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Über dieses Buch

This twelfth volume in the Poincaré Seminar Series presents a complete and interdisciplinary perspective on the concept of Chaos, both in classical mechanics in its deterministic version, and in quantum mechanics. This book expounds some of the most wide ranging questions in science, from uncovering the fingerprints of classical chaotic dynamics in quantum systems, to predicting the fate of our own planetary system. Its seven articles are also highly pedagogical, as befits their origin in lectures to a broad scientific audience. Highlights include a complete description by the mathematician É. Ghys of the paradigmatic Lorenz attractor, and of the famed Lorenz butterfly effect as it is understood today, illuminating the fundamental mathematical issues at play with deterministic chaos; a detailed account by the experimentalist S. Fauve of the masterpiece experiment, the von Kármán Sodium or VKS experiment, which established in 2007 the spontaneous generation of a magnetic field in a strongly turbulent flow, including its reversal, a model of Earth’s magnetic field; a simple toy model by the theorist U. Smilansky – the discrete Laplacian on finite d-regular expander graphs – which allows one to grasp the essential ingredients of quantum chaos, including its fundamental link to random matrix theory; a review by the mathematical physicists P. Bourgade and J.P. Keating, which illuminates the fascinating connection between the distribution of zeros of the Riemann ζ-function and the statistics of eigenvalues of random unitary matrices, which could ultimately provide a spectral interpretation for the zeros of the ζ-function, thus a proof of the celebrated Riemann Hypothesis itself; an article by a pioneer of experimental quantum chaos, H-J. Stöckmann, who shows in detail how experiments on the propagation of microwaves in 2D or 3D chaotic cavities beautifully verify theoretical predictions; a thorough presentation by the mathematical physicist S. Nonnenmacher of the “anatomy” of the eigenmodes of quantized chaotic systems, namely of their macroscopic localization properties, as ruled by the Quantum Ergodic theorem, and of the deep mathematical challenge posed by their fluctuations at the microscopic scale; a review, both historical and scientific, by the astronomer J. Laskar on the stability, hence the fate, of the chaotic Solar planetary system we live in, a subject where he made groundbreaking contributions, including the probabilistic estimate of possible planetary collisions. This book should be of broad general interest to both physicists and mathematicians.



The Lorenz Attractor, a Paradigm for Chaos

It is very unusual for a mathematical or physical idea to disseminate into the society at large. An interesting example is chaos theory, popularized by Lorenz’s butterfly effect:

“does the flap of a butterfly’s wings in Brazil set off a tornado in Texas?”

A tiny cause can generate big consequences! Mathematicians (and non mathematicians) have known this fact for a long time! Can one adequately summarize chaos theory is such a simple minded way? In this review paper, I would like first of all to sketch some of the main steps in the historical development of the concept of chaos in dynamical systems, from the mathematical point of view. Then, I would like to present the present status of the Lorenz attractor in the panorama of the theory, as we see it Today.

Étienne Ghys

Chaotic Dynamos Generated by Fully Turbulent Flows

We report experimental results on the generation of a magnetic field by the motion of an electrically conducting liquid metal. We recall that this dynamo effect is a canonical example of an instability process that occurs on a strongly turbulent flow. Although the magnetic field is driven by a turbulent velocity field that involves a wide range of interacting scales, we observe that its dynamics results from a small number of interacting modes. We present a model that describes both periodic and random reversals of the magnetic field and compare it with the experimental results and direct numerical simulations.

Stéphan Fauve

Discrete Graphs – A Paradigm Model for Quantum Chaos

The research in Quantum Chaos attempts to uncover the fingerprints of classical chaotic dynamics in the corresponding quantum description. To get to the roots of this problem, various simplified models were proposed and used. Here a very simple model of a random walker on large


-regular graphs, and its quantum analogue are proposed as a paradigm which shares many salient features with realistic models – namely the affinity of the spectral statistics with random matrix theory, the role of cycles and their statistics, and percolation of level sets of the eigenvectors. These concepts will be explained and reviewed with reference to the original publications for further details.

Uzy Smilansky

Quantum Chaos, Random Matrix Theory, and the Riemann ζ-function

We review some connections between quantum chaos and the theory of the Riemann zeta function and the primes. Specifically, we give an overview of the similarities between the semiclassical trace formula that connects quantum energy levels and classical periodic orbits in chaotic systems and an analogous formula that connects the Riemann zeros and the primes. We also review the role played by Random Matrix Theory in both quantum chaos and the theory of the zeta function. The parallels we review are conjectural and still far from being understood, but the ideas have led to substantial progress in both areas.

Paul Bourgade, Jonathan P. Keating

Chaos in Microwave Resonators

Chaotic billiards are a paradigm of quantum chaos studied theoretically in numerous papers. In flat microwave resonators with cross-sections mimicking the billiard shape there is a one-to-one correspondence between the stationary Schrödinger equation and the Helmholtz equation. This allows an experimental access to questions hitherto studied exclusively theoretically. In the article various aspects of quantum chaos are presented and illustrated by experimental results. It continues with a discussion of random matrices and the universal features of wave functions of chaotic billiards. Next, semiclassical quantum mechanics is introduced, establishing a link between the quantummechanical Green function and the classical trajectories. The article ends with a presentation of recent applications of wave-chaos research.

Hans-Jürgen Stöckmann

Anatomy of Quantum Chaotic Eigenstates

We present one aspect of “Quantum Chaos”, namely the description of high frequency eigenmodes of a quantum system, the classical limit of which is chaotic (we call such eigenmodes “chaotic eigenmodes”). A paradigmatic example is provided by the eigenmodes of the Laplace–Beltrami operator on a compact Riemannian manifold of negative curvature: the corresponding classical dynamics is the geodesic flow on the manifold, which is strongly chaotic (Anosov). Other well-studied classes of examples include certain Euclidean domains (“chaotic billiards”), or quantized chaotic symplectomorphisms of the two-dimensional torus.

We propose several levels of description, some of them allowing for mathematical rigor, others being more heuristic.

The macroscopic distribution of the eigenstates makes use of semiclassical measures, which are probability measures invariant w.r.t. the classical dynamics; these measures reflect the asymptotic “shape” of a sequence of high frequency eigenmodes. The quantum ergodicity theorem states that the vast majority of the eigenstates is associated with the “flat” (Liouville) measure. A major open problem addresses the existence of “exceptional” eigenmodes admitting different macroscopic properties.

The microscopic description deals with the structure of the eigenfunctions at the scale of their wavelengths. It is mainly of statistical nature: it addresses, for instance, the value distribution of the eigenfunctions, their shortdistance correlation functions, the statistics of their nodal sets or domains. This microscopic description mostly relies on a random state (or random wave) Ansatz for the chaotic eigenmodes, which is far from being mathematically justified, but already offers interesting challenges for probabilists and harmonic analysts.

Stéphane Nonnenmacher

Is the Solar System Stable?

Since the formulation of the problem by Newton, and during three centuries, astronomers and mathematicians have sought to demonstrate the stability of the Solar System. As mentioned by Poincaré, several demonstrations of the stability of the Solar System have been published. By Laplace and Lagrange in the first place, then by Poisson, and more recently by Arnold. Others came after again.

Were the old demonstrations insufficient, or are the new ones unnecessary?

These rigorous demonstrations are in fact various approximations of idealized systems, but thanks to the numerical experiments of the last two decades, we know now that the motion of the planets in the Solar System is chaotic. This prohibits any accurate prediction of the planetary trajectories beyond a few tens of millions of years. The recent simulations even show that planetary collisions or ejections are possible on a period of less than 5 billion years, before the end of the life of the Sun.

Jacques Laskar
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