Smoothed wave functions of chaotic quantum systems

It is shown that wave functions of quantum systems as ħ → 0 have an extra density near unstable periodic trajectories of the classical problem. The averaged wave function square is represented as the sum over a finite number of periodic trajectories. The contribution of each trajectory is expressed through the elements of the monodromy matrix of the trajectory. The results are compared with the numerical calculations of the wave functions for the stadium billiard.     

Chaotic spectroscopy

The spectra of quantized chaotic billiards from the point of view of scattering theory are discussed. It is shown how the spectral and resonance density functions both fluctuate about a common mean. A semiclassical treatment explains this in terms of classical scattering trajectories and periodic orbits of the Poincare scattering map. It is shown that this formalism provides an alternative derivation and a new interpretation of Gutzwiller's periodic orbits sum for the spectral density. Moreover, it is a convenient starting point for a derivation of a Riemann-Siegel "look alike" expression for the secular equation in terms of periodic orbits of finite length.     

No Quantum Ergodicity for Star Graphs

We investigate statistical properties of the eigenfunctions of the Schrödinger operator on families of star graphs with incommensurate bond lengths. We show that these eigenfunctions are not quantum ergodic in the limit as the number of bonds tends to infinity by finding an observable for which the quantum matrix elements do not converge to the classical average. We further show that for a given fixed graph there are subsequences of eigenfunctions which localise on pairs of bonds. We describe how to construct such subsequences explicitly. These structures are analogous to scars on short unstable periodic orbits.     

Phase space scars and quantum billiards

A semiclassical expression is derived for the spectral Wigner function of ergodic billiards in terms of a sum over contributions from classical periodic orbits. It represents a generalization of a similar formula by Berry, which does not immediately apply to billiard systems. These results are a natural generalization of Gutzwiller's trace formula for the density of states. Our theory clarifies the origin of scars in the eigenfunctions of billiard systems. However, in its present form, it is unable to predict what states will be dominated by individual periodic orbits. Finally, we compare some of the predictions of our theory with numerical results from the stadium. Within the limitations of numerical resolution, we find agreement between the two.     

Non-ergodicity of Two Particles Interacting via a Smooth Potential

We examine two point particles interacting via a smooth Lennard-Jones-type potential of finite range on a two-dimensional torus. We find situations under which this system contains a stable, elliptic periodic orbit and hence is not ergodic. This result is in contrast to the case of hard spheres interacting via inelastic collision, which are always ergodic for two particles, are conjectured to be ergodic for arbitrarily many particles, and can never contain elliptic periodic orbits.     

Equilibrium states and ergodic properties of infinite systems

We discuss the ergodic theoretic structure of infinite classical systems and present results on the ergodic properties of some simple model systems, e.g., ideal gas, Lorentz gas, Harmonic crystal. (The ergodic properties of the latter system are shown to be related in a simple way to the spectrum of the force matrix; when the spectrum is absolutely continuous, as in the translation-invariant crystal, the flow is Bernoulli.) We argue that ergodic properties, suitably refined by the inclusion of space translations, and other structure, are important for an understanding of nonequilibrium properties of macroscopic systems [1–5]. Possible additional structures include requirements of stability for the stationary state. We shall present results on the classical analog of the work by Haag, Kastler, and Trych-Pohlmeyer [6], Araki [7], and others [8]. The existence of a time evolution and equilibrium states for various anharmonic crystal systems will also be discussed [9].Supported in part by AFOSR Grant No. 73-2430B.     

Complex Classical Mechanics of a QES Potential

We study a combined parity(P) and time reversal(T) invariant non-Hermitian quasi-exactly solvable(QES) potential, which exhibits PT phase transition, in the complex plane classically to demonstrate different quantum effects. The particle with real energy makes closed orbits around one of the periodic wells of the complex potential depending on the initial condition. However interestingly the particle escapes to an open orbits even with real energy if it is placed beyond a certain distance from the center of the well. On the other hand when the particle energy is complex the trajectory is open and the particle tunnels back and forth between two wells which are separated by a classically forbidden path. The tunneling time is calculated for different pair of wells and is shown to vary inversely with the imaginary component of energy. Our study reveals that spontaneous PT symmetry breaking does not affect the qualitative features of the particle trajectories in the analogous complex classical model.     

Complex Classical Mechanics of a QES Potential

We study a combined parity (P) and time reversal (T) invariant non-Hermitian quasi-exactly solvable (QES) potential, which exhibits PT phase transition, in the complex plane classically to demonstrate different quantum effects. The particle with real energy makes closed orbits around one of the periodic wells of the complex potential depending on the initial condition. However interestingly the particle escapes to an open orbits even with real energy if it is placed beyond a certain distance from the center of the well. On the other hand when the particle energy is complex the trajectory is open and the particle tunnels back and forth between two wells which are separated by a classically forbidden path. The tunneling time is calculated for different pair of wells and is shown to vary inversely with the imaginary component of energy. Our study reveals that spontaneous PT symmetry breaking does not affect the qualitative features of the particle trajectories in the analogous complex classical model.     

Periodic orbits of nonscaling Hamiltonian systems from quantum mechanics

Quantal (E,tau) plots are constructed from the eigenvalues of the quantum system. We demonstrate that these representations display the periodic orbits of the classical system, including bifurcations and the transition from stable to unstable. (c) 1995 American Institute of Physics.     

Invariant polygons in systems with grazing-sliding

The paper investigates generic three-dimensional nonsmooth systems with a periodic orbit near grazing-sliding. We assume that the periodic orbit is unstable with complex multipliers so that two dominant frequencies are present in the system. Because grazing-sliding induces a dimension loss and the instability drives every trajectory into sliding, the system has an attractor that consists of forward sliding orbits. We analyze this attractor in a suitably chosen Poincare section using a three-parameter generalized map that can be viewed as a normal form. We show that in this normal form the attractor must be contained in a finite number of lines that intersect in the vertices of a polygon. However the attractor is typically larger than the associated polygon. We classify the number of lines involved in forming the attractor as a function of the parameters. Furthermore, for fixed values of parameters we investigate the one-dimensional dynamics on the attractor.     

Trapping of quantum particles for a class of time-periodic potentials. A semi-classical approach

We study the time evolution for Schrödinger operators with time-periodic potentials when the classical equations of motion possess periodic orbits. We exhibit a class of time-periodic potentials such that for initial states suitably localized around these periodic orbits, then at the dominant order of the semi-classical approximation, the system is trapped forever at sufficiently large frequency. An estimation of the correction to the semi-classical approximation is given, which yields a minimum “trapping time” for these systems.     

Three-dimensional molecular wave packets: Calculation of revival times from periodic orbits

We present a theoretical analysis of revivals and fractional revivals of three-dimensional wave packets, which describe the coupled vibrational motion of phosphaethyne (HCP) in its ground electronic state. The wave packets studied are chosen to evolve along the periodic orbits, which quantize the states in the three fundamental progressions. The revival times T_rev are found to depend strongly on the particular mode excited and on the mean excitation energy. Based on a semiclassical analysis, T_rev is shown to be determined by the dependence of the period of the orbits on the classical action along them.     

周期轨道的量子化与量子能谱中的长程关联

用二维可积系统的半经典量子化方案和二维无关联振子系统的量子能级与周期轨道之间的对应关系,讨论了一组量子能级之间具有长程关联的内在机制,在二维无关联振子系统中,发现了具有相同拓扑M(M1,M2)的周期轨道相对应的量子能级之间存在着长程关联,并以二维4次无关联振子系统为例做了具体说明.     

Classical limit of the quantized hyperbolic toral automorphisms

The canonical quantization of any hyperbolic symplectomorphismA of the 2-torus yields a periodic unitary operator on aN-dimenional Hilbert space,N=1/h. We prove that this quantum system becomes ergodic and mixing at the classical limit (N,N prime) which can be interchanged with the time-average limit. The recovery of the stochastic behaviour out of a periodic one is based on the same mechanism under which the uniform distribution of the classical periodic orbits reproduces the Lebesgue measure: the Wigner functions of the eigenstates, supported on the classical periodic orbits, are indeed proved to become uniformly speread in phase space.     

A numerical method for determining bifurcation curves of mappings

A shooting type iterative method for determining bifurcation curves of mappings is developed in this paper. By using this method we are able to calculate bifurcation curves for orbits of periods ≤ 5 of a Hénon-like map in all details. We find an interesting phenomenon that two or more stable periodic orbits of the same or different periods may coexist for some combinations of parameter values. Two stable p5 orbits and their attracting regions are shown for (μ, b) = (1.536, 0.153).     

Eigenvalues and eigenfunctions of a clover plate

We report a numerical study of the flexural modes of a plate using semi-classical analysis developed in the context of quantum systems. We first introduce the Clover billiard as a paradigm for a system inside which rays exhibit stable and chaotic trajectories. The resulting phase space explored by the ray trajectories is illustrated using the Poincare surface of section, and shows that it has both integrable and chaotic regions. Examples of the stable and the unstable periodic orbits in the geometry are presented. We numerically solve the biharmonic equation for the flexural vibrations of the Clover shaped plate with clamped boundary conditions. The first few hundred eigenvalues and the eigenfunctions are obtained using a boundary elements method. The Fourier transform of the eigenvalues show strong peaks which correspond to ray periodic orbits. However, the peaks corresponding to the shortest stable periodic orbits are not stronger than the peaks associated with unstable periodic orbits. We also perform statistics on the obtained eigenvalues and the eigenfunctions. The eigenvalue spacing distribution P(s) shows a strong peak and therefore deviates from both the Poisson and the Wigner distribution of random matrix theory at small spacings because of the C_4v symmetry of the Clover geometry. The density distribution of the eigenfunctions is observed to agree with the Porter-Thomas distribution of random matrix theory. Received 12 February 2001 and Received in final form 17 April 2001     

Integrability and ergodicity of classical billiards in a magnetic field

We consider classical billiards in plane, connected, but not necessarily bounded domains. The charged billiard ball is immersed in a homogeneous, stationary magnetic field perpendicular to the plane. The part of dynamics which is not trivially integrable can be described by a bouncing map. We compute a general expression for the Jacobian matrix of this map, which allows us to determine stability and bifurcation values of specific periodic orbits. In some cases, the bouncing map is a twist map and admits a generating function. We give a general form for this function which is useful to do perturbative calculations and to classify periodic orbits. We prove that billiards in convex domains with sufficiently smooth boundaries possess invariant tori corresponding to skipping trajectories. Moreover, in strong field we construct adiabatic invariants over exponentially large times. To some extent, these results remain true for a class of nonconvex billiards. On the other hand, we present evidence that the billiard in a square is ergodic for some large enough values of the magnetic field. A numerical study reveals that the scattering on two circles is essentially chaotic.     

The Norwegian Façade Insulation Study: the efficacy of façade insulation in reducing noise annoyance due to road traffic

The ergodic propriety of a room has strong effects on its reverberation. If the room is ergodic, the reverberation can be broken up in two steps: a deterministic process followed by a stochastic one. The late reverberation can be then modeled by a reverberation algorithm instead of more computationally consuming methods. In this study, the free path temporal distribution obtained by ray-tracing is used as an indicator of the room's mixing: the energetic average of the path lengths is computed at each time step. Ergodic rooms are thus characterized by rapidly convergent distributions. The free path value becomes independent of time. On the other hand, path selection mechanism and orbits are observed in non-ergodic rooms. The transition time from the deterministic process to the stochastic one is also studied through the evaluation of the room's time constant. It is shown that its value depends only on the mean free path and the boundaries scattering value. An empirical expression is obtained which agrees well with simulations carried out in a concert hall. This transition time from a deterministic model to a stochastic one can be used to speed up the acoustical predictions and auralizations in ergodic rooms.     

耦合不连续系统同步转换过程中的多吸引子共存

本文研究了耦合不连续系统的同步转换过程中的动力学行为, 发现由混沌非同步到混沌同步的转换过程中特殊的多吸引子共存现象. 通过计算耦合不连续系统的同步序参量和最大李雅普诺夫指数随耦合强度的变化, 发现了较复杂的同步转换过程: 临界耦合强度之后出现周期非同步态(周期性窗口); 分析了系统周期态的迭代轨道,发现其具有两类不同的迭代轨道: 对称周期轨道和非对称周期轨道, 这两类周期吸引子和同步吸引子同时存在, 系统表现出对初值敏感的多吸引子共存现象. 分析表明, 耦合不连续系统中的周期轨道是由于局部动力学的不连续特性和耦合动力学相互作用的结果. 最后, 对耦合不连续系统的同步转换过程进行了详细的分析, 结果表明其同步呈现出较复杂的转换过程.