Material Theories

Material theories is a series of workshops concerned with a broad range of topics related to the mechanics and mathematics of materials. As such, this edition brought together researchers from diverse fields converging toward the interaction between mathematics, mechanics, and material science

Shell structure and orbit bifurcations in finite fermion systems

We first give an overview of the shell-correction method which was developed by V. M. Strutinsky as a practicable and efficient approximation to the general selfconsistent theory of finite fermion systems suggested by A. B. Migdal and collaborators. Then we present in more detail a semiclassical theory of shell effects, also developed by Strutinsky following original ideas of M. Gutzwiller. We emphasize, in particular, the influence of orbit bifurcations on shell structure. We first give a short overview of semiclassical trace formulae, which connect the shell oscillations of a quantum system with a sum over periodic orbits of the corresponding classical system, in what is usually called the "periodic orbit theory". We then present a case study in which the gross features of a typical double-humped nuclear fission barrier, including the effects of mass asymmetry, can be obtained in terms of the shortest periodic orbits of a cavity model with realistic deformations relevant for nuclear fission. Next we investigate shell structures in a spheroidal cavity model which is integrable and allows for far-going analytical computation. We show, in particular, how period-doubling bifurcations are closely connected to the existence of the so-called "superdeformed" energy minimum which corresponds to the fission isomer of actinide nuclei. Finally, we present a general class of radial power-law potentials which approximate well the shape of a Woods-Saxon potential in the bound region, give analytical trace formulae for it and discuss various limits (including the harmonic oscillator and the spherical box potentials).Comment: LaTeX, 67 pp., 30 figures; revised version (missing part at end of 3.1 implemented; order of references corrected

Random Matrix Theories in Quantum Physics: Common Concepts

We review the development of random-matrix theory (RMT) during the last decade. We emphasize both the theoretical aspects, and the application of the theory to a number of fields. These comprise chaotic and disordered systems, the localization problem, many-body quantum systems, the Calogero-Sutherland model, chiral symmetry breaking in QCD, and quantum gravity in two dimensions. The review is preceded by a brief historical survey of the developments of RMT and of localization theory since their inception. We emphasize the concepts common to the above-mentioned fields as well as the great diversity of RMT. In view of the universality of RMT, we suggest that the current development signals the emergence of a new "statistical mechanics": Stochasticity and general symmetry requirements lead to universal laws not based on dynamical principles.Comment: 178 pages, Revtex, 45 figures, submitted to Physics Report

Integrability and Chaotic Behavior in Mechanical Billiard Systems

This thesis is devoted to the study of mathematical billiards in the presence of non-constant potentials and their integrability and chaotic behavior. Classical examples of integrable billiards are free billiards in circles and ellipses. In the presence of specific potentials (such as Kepler potential and harmonic (Hooke) potential), there are various known integrable billiard systems. These integrable examples have been found independently in different contexts. In Chapter 2, we illustrate how some of these integrable billiard systems are related to each other by conformal transformations. As an application, we obtain infinitely many billiard systems defined in central force problems which are integrable on a particular energy level. We then explain that the classical Hooke-Kepler correspondence extends to the correspondence between integrable Hooke and Kepler billiards. As a result, we show that any focused conic sections give rise to integrable Kepler billiards which give new examples of integrable Kepler billiards. The conformal transformation technique is applied to Stark-type problems and Euler\u27s two-center problem and provides new examples of integrable mechanical billiards. In Chapter 3 we show that integrable Kepler and Hooke billiard systems on the plane have the corresponding integrable billiard systems on surfaces of constant curvatures. We also establish the integrability of a class of billiard systems defined in the Lagrangian problem, which is the superposition of two Kepler problems and a Hooke problem, on the sphere, in the plane, and in the hyperbolic plane. These results are obtained by the method of projective dynamics and projective billiards. A toy model of billiard systems with a central force problem in the plane and with a line as the reflection wall was proposed by L. Boltzmann to illustrate his ergodic hypothesis. Later, it has been found that not all such systems are ergodic, and it becomes a question whether some of such systems are ergodic. In Chapter 4, we compute the billiard mappings of Boltzmann\u27s billiard systems, and we present some numerical studies on their chaotic behavior and ergodicity. We found some numerical evidence suggesting that some of these systems might be ergodic

Rydberg Composites

We introduce the Rydberg composite, a new class of Rydberg matter where a single Rydberg atom is interfaced with a dense environment of neutral ground state atoms. The properties of the composite depend on both the Rydberg excitation, which provides the gross energetic and spatial scales, and the distribution of ground state atoms within the volume of the Rydberg wave function, which sculpt the electronic states. The latter range from the "trilobites," for small numbers of scatterers, to delocalized and chaotic eigenstates, for disordered scatterer arrays, culminating in the dense scatterer limit in symmetry-dominated wave functions which promise good control in future experiments. We discuss one-, two-, and three-dimensional arrangements of scatterers using different theoretical methods, enabling us to obtain scaling behavior for the regular spectrum and measures of chaos and delocalization in the disordered regime. We also show that analogous quantum dot composites can elucidate in particular the dense scatterer limit. Thus, we obtain a systematic description of the composite states. The two-dimensional monolayer composite possesses the richest spectrum with an intricate band structure in the limit of homogeneous scatterers, experimentally accessible with pancake-shaped condensates