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Relaxation methods (Mathematics) --- Relaxation, Méthodes de (mathématiques) --- Analyse numérique --- Numerical analysis. --- Equations aux derivees partielles --- Boltzmann equation --- Methodes numeriques --- Equations aux derivees partielles --- Boltzmann equation --- Methodes numeriques

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Polymers. --- Chemical kinetics. --- Aging tests materials --- Materials --- Creep properties --- Mechanical properties --- Relaxation mechanics --- Electric property --- Cracking fracturing --- Impact tests --- Glass fibers --- Extrapolation --- Dilatometry --- Tension tests --- Boltzmann equation --- Aging tests materials --- Materials --- Creep properties --- Mechanical properties --- Relaxation mechanics --- Electric property --- Cracking fracturing --- Impact tests --- Glass fibers --- Extrapolation --- Dilatometry --- Tension tests --- Boltzmann equation

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The PIDM model consists in two types of dark matter, one component of which accounts for a small portion of the total amount of dark matter and is self-interacting. The aim is to establish the characteristics of this subdominant dark-matter part. We first present the dark matter concept by reviewing the historical evidence of dark matter as well as the different sorts of dark matter. Then we introduce the concordance model in cosmology and evoke its small-scales defects. Dark-matter models are then mentioned to solve these problems, one of which being the PIDM model. Next, starting from the Boltzmann equation, we derive a formula allowing to compute the relic density of a species and we test it for the WIMP model. Then we derive a precise expression for the thermally averaged annihilation cross section. In doing so, we determine the characteristics of the subdominant dark-matter particles by constraining the thermally averaged annihilation cross section so that its abundance remains small compared to the total dark-matter abundance as stated by the PIDM model.

dark matter --- subdominant --- self-interacting --- abundance --- relic density --- WIMP --- PIDM --- freeze-out --- Boltzmann equation --- thermally averaged annihilation cross section --- kinetic mixing --- Physique, chimie, mathématiques & sciences de la terre > Aérospatiale, astronomie & astrophysique

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The book is devoted to the fundamental aspects of the non-equilibrium statistical mechanics of many-particle systems. The concept of Zubarev’s approach, which generalizes the notion of Gibbs’ ensembles, and introduces a nonequilibrium statistical operator, providing an adequate basis for dealing with strongly correlated systems that are governed by nonperturbative phenomena, such as the formation of bound states, quantum condensates and the instability of the vacuum. Besides a general introduction to the formalism, this book contains contributions devoted to the applications of Zubarev’s method to the solution of modern problems in different fields of physics: transport theory, hydrodynamics, high-energy physics, quark-gluon plasma and hadron production in heavy-ion collisions. The book provides valuable information for researchers and students in these fields, requiring powerful concepts to solve fundamental problems of non-equilibrium phenomena in strongly

Research & information: general --- relativistic fluid dynamics --- statistical operator --- non-equilibrium states --- transport coefficients --- correlation functions --- open quantum system --- master equation --- non-equilibrium statistical operator --- relevant statistical operator --- quasi-temperature --- dynamic correlations --- QCD matter --- phase transition --- critical point --- nonequilibrium thermo-field dynamics --- kinetics --- hydrodynamics --- kinetic equations --- bound states --- quark-gluon plasma --- out-of-equilibrium quantum field theory --- dimensional renormalization --- finite-time-path formalism --- Boltzmann equation --- gluon saturation --- pion enhancement --- ALICE --- LHC --- thermalization --- hadronization --- Gibbs equilibrium statistical mechanics --- Bogoliubov’s quasi-averages --- pressure fluctuations --- relativistic ideal gas --- kinetic theory --- particle production --- Schwinger effect --- Zitterbewegung --- low density approximation --- quantum statistical mechanics --- relativistic hydrodynamics --- Kubo formulae --- graphene --- dynamic critical phenomena --- high-field and nonlinear effects --- QCD --- gluons --- Bose-Einstein condensate --- Fokker-Planck equation --- relaxation time approximation --- linear response theory --- permittivity, dynamical conductivity, absorption coefficient, dynamical collision frequency --- ordered lattice, disordered lattice --- Umklapp process --- interband transitions --- finite temperature field theory --- path integrals --- quantum fields in curved spacetime --- symmetries --- quantum anomalies --- irreversibility --- entropy --- electrical conductivity --- Zubarev operator --- Unruh effect --- acceleration --- Zubarev formalism --- pion chemical potential --- relativistic fluid dynamics --- statistical operator --- non-equilibrium states --- transport coefficients --- correlation functions --- open quantum system --- master equation --- non-equilibrium statistical operator --- relevant statistical operator --- quasi-temperature --- dynamic correlations --- QCD matter --- phase transition --- critical point --- nonequilibrium thermo-field dynamics --- kinetics --- hydrodynamics --- kinetic equations --- bound states --- quark-gluon plasma --- out-of-equilibrium quantum field theory --- dimensional renormalization --- finite-time-path formalism --- Boltzmann equation --- gluon saturation --- pion enhancement --- ALICE --- LHC --- thermalization --- hadronization --- Gibbs equilibrium statistical mechanics --- Bogoliubov’s quasi-averages --- pressure fluctuations --- relativistic ideal gas --- kinetic theory --- particle production --- Schwinger effect --- Zitterbewegung --- low density approximation --- quantum statistical mechanics --- relativistic hydrodynamics --- Kubo formulae --- graphene --- dynamic critical phenomena --- high-field and nonlinear effects --- QCD --- gluons --- Bose-Einstein condensate --- Fokker-Planck equation --- relaxation time approximation --- linear response theory --- permittivity, dynamical conductivity, absorption coefficient, dynamical collision frequency --- ordered lattice, disordered lattice --- Umklapp process --- interband transitions --- finite temperature field theory --- path integrals --- quantum fields in curved spacetime --- symmetries --- quantum anomalies --- irreversibility --- entropy --- electrical conductivity --- Zubarev operator --- Unruh effect --- acceleration --- Zubarev formalism --- pion chemical potential

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This book describes the latest advances in the theory of mean field games, which are optimal control problems with a continuum of players, each of them interacting with the whole statistical distribution of a population. While originating in economics, this theory now has applications in areas as diverse as mathematical finance, crowd phenomena, epidemiology, and cybersecurity.Because mean field games concern the interactions of infinitely many players in an optimal control framework, one expects them to appear as the limit for Nash equilibria of differential games with finitely many players, as the number of players tends to infinity. This book rigorously establishes this convergence, which has been an open problem until now. The limit of the system associated with differential games with finitely many players is described by the so-called master equation, a nonlocal transport equation in the space of measures. After defining a suitable notion of differentiability in the space of measures, the authors provide a complete self-contained analysis of the master equation. Their analysis includes the case of common noise problems in which all the players are affected by a common Brownian motion. They then go on to explain how to use the master equation to prove the mean field limit.This groundbreaking book presents two important new results in mean field games that contribute to a unified theoretical framework for this exciting and fast-developing area of mathematics.

Convergence. --- Mean field theory. --- Many-body problem --- Statistical mechanics --- Functions --- A priori estimate. --- Approximation. --- Bellman equation. --- Boltzmann equation. --- Boundary value problem. --- C0. --- Chain rule. --- Compact space. --- Computation. --- Conditional probability distribution. --- Continuous function. --- Convergence problem. --- Convex set. --- Cooperative game. --- Corollary. --- Decision-making. --- Derivative. --- Deterministic system. --- Differentiable function. --- Directional derivative. --- Discrete time and continuous time. --- Discretization. --- Dynamic programming. --- Emergence. --- Empirical distribution function. --- Equation. --- Estimation. --- Euclidean space. --- Folk theorem (game theory). --- Folk theorem. --- Heat equation. --- Hermitian adjoint. --- Implementation. --- Initial condition. --- Integer. --- Large numbers. --- Linearization. --- Lipschitz continuity. --- Lp space. --- Macroeconomic model. --- Markov process. --- Martingale (probability theory). --- Master equation. --- Mathematical optimization. --- Maximum principle. --- Method of characteristics. --- Metric space. --- Monograph. --- Monotonic function. --- Nash equilibrium. --- Neumann boundary condition. --- Nonlinear system. --- Notation. --- Numerical analysis. --- Optimal control. --- Parameter. --- Partial differential equation. --- Periodic boundary conditions. --- Porous medium. --- Probability measure. --- Probability theory. --- Probability. --- Random function. --- Random variable. --- Randomization. --- Rate of convergence. --- Regime. --- Scientific notation. --- Semigroup. --- Simultaneous equations. --- Small number. --- Smoothness. --- Space form. --- State space. --- State variable. --- Stochastic calculus. --- Stochastic control. --- Stochastic process. --- Stochastic. --- Subset. --- Suggestion. --- Symmetric function. --- Technology. --- Theorem. --- Theory. --- Time consistency. --- Time derivative. --- Uniqueness. --- Variable (mathematics). --- Vector space. --- Viscosity solution. --- Wasserstein metric. --- Weak solution. --- Wiener process. --- Without loss of generality.