among which island modes and chaotic modes are expected to be the most visible. In the case of island modes, a semi-analytic formula describing the asymptotic behavior of island modes has been obtained previously. We study the properties of high frequency chaotic p-modes in a polytropic model. Unexpected peaks appear in the frequency autocorrelations of the spectra. …
Dynamical localization in chaotic systems: spectral statistics and localization measure in the kicked rotator as a paradigm for time-dependent and time-independent systems
The existing periodic orbit theory of spectral correlations for classically chaotic systems relies on the Riemann-Siegel-like representation of the spectral determinants which is still largely hypothetical. We suggest a simpler derivation using analytic continuation of the periodic-orbit expansion of the pertinent generating function from the convergence border to physically important real values of its arguments. As examples we consider chaotic systems without time reversal as well as the Riemann zeta function and Dirichlet L-functions zeros.
Brains process information through the collective dynamics of large neural networks. Collective chaos was suggested to underlie the complex ongoing dynamics observed in cerebral cortical circuits and determine the impact and processing of incoming information streams. In dissipative systems, chaotic dynamics takes place on a subset of phase space of reduced dimensionality and is organized by a complex tangle of stable, neutral and unstable manifolds. Key topological invariants of this phase space structure such as attractor dimension, and Kolmogorov-Sinai entropy so far remained elusive. Here …
Trace formula for chaotic dielectric resonators tested with microwave experiments
我们测量了两个体育场形介质微波谐振器的谐振谱，并测试了Bogomolny等人提出的混沌介质谐振器的半经典迹公式[Phys.Rev.E78，056202（2008）]。我们发现实验数据和痕量公式的预测之间有很好的定性一致性。偏差可归因于根据先前实验[Phys. Rev. E 81，066215（2010）]测得的光谱中缺少共振。数值长度谱的研究表明，与迹线公式有良好的定性和定量一致性。然而，它表明需要对迹线公式进行高阶修正。曲率修正的应用程序的菲涅尔反射系数进入跟踪公式产生了更好的协议，但偏差仍然存在，表明进一步调查的必要性。
We study the quantum localization in the chaotic eigenstates of a billiard with mixed-type phase space, 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. In quantum systems with discrete energy spectrum the Heisenberg time $t_H =2π\hbar/ΔE$, where $ΔE$ is the mean level spacing, is an important time scale.The classical transport time scale $t_T$ in relation to the Heisenberg time scale $t_H$ (their ratio is the parameter $…
