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Journal of Physics A Mathematical and General

The chaotic analytic function

J H Hannay

Published Dec 11, 1998• 33 citations
Publisher

Abstract

In contrast to a stationary Gaussian random function of a real variable which is free to have any correlation function, the closest analogous analytic random function in the complex plane has no true freedom - it is (statistically) unique. Since it has arisen only recently, as an apparently universal feature in the physical context of quantum chaos, I refer to it here as `the chaotic analytic function'. I note that it is implied by the assumption that a quantum chaotic wavefunction has Gaussian randomness and has a constant value for the average of its Wigner function in phase space. Interpreted literally this shows that the chaotic analytic function is the Bargmann function of a pure `white noise' wavefunction. More physically, if `constant' is replaced by `smooth on the scale of a Planck area', these assumptions are the semiclassical ones made by Berry for chaotic eigenstates. The analysis shows that the chaotic analytic function is still obtained semiclassically.

DOI
10.1088/0305-4470/31/49/001

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In contrast to a stationary Gaussian random function of a real variable which is free to have any correlation function, the closest analogous analytic random function in the complex plane has no true freedom - it is (statistically) unique. Since it has arisen only recently, as an apparently universal feature in the physical context of quantum chaos, I refer to it here as the chaotic analytic function'. I note that it is implied by the assumption that a quantum chaotic wavefunction has Gaussian randomness and has a constant value for the average of its Wigner function in phase space. Interpreted literally this shows that the chaotic analytic function is the Bargmann function of a pure white noise' wavefunction. More physically, if constant' is replaced by smooth on the scale of a Planck area', these assumptions are the semiclassical ones made by Berry for chaotic eigenstates. The analysis shows that the chaotic analytic function is still obtained semiclassically.
In contrast to the classical Hamiltonian mechanics whose chaotic behaviors are now well established, quantum mechanics is still obscure about the origin of its random behavior. It is shown in this paper that in case of the stadium billiard, the fact that the nodal lines are smooth but are randomly oscillating, as first described by Heller, and MacDonald and Kaufman, can be interpreted as due to the irregular pseudochaotic behavior of the phase. To do this, we employ the path integral representation of propagator and the reinterpreted Bohm's formulation of quantum mechanics. Thus we reach the conclusion that although the chaotic behavior in quantum mechanics is suppressed by the Einstein quantization condition, it is observed in the properly defined phase of the propagator, or of the wave function, as a pseudochaos. In other words quantum mechanics has the counterpart of classical chaos in the phase of the propagator, and consequently, the wave function is also, pseudochaotic (random) with respect to q , but it does not exhibit exponential instability. The essential feature of quantum chaos manifests itself in the vanishings of the average of the wave function and the two-point correlation of the wave function at high energies. They are closely related to ergodicity and quantum measurement, and in particular new light will be shed on the measurement theory. The basic assumptions in the random matrix theory are discussed, confirming the universality of level spacing distribution.
The apparent randomness of chaotic eigenstates in interacting quantum systems hides subtle correlations dynamically imposed by their finite energy per particle. These correlations are revealed when Berry's approach for chaotic eigenfunctions in single-particle systems is lifted into many-body space. We achieve this by a many-body semiclassical analysis, appropriate for the mesoscopic regime of a large but finite number of particles. We thereby identify as signatures of chaotic many-body eigenstates the universality of both their cross-correlations and the Gaussian distribution of expansion coefficients. Combined, these two features imprint characteristic features to the morphology of eigenstates that we check against extensive quantum simulations. The universality of eigenstate correlations for fixed energy density is hence a further signature of many-body quantum chaos that, while consistent with the eigenstate thermalization hypothesis, lies beyond random matrix theory.
We elucidate the basic physical mechanisms responsible for the quantum-classical transition in one-dimensional, bounded chaotic systems subject to unconditioned environmental interactions. We show that such a transition occurs due to the dual role of noise in regularizing the semiclassical Wigner function and averaging over fine structures in classical phase space. The results are interpreted in the context of applying recent advances in the theory of measurement and open systems to the semiclassical quantum regime. We use these methods to show how a local semiclassical picture is stabilized and can then be approximated by a classical distribution at later times. The general results are demonstrated explicitly via high-resolution numerical simulations of the quantum master equation for a chaotic Duffing oscillator.
The quantisation of the two-dimensional toric and spherical phase spaces is considered in analytic coherent state representations. Every pure quantum state admits therein a finite multiplicative parametrisation by the zeros of its Husimi function. For eigenstates of quantised systems, this description explicitly reflects the nature of the underlying classical dynamics: in the semiclassical regime, the distribution of the zeros in the phase space becomes one-dimensional for integrable systems, and highly spread out (conceivably uniform) for chaotic systems. This multiplicative representation thereby acquires a special relevance for semiclassical analysis in chaotic systems.
Quantum Chaos provides a comprehensive overview of our understanding of chaotic behaviour in a wide variety of quantum and semiclassical systems, and describes both experimental and theoretical investigations. A general introduction sets out the main features of chaos in quantum systems. Thereafter, in an authoritative collection of papers, prominent scientists put forward their particular interpretations of quantum chaos, with reference to a broad range of interesting physical systems. As yet, there is no universally accepted definition of quantum chaos. However, by dealing with such a wide range of topics from different branches of physics, this book provides an overview of this rapidly expanding field, and will be of great interest to graduate students and researchers in many areas of physics and chemistry.
We compare the influence of perturbations of classically regular respectively chaotic Hamiltonians on their eigenfunctions. A generic measure of the perturbation strength is given by the average spreading width, which semiclassically is a phase-space integral. Thus, contrary to common assumption, the spreading width cannot be an indicator of regularity or chaos. The distribution of expansion coefficients of perturbed eigenfunctions in terms of unperturbed ones is markedly different for the two cases and may serve as a quantum signature of chaos referring to states rather than to spectra.