This review is devoted to the problem of thermalization in a small isolated conglomerate of interacting constituents. A variety of physically important systems of intensive current interest belong to this category: complex atoms, molecules (including biological molecules), nuclei, small devices of condensed matter and quantum optics on nano-and micro-scale, cold atoms in optical lattices, ion traps. Physical implementations of quantum computers, where there are many interacting qubits, also fall into this group. Statistical regularities come into play through inter-particle interactions, which have two fundamental components: mean field, that along with external conditions, forms the regular component of the dynamics, and residual interactions responsible for the complex structure of the actual stationary states. At sufficiently high level density, the stationary states become exceedingly complicated superpositions of simple quasiparticle excitations. At this stage, regularities typical of quantum chaos emerge and bring in signatures of thermalization. We describe all the stages and the results of the processes leading to thermalization, using analytical and massive numerical examples for realistic atomic, nuclear, and spin systems, as well as for models with random parameters. The structure of stationary states, strength functions of simple configurations, and concepts of entropy and temperature in application to isolated mesoscopic systems are discussed in detail. We conclude with a schematic discussion of the time evolution of such systems to equilibrium.

We study the onset of chaos and statistical relaxation in two isolated dynamical quantum systems of interacting spins 1/2, one of which is integrable and the other chaotic. Our approach to identifying the emergence of chaos is based on the level of delocalization of the eigenstates with respect to the energy shell, the latter being determined by the interaction strength between particles or quasiparticles. We also discuss how the onset of chaos may be anticipated by a careful analysis of the Hamiltonian matrices, even before diagonalization. We find that despite differences between the two models, their relaxation processes following a quench are very similar and can be described analytically with a theory previously developed for systems with two-body random interactions. Our results imply that global features of statistical relaxation depend on the degree of spread of the eigenstates within the energy shell and may happen to both integrable and nonintegrable systems.

We study the transition to chaos and the emergence of statistical relaxation in isolated dynamical quantum systems of interacting particles. Our approach is based on the concept of delocalization of the eigenstates in the energy shell, controlled by the Gaussian form of the strength function. We show that, although the fluctuations of the energy levels in integrable and nonintegrable systems are different, the global properties of the eigenstates are quite similar, provided the interaction between particles exceeds some critical value. In this case, the statistical relaxation of the systems is comparable, irrespective of whether or not they are integrable. The numerical data for the quench dynamics manifest excellent agreement with analytical predictions of the theory developed for systems of two-body interactions with a completely random character.

We study analytically and numerically the classical diffusive process which takes place in a chaotic billiard. This allows to estimate the conditions under which the statistical properties of eigenvalues and eigenfunctions can be described by Random Matrix Theory. In particular the phenomenon of quantum dynamical localization should be observable in real experiments.PACS numbers: 05.45.+b, 05.20.-y One of the main modifications that quantum mechanics introduces in our classical picture of deterministic chaos is "quantum dynamical localization" which results e.g. in the suppression of chaotic diffusive-like process which may take place in systems under external periodic perturbations. This phenomenon, first pointed out in the model of quantum kicked rotator [1], is now firmly established and observed in several laboratory experiments [2].For conservative Hamiltonian systems the question of localization is much less investigated. The situation here is much more intriguing : from one hand, in a conservative system, one may argue that there is always localization due to the finite number of unperturbed basis states effectively coupled by the perturbation; on the other hand a large amount of numerical evidence indicates that quantization of classically chaotic systems leads to results which appear in agreement with the predictions of Random Matrix Theory (RMT) [3].Recently the problem of localization in conservative systems has been explicitely investigated. In particular, on the base of Wigner band random matrix model, conditions for localization were explicitely given together with the relation between localization and level spacing distribution [4].Billiards are very important models in the study of conservative dynamical systems since they provide clear mathematical examples of classical chaos and their quantum properties have been extensively studied theoretically and experimentally. Moreover they are becoming increasingly relevant for the study of optical processes in microcavities which may lead to possible applications such as the design of novel microlasers or other optical devices [5].In this paper we focus our attention on a two dimensional chaotic billiard: the Bunimovich stadium, and study the classical diffusive process which takes place in angular momentum. This will allow us to predict the conditions for quantum localization and therefore the conditions under which the standard Random Matrix Theory is not applicable.We consider the motion of a particle having mass m, velocity v and elastically bouncing inside the stadium shown in Fig 1. We denote with R the radius of the semicircles and with 2a the length of the straight segments. The total energy is E = m v 2 /2.The statistical properties of the billiard are controlled by the dimensionless parameter ǫ = a/R and, for any ǫ > 0, the motion is ergodic, mixing and exponentially unstable with Lyapunov exponent Λ which, for small ǫ, is given by [6] Λ ∼ ǫ 1/2 .For the analysis of classical dynamics, a typical choice of canonical variables is (s, v t ) whe...