In the preceding chapters we have learnt that random matrix theory is perfectly able to explain the universal properties of the spectra of chaotic systems, and this in spite of the oversimplifying assumptions applied. On the one hand it is very satisfactory that one single theory can cope with such a variety of systems as nuclei, mesoscopic structures, or microwave billiards, on the other hand this is a bit disappointing. If there is no possibility of discriminating between the spectra of a nucleus and a quantum dot, then there is little hope of learning anything of relevance about it.

Fortunately, random matrix theory is only one side of the coin. We have already come across some examples demonstrating its limits of validity. Remember the spectral level dynamics where bouncing balls disturbed the otherwise universal Gaussian velocity distribution (see Section 5.2.2). Another example is the scarring phenomenon observed in many wave functions. Here obviously closed classical orbits have left their fingerprints in the amplitude patterns. We cannot expect that the universal random matrix theory can correctly account for individual features such as periodic orbits.

We now come to an alternative approach to analysing the spectra. As we know from the correspondence principle, in the semiclassical limit quantum mechanics eventually turns into classical mechanics. That is why classical dynamics must be hidden somewhere in the spectra, at least in the limit of high quantum numbers. In the introduction we have already discussed this connection for a particle in a one-dimensional box.

In many spectroscopic experiments the energy levels of a system are determined as a function of an external parameter. Take as an example a hydrogen atom in a magnetic field. As long as the magnetic interaction is small compared to the spacing between neighbouring levels, a level with angular momentum quantum number L splits equidistantly into its 2L + 1 magnetic sublevels. A qualitatively different behaviour is found if the magnetic Zeeman interaction is of the same order of magnitude as the mean level spacing. In this range any regularity in the magnetic splitting is lost, and the energy levels move erratically as a function of the magnetic field. In billiard systems a shape parameter such as one length may take the role of the magnetic field. In chaotic systems degeneracies no longer occur, apart from accidental ones having the measure zero. Consequently crossings of eigenvalues are never observed when the parameter is changed, and they are converted into anti-crossings. For two-level systems this is an effect well-known in elementary quantum mechanics. The resulting motion of the eigenvalues dependent on the external parameter strongly resembles the dynamics of the particles of a one-dimensional gas with repulsive interaction.

The dynamical concept was introduced by Pechukas [Pec83] and further developed by Yukawa [Yuk85], and is therefore called the Pechukas–Yukawa model in the following. The major part of this chapter will treat the consequences of this model.

It is impossible to study a system without disturbing it by the measuring process. To determine the spectrum of a microwave billiard, for example, we have to drill a hole into its wall, introduce a wire, and radiate a microwave field. We learnt in Section 3.2.2 that the spectrum obtained for a rectangular microwave cavity is no longer integrable, but has become pseudointegrable by the presence of the antenna. The measurement thus unavoidably yields an unwanted combination of the system's own properties and those of the measuring apparatus. The mathematical tool to treat the coupling between the system and the environment is provided by scattering theory, which was originally developed in nuclear physics [Mah69, Lew91]. This theory has been successfully applied to mesoscopic systems and microwave billiards as well.

In this chapter scattering theory will be introduced with special emphasis on billiard systems. We shall also discuss the amplitude distributions of wave functions in chaotic systems, since they enter into the calculation of the distribution of scattering matrix elements. The perturbing bead method, used to get information on field distributions in microwave resonators, can also be described by scattering theory. In the last section we shall touch on the discussion of mesoscopic systems, which can be linked to scattering theory via the Landauer formula [Lan57] expressing conduction through mesoscopic devices in terms of transmission probabilities.

In their studies of the kicked rotator Casati et al. [Cas79] discovered that the quantization of classically chaotic systems can lead to the suppression of chaos. This is one of the most spectacular results in quantum chaos research. The classical dynamics of the kicked rotator is described by the famous standard map, and it has been known for some time that for sufficiently large kick strengths the classical system shows diffusive behaviour. But for its quantum mechanical counterpart the diffusion is stopped after some time by a process called dynamical localization. The dynamical localization of electrons has also been observed experimentally in the microwave ionization of highly excited hydrogen atoms [Gal88, Bay89]. For atoms the effect has also been demonstrated recently [Moo94].

All the examples mentioned belong to the Floquet systems, i.e. periodic time dependences are involved. Soon after the discovery of dynamical localization Fishman et al. [Fis82] found that there is a close relation between Floquet systems and the Anderson model, which has been used for a long time to study the influence of disorder on electronic wave functions in crystalline lattices. From this equivalence it became clear that dynamical localization is just a special case of the well-known Anderson localization.

For this reason Floquet systems and the Anderson model are discussed here together. For the Anderson model, however, only those aspects will be considered which are of relevance for dynamical localization.

Until about 1990 only a very small number of experiments on the quantum mechanics of chaotic systems existed, apart from the early studies of nuclear spectra [Por65]. In this context the experiments on hydrogen atoms in strong microwave fields by Bayfield and Koch [Bay74], and in strong magnetic fields by Welge and his group [Hol86, Mai86] have to be mentioned in particular. The studies of irregularly shaped microwave cavities by Stöckmann and Stein [Stö90] have introduced a new type of quantum chaos research. The microwave billiards and a number of variants to be discussed in this chapter are analogue systems, as they use the equivalence of the Helmholtz equation and the time independent Schrödinger equation. Whether there is a complete correspondence with quantum mechanics or not depends on the respective boundary conditions. As most of the phenomena discussed in the following are common to all types of waves, this does not reduce the conclusiveness of the analogue experiments.

Starting with a historical review, the state of the art in billiard experiments is presented with emphasis on a general survey and the technical background. The results and their quantum mechanical implications will be presented later. The hydrogen experiments, too, will be described in the proper context. The discussion of mesoscopic systems is restricted to billiard-like structures such as antidot lattices [Wei91], quantum dots [Mar92], and tunnelling devices [Fro94].

Random matrix theory was developed in the nineteen fifties and sixties by Wigner, Dyson, Mehta and others. Originally conceived to bring some order into the spectra of complex nuclei, the interest in random matrix theory was renewed enormously when Bohigas, Giannoni and Schmit [Boh84] conjectured that it should be applicable to the spectra of all chaotic systems. In the following years overwhelming evidence has been obtained that this conjecture is true. The most important works up to 1965 together with a summarizing text have been reprinted in a book by Porter [Por65] which even up to this day belongs to the standard literature on the topic. The state of the art up to 1980 is compiled in an excellent review article by Brody and others [Bro81]. For the newcomer the most recent survey by Bohigas [Boh89] is recommended.

Two monographs on the subject have to be mentioned. The first one, Random Matrices, was written by Mehta [Meh91], one of the pioneers in the field. The first edition appeared in 1967, the second enlarged one in 1991. Probably there is no really important topic on random matrices which cannot be found there. It is, however, a book for specialists. For those who are only interested in the basic principles, it is much too detailed. The other monograph, Quantum Signatures of Chaos by Haake [Haa91a], has now, a few years after its first appearance, become the standard introduction into the field.

From the very beginning classical nonlinear dynamics has enjoyed much popularity even among the noneducated public as is documented by numerous articles in well-renowned magazines, including nonscientific papers. For its nonclassical counterpart, the quantum mechanics of chaotic systems, termed in short ‘quantum chaos’, the situation is completely different. It has always been considered as a more or less mysterious topic, reserved to a small exclusive circle of theoreticians. Whereas the applicability of classical nonlinear dynamics to daily life is comprehensible for a complete outsider, quantum chaos, on the other hand, seems to be of no practical relevance at all. Moreover, in classical nonlinear dynamics the theory is supported by numerous experiments, mainly in hydrodynamics and laser physics, whereas quantum chaos at first sight seems to be the exclusive domain of theoreticians. In the beginning the only experimental contributions came from nuclear physics [Por65]. This preponderance of theory seems to have suppressed any experimental effort for nearly two decades. The situation gradually changed in the middle of the eighties, since when numerous experiments have been performed. An introductory presentation also suited to the experimentalist with no or only little basic knowledge is still missing.

It is the intention of this monograph to demonstrate that there is no reason to be afraid of quantum chaos. The underlying ideas are very simple. It is essentially the mathematical apparatus that makes things difficult and often tends to obscure the physical background.

In the preceding chapter we derived semiclassical expressions for the quantum mechanical propagator, the Green function, and the density of states. Now we are going to discuss a number of consequences and applications, in particular of the Gutzwiller trace formula. Anybody working in the field should be familiar with the basic ideas applied in the derivation, but an understanding of the implications of periodic orbit theory is also possible for readers not willing to enter too deeply into the details. For this reason knowledge of the preceding chapter should be dispensable for the major part of the present one.

We start with a description of the techniques applied to extract the contributions of the different periodic orbits from the spectra and wave functions. The spectra of billiards and hydrogen atoms in strong magnetic fields suggest a semiclassical interpretation. The most spectacular manifestation of periodic orbits is the scarring phenomenon, found in many wave functions of chaotic billiards.

As the quantum mechanical spectrum is uniquely determined by the periodic orbits, i.e. by individual system properties, the success of random matrix theory in description of numerous spectral correlations requires an explanation. This is given by Berry in his semiclassical theory of spectral rigidity [Ber85].

It is comparatively easy to analyse a given spectrum in terms of periodic orbit theory. This is not true for the reverse procedure, namely the calculation of a spectrum from the periodic orbits of the system.

This monograph is based on the script of a lecture series on the quantum mechanics of classically chaotic systems given by the author at the University of Marburg during the summer term 1995. The lectures were attended by students with basic knowledge in quantum mechanics, including members of the author's own group working on microwave analogous experiments on quantum chaotic questions.

When preparing the lectures the author became aware that a comprehensive textbook, covering both the theoretical and the experimental aspects, was not available. The present monograph is intended to fill this gap.

The basic concepts of the quantum mechanics of classically chaotic systems, termed ‘quantum chaos’ for short, are easy to grasp by any student of physics. The mathematical apparatus needed, however, often tends to obscure the physical background. That is why the theoretical results will be illustrated by real experimental or numerical data whenever possible.

Chapter 1 gives a short introduction on the essential ideas of semiclassical quantum mechanics, which is illustrated by two examples taken from the microwave billiards and the kicked rotator.

Chapter 2 treats the different types of billiard experiments. Methods to study vibrating solids and liquids are presented. The main part of this chapter deals with microwave techniques, as by far the most experiments have been performed in microwave billiards. The chapter ends with a discussion of mesoscopic billiards including quantum corrals.

Chapter 3 introduces random matrix theory. It was developed in the sixties, and is treated in several reviews and monographs.

The escape-rate formalism

One of the most interesting developments in the theory of irreversible processes is a connection, first explored by Gaspard and Nicolis, between the dynamical properties of open systems and the hydrodynamic or transport properties of such systems. We will explore this connection in Chapter 13, but first we must consider the dynamical properties of open systems. We consider a system to be open if the phase space of the system has physical boundaries and there is a mapping or transformation which can take phase points inside the boundaries to phase points outside the boundaries. Further, we will assume that the boundaries are such that once a phase point passes a boundary, it can never return to the bounded system. Thus the boundaries on the phase space region may be considered to be absorbing. To get some idea of the motivation for considering open systems, we might imagine a Brownian particle diffusing in a fluid inside a container with absorbing boundaries. The motion of the particle is really deterministic and can be described – microscopically – by some transformation in the phase space of the entire system. Now, when we describe the motion of the phase point, we lose the Brownian particle whenever it encounters the boundary of the container. If we were to describe the motion macroscopically, we would solve the diffusion equation for the probability density of the Brownian particle, in the fluid, with absorbing boundaries. The probability of finding the particle inside the container is an exponentially decreasing function of time with decay coefficient depending on the diffusion coefficient of the Brownian particle in the fluid, and on the geometry of the container.

We have just seen that in a system subjected to an external field and a thermostat which maintains the system's kinetic energy at a constant value there is a phase space contraction taking place. Prior to that, we showed that a fractal repeller can form in the phase-space for a Hamiltonian system with absorbing boundaries such that trajectories hitting the boundary never re-appear in the system. Some questions naturally arise as we think about these systems, such as:

1. To what kind of structure is the phase-space for a thermostatted system contracting?

2. What are the properties of such a structure? Is it a smooth hypersurface, say, or does it have a more complicated structure?

3. How do we describe fractal attractors and/or repellers and compute the properties of trajectories which are confined to them, since these objects are typically of zero Lebesgue measure in phasespace?

In this chapter, we will show that for each of these situations there is an appropriate measure that can be used to describe the resulting sets and to compute the properties of trajectories which are confined to these sets. Throughout our discussions we will suppose that the dynamics is hyperbolic. In the thermostatted case, the system contracts onto an attractor, and the attractor is characterized by an invariant measure which is smooth along unstable directions and fractal in the stable directions. Such measures are known as SRB (Sinai–Ruelle–Bowen) measures.

For a system with escape, the invariant measure on the repeller is different from that on an attractor, because escape takes place along the expanding directions.

Now we want to discuss a number of topics that are essential for an understanding of dynamical systems theory and which also play a role in a more detailed discussion of the relation between transport theory and dynamical systems theory. We begin with a discussion of the Kolmogorov-Sinai (KS) entropy, which is a characteristic property of those deterministic dynamical systems with ‘randomness’ properties similar to Bernoulli shifts discussed earlier. The KS entropy is essential for formulating the escape-rate expressions for transport coefficients, to be discussed in Chapters 11 and 12.

Heuristic considerations

Let us return for a moment to the Arnold cat map discussed in the previous chapter. There we illustrated an initial set, A, say, that is located in the lower left-hand corner of the unit square (see Fig. 8.3). As this set evolves under the action of the map TA, the set becomes longer and thinner so that after three iterations, the set has begun to fold back across the unit square, and after ten iterations, the set is so stretched out that it appears to cover the unit square uniformly (see Figs 8.5 and 8.6, respectively). Since the initial set A is getting stretched along the unstable direction, at every iteration we learn more about the initial location of the points within the initial set A That is, suppose that we can distinguish two points on the unit square only if they are separated by a distance δ, the resolution parameter, and suppose further that the characteristic dimension of the initial set, A, is of the order of δ.

Linear response theory

Linear response theory describes the changes that a small applied external field induces in the macroscopic properties of a system in equilibrium. The external field is supposed to be turned on at some initial time, when the system is in equilibrium, and then treated as a perturbation. As an example of linear response theory, we show how to use it to obtain the time–correlation-function expression – often called the Green–Kubo expression – for the electrical conductivity of a system that contains charged particles. If the applied electric field is small enough that heating effects can be ignored, then Ohm's law can be expressed as Je=σE, where Je is the electrical current density, E is the applied electric field, and σ is the electrical conductivity that we wish to compute. The time-correlation formula is an example of a set of formulae which relate transport coefficients in a fluid to time integrals of timecorrelation- functions. In the last section of this chapter, we will give an example of the derivation of such formulae for the case of tagged particle diffusion. First, we wish to examine one particular derivation of the formula for the electrical conductivity which has provoked a great deal of very instructive discussion, which, in turn, is closely connected to the general theme of this book.

We began this excursion into the dynamical systems approach to nonequilibrium statistical mechanics with a discussion of the Boltzmann transport equation. We end this excursion with the Boltzmann equation, but now we are going to use it to compute some Lyapunov exponents. The fact that the Boltzmann equation begins and ends this book may serve to illustrate both the power and the beauty of this equation, sitting at the heart of our understanding of irreversible phenomena.

The Lorentz gas as a billiard system

We are going to calculate the positive Lyapunov exponent for a two-dimensional hard-disk Lorentz gas. To do so, we will combine ideas of Boltzmann with those of Sinai, thus completing, in some sense, the transition from molecular chaos to dynamical chaos, and showing the deep connection between them. Imagine then a collection of hard disks of radius a placed at random in the plane at low density, i.e., na2 « 1, where n is the number density of the disks (see Fig. 18.1). Next, imagine a point particle moving with speed v in this array. The particle moves freely between collisions with the disks and makes specular collisions with the disks from time to time, preserving its speed and energy, but not its momentum upon collision. Sinai has considered some of the mathematical properties of this system, and has proved that it is mixing and ergodic. The moving particle has four degrees of freedom – two coordinates and two momenta – but the energy is conserved. Therefore, the phase-space of the moving particle is three-dimensional.