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This book contains a unique survey of the mathematically rigorous results about the quantum-mechanical many-body problem that have been obtained by the authors in the past seven years. It addresses a topic that is not only rich mathematically, using a large variety of techniques in mathematical analysis, but is also one with strong ties to current experiments on ultra-cold Bose gases and Bose-Einstein condensation. The book provides a pedagogical entry into an active area of ongoing research for both graduate students and researchers. It is an outgrowth of a course given by the authors for graduate students and post-doctoral researchers at the Oberwolfach Research Institute in 2004. The book also provides a coherent summary of the field and a reference for mathematicians and physicists active in research on quantum mechanics.

Bose-Einstein condensation. --- Bose-Einstein gas --- Mathematics. --- Bose gas --- Gas, Bose-Einstein --- Photons --- Quantum statistics --- Bose condensed fluids --- Bose condensed liquids --- Bose fluids --- Bose liquids --- Einstein condensation --- Bosons --- Condensation --- Superfluidity --- Mathematical physics. --- Statistical physics. --- Condensed Matter Physics. --- Applications of Mathematics. --- Mathematical Methods in Physics. --- Complex Systems. --- Statistical Physics and Dynamical Systems. --- Physics --- Mathematical statistics --- Physical mathematics --- Math --- Science --- Statistical methods --- Mathematics --- Condensed matter. --- Applied mathematics. --- Engineering mathematics. --- Physics. --- Dynamical systems. --- Dynamical systems --- Kinetics --- Mechanics, Analytic --- Force and energy --- Mechanics --- Statics --- Natural philosophy --- Philosophy, Natural --- Physical sciences --- Dynamics --- Engineering --- Engineering analysis --- Mathematical analysis --- Condensed materials --- Condensed media --- Condensed phase --- Materials, Condensed --- Media, Condensed --- Phase, Condensed --- Liquids --- Matter --- Solids

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This work presents a series of experiments with ultracold one-dimensional Bose gases, which establish said gases as an ideal model system for exploring a wide range of non-equilibrium phenomena. With the help of newly developed tools, like full distributions functions and phase correlation functions, the book reveals the emergence of thermal-like transient states, the light-cone-like emergence of thermal correlations and the observation of generalized thermodynamic ensembles. This points to a natural emergence of classical statistical properties from the microscopic unitary quantum evolution, and lays the groundwork for a universal framework of non-equilibrium physics. The thesis investigates a central question that is highly contested in quantum physics: how and to which extent does an isolated quantum many-body system relax? This question arises in many diverse areas of physics, and many of the open problems appear at vastly different energy, time and length scales, ranging from high-energy physics and cosmology to condensed matter and quantum information. A key challenge in attempting to answer this question is the scarcity of quantum many-body systems that are both well isolated from the environment and accessible for experimental study.

Physics. --- Quantum Gases and Condensates. --- Thermodynamics. --- Statistical Physics, Dynamical Systems and Complexity. --- Physique --- Thermodynamique --- Physics --- Physical Sciences & Mathematics --- Atomic Physics --- Nonequilibrium statistical mechanics. --- Bose-Einstein gas. --- Many-body problem. --- n-body problem --- Problem of many bodies --- Problem of n-bodies --- Bose gas --- Gas, Bose-Einstein --- Non-equilibrium statistical mechanics --- Phase transformations (Statistical physics). --- Condensed materials. --- Condensed matter. --- Statistical physics. --- Dynamical systems. --- Mechanics, Analytic --- Photons --- Quantum statistics --- Statistical mechanics --- Complex Systems. --- Statistical Physics and Dynamical Systems. --- Mathematical statistics --- Chemistry, Physical and theoretical --- Dynamics --- Mechanics --- Heat --- Heat-engines --- Quantum theory --- Statistical methods --- Dynamical systems --- Kinetics --- Mathematics --- Force and energy --- Statics --- Condensed materials --- Condensed media --- Condensed phase --- Materials, Condensed --- Media, Condensed --- Phase, Condensed --- Liquids --- Matter --- Solids --- Phase changes (Statistical physics) --- Phase transitions (Statistical physics) --- Phase rule and equilibrium --- Statistical physics