Hydrodynamic Scales Of Integrable Many Body Systems

This book provides a broad introduction to integrable systems with many degrees of freedom. Within a much larger orbit, discussed are models such as the classical Toda lattice, Calogero fluid, and Ablowitz-Ladik discretized nonlinear Schrödinger equation. On the quantum mechanical side, featured are the Lieb-Liniger delta-Bose gas and the quantum Toda lattice. As a genuinely novel twist, the study deals with random initial data described by generalized Gibbs ensembles with parameters of slow spatial variation. This is the hydrodynamic scale, in spirit similar to the ballistic Euler scale of nonintegrable simple fluids. While integrable microscopic particle models are very diverse, the central theme of this book is to elucidate their structural similarity on hydrodynamic scales.

The book is devoted to the study of the correlation effects in many-particle systems. It presents the advanced methods of quantum statistical mechanics (equilibrium and nonequilibrium), and shows their effectiveness and operational ability in applications to problems of quantum solid-state theory, quantum theory of magnetism and the kinetic theory. The book includes description of the fundamental concepts and techniques of analysis following the approach of N N Bogoliubov's school, including recent developments. It provides an overview that introduces the main notions of quantum many-particle physics with the emphasis on concepts and models. This book combines the features of textbook and research monograph. For many topics the aim is to start from the beginning and to guide the reader to the threshold of advanced researches. Many chapters include also additional information and discuss many complex research areas which are not often discussed in other places. The book is useful for established researchers to organize and present the advanced material disseminated in the literature. The book contains also an extensive bibliography. The book serves undergraduate, graduate and postgraduate students, as well as researchers who have had prior experience with the subject matter at a more elementary level or have used other many-particle techniques.

This is the second part of a two-volume textbook on the modern statistical theory of nonequilibrium processes. The general method of nonequilibrium statistical ensembles developed in the first volume is applied to various problems in linear response theory, relaxation phenomena, hydrodynamics, and the dynamical theory of fluctuations. Some active areas of research are considered, including relaxation processes in highly nonequilibrium systems, kinetic processes in lasers, nonequilibrium many-particle correlations in the Green’s function formalism, superfluid statistical hydrodynamics, large-scale fluctuations in nonequilibrium systems, and statistical mechanics of turbulence. Exercises and problems for readers are also included. The book is self-contained and accessible to students having heard the standard course in statistical physics. It is also of interest for specialists working in solid state physics, chemical physics, and physics of plasma and fluids.

This book includes the joint proceedings of the International Conference on Particle Systems and PDEs VI, VII and VIII. Particle Systems and PDEs VI was held in Nice, France, in November/December 2017, Particle Systems and PDEs VII was held in Palermo, Italy, in November 2018, and Particle Systems and PDEs VIII was held in Lisbon, Portugal, in December 2019. Most of the papers are dealing with mathematical problems motivated by different applications in physics, engineering, economics, chemistry and biology. They illustrate methods and topics in the study of particle systems and PDEs and their relation. The book is recommended to probabilists, analysts and to those mathematicians in general, whose work focuses on topics in mathematical physics, stochastic processes and differential equations, as well as to those physicists who work in statistical mechanics and kinetic theory.

The quantum inverse scattering method is a means of finding exact solutions of two-dimensional models in quantum field theory and statistical physics (such as the sine-Go rdon equation or the quantum non-linear Schrödinger equation). These models are the subject of much attention amongst physicists and mathematicians.The present work is an introduction to this important and exciting area. It consists of four parts. The first deals with the Bethe ansatz and calculation of physical quantities. The authors then tackle the theory of the quantum inverse scattering method before applying it in the second half of the book to the calculation of correlation functions. This is one of the most important applications of the method and the authors have made significant contributions to the area. Here they describe some of the most recent and general approaches and include some new results.The book will be essential reading for all mathematical physicists working in field theory and statistical physics.

This thesis is a contribution at the intersection of a number of active fields in theoretical and experimental condensed matter, particularly those concerned with disordered systems, integrable models, lattice gauge theories, and non-equilibrium quantum dynamics. It contributes an important new facet to our understanding of relaxation in isolated quantum systems by conclusively demonstrating localization without disorder for the first time, answering a long-standing question in this field. This is achieved by introducing a family of models – intimately related to paradigmatic condensed matter models – and studying their non-equilibrium dynamics through a combination of exact analytical mappings and an array of numerical techniques. This thesis also makes contributions relevant to the theory of quantum chaotic behaviour by calculating novel, and often intractable, entanglement measures and out-of-time-ordered correlators. A concrete and feasible proposal is also made for the experimental realization and dynamical study of the family of models, based on currently available technologies.

This book deals with one of the fundamental problems of nonequilibrium statistical mechanics: the explanation of large-scale dynamics (evolution differential equations) from models of a very large number of interacting particles. This book addresses both researchers and students. Much of the material presented has never been published in book-form before.