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We show that a mobility edge exists in 1D random potentials provided specific long-range correlations. Our approach is based on the relation between binary correlator of a site potential and the localization length. We give the algorithm to construct numerically potentials with mobility edge at any given energy inside allowed zone. Another natural way to generate such potentials is to use chaotic trajectories of non-linear maps. Our numerical calculations for few particular potentials demonstrate the presence of mobility edges in 1D geometry.PACS numbers: 72.15. Rw, 03.65.BZ, 72.10.Bg It is commonly believed that there is no mobility edge in 1D models with random-like potentials. This is based on the fact that for random potentials all eigenstates are exponentially localized, no matter how weak the randomness is [1]. On the other hand, for potentials with "correlated disorder" the localization length diverges for specific values of energy (see, e.g. [2,3]). The well-studied model of this kind is the so-called "random dimer" [4] for which the potential has peculiar short-range correlations. Though there is no mobility edge for such potentials, this example shows a highly non-trivial role of correlations. In this Letter we study the relation between correlations in the site-potential of 1D tight-binding model and localization properties of eigenstates, and give examples of the potentials with mobility edges inside the energy band.The model under consideration is the discrete Shrödinger equation for stationary eigenstates ψ n (E) ,where E is the eigenenergy and ǫ n is the site-potential.To study the origin of delocalized states in longcorrelated random potentials, we suggest a simple and clear approach based on the representation of the quantum model (1) in terms of classical two-dimensional Hamiltonian map,

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.

A brief review of the Fermi-Pasta-Ulam (FPU) paradox is given, together with its suggested resolutions and its relation to other physical problems. We focus on the ideas and concepts that have become the core of modern nonlinear mechanics, in their historical perspective. Starting from the first numerical results of FPU, both theoretical and numerical findings are discussed in close connection with the problems of ergodicity, integrability, chaos and stability of motion. New directions related to the Bose-Einstein condensation and quantum systems of interacting Boseparticles are also considered.

The approach is developed for the description of isolated Fermi-systems with finite number of particles, such as complex atoms, nuclei, atomic clusters etc. It is based on statistical properties of chaotic excited states which are formed by the interaction between particles. New type of "microcanonical" partition function is introduced and expressed in terms of the average shape of eigenstates F (E k , E) where E is the total energy of the system. This partition function plays the same role as the canonical expression exp(−E (i) /T ) for open systems in thermal bath. The approach allows to calculate mean values and non-diagonal matrix elements of different operators. In particular, the following problems have been considered: distribution of occupation numbers and its relevance to the canonical and Fermi-Dirac distributions; criteria of equilibrium and thermalization; thermodynamical equation of state and the meaning of temperature, entropy and heat capacity, increase of effective temperature due to the interaction. The problems of spreading widths and shape of the eigenstates are also studied.

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