Some exactly solvable models of urn process theory

International audienceWe establish a fundamental isomorphism between discrete-time balanced urn processes and certain ordinary differential systems, which are nonlinear, autonomous, and of a simple monomial form. As a consequence, all balanced urn processes with balls of two colours are proved to be analytically solvable in finite terms. The corresponding generating functions are expressed in terms of certain Abelian integrals over curves of the Fermat type (which are also hypergeometric functions), together with their inverses. A consequence is the unification of the analyses of many classical models, including those related to the coupon collector's problem, particle transfer (the Ehrenfest model), Friedman's "adverse campaign'' and Pólya's contagion model, as well as the OK Corral model (a basic case of Lanchester's theory of conflicts). In each case, it is possible to quantify very precisely the probable composition of the urn at any discrete instant. We study here in detail "semi-sacrificial'' urns, for which the following are obtained: a Gaussian limiting distribution with speed of convergence estimates as well as a characterization of the large and extreme large deviation regimes. We also work out explicitly the case of $2$-dimensional triangular models, where local limit laws of the stable type are obtained. A few models of dimension three or greater, e.g., "autistic'' (generalized Pólya), cyclic chambers (generalized Ehrenfest), generalized coupon-collector, and triangular urns, are also shown to be exactly solvable

New Trends in Statistical Physics of Complex Systems

A topical research activity in statistical physics concerns the study of complex and disordered systems. Generally, these systems are characterized by an elevated level of interconnection and interaction between the parts so that they give rise to a rich structure in the phase space that self-organizes under the control of internal non-linear dynamics. These emergent collective dynamics confer new behaviours to the whole system that are no longer the direct consequence of the properties of the single parts, but rather characterize the whole system as a new entity with its own features, giving rise to the birth of new phenomenologies. As is highlighted in this collection of papers, the methodologies of statistical physics have become very promising in understanding these new phenomena. This volume groups together 12 research works showing the use of typical tools developed within the framework of statistical mechanics, in non-linear kinetic and information geometry, to investigate emerging features in complex physical and physical-like systems. A topical research activity in statistical physics concerns the study of complex and disordered systems. Generally, these systems are characterized by an elevated level of interconnection and interaction between the parts so that they give rise to a rich structure in the phase space that self-organizes under the control of internal non-linear dynamics. These emergent collective dynamics confer new behaviours to the whole system that are no longer the direct consequence of the properties of the single parts, but rather characterize the whole system as a new entity with its own features, giving rise to the birth of new phenomenologies. As is highlighted in this collection of papers, the methodologies of statistical physics have become very promising in understanding these new phenomena. This volume groups together 12 research works showing the use of typical tools developed within the framework of statistical mechanics, in non-linear kinetic and information geometry, to investigate emerging features in complex physical and physical-like systems

Towards the spectral properties and phase structure of QCD

In this thesis we explore a multitude of aspects concerning strongly coupled quantum field theories, with a special focus on QCD. The first part of the thesis is concerned with formal developments, with the noteworthy highlight of enabling the use of hydrodynamic numerical methods in Functional Renormalization Group equations. This lead to the subsequent discovery of discontinuous solutions for the effective potential in the vicinity of first order phase transitions

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

Bose-Einstein condensates with long-range dipolar interactions

Bose-Einstein condensation is a phase transition which atoms undergo when cooled near absolute zero temperature Since the theoretical prediction in 1924, and the spectacular experimental confirmation of Bose-Einstein condensation in 1995, a rich new field in physics has emerged studying ultracold degenerate quantum gases. Although these ultracold gases are very dilute, their properties are nevertheless strongly influenced by interatomic interactions. Usually, these interactions are dominated by short range, isotropic contact interactions. In contrast, the recently realised Bose-Einstein Condensate (BEC) of Chromium atoms contains long-range, anisotropic dipolar interactions leading to interesting new physics. In this graduation project, stationary states of such dipolar BECs in harmonic traps are investigated for various experimentally relevant parameters. Furthermore, the elementary excitations of the BEC are calculated, as well as its response to a rotating perturbation. Finally, some more advanced topics such as vortex interactions and condensate response to impurities are investigated. Bose-Einstein condensation is a phase transition which atoms undergo when cooled near absolute zero temperature Since the theoretical prediction in 1924, and the spectacular experimental confirmation of Bose-Einstein condensation in 1995, a rich new field in physics has emerged studying ultracold degenerate quantum gases. Although these ultracold gases are very dilute, their properties are nevertheless strongly influenced by interatomic interactions. Usually, these interactions are dominated by short range, isotropic contact interactions. In contrast, the recently realised Bose-Einstein Condensate (BEC) of Chromium atoms contains long-range, anisotropic dipolar interactions leading to interesting new physics. In this graduation project, stationary states of such dipolar BECs in harmonic traps are investigated for various experimentally relevant parameters. Furthermore, the elementary excitations of the BEC are calculated, as well as its response to a rotating perturbation. Finally, some more advanced topics such as vortex interactions and condensate response to impurities are investigated