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Intensive studies on light–matter interactions and technological breakthroughs, especially conducted in the field of dressed photon research, have led to a growing concern regarding unsettled off-shell quantum field interactions. The Special Issue, entitled “Quantum Fields and Off-Shell Sciences”, was organized to promote the progress of such research activities from a wider perspective, not limited to dressed photon studies. This book contains excellent papers that were published in this Special Issue. It will provide scientific and technical information on the quantum fields and off-shell sciences to scientists, engineers, and students who are and will be engaged in this field.

Research & information: general --- Physics --- dressed photon --- dressed photon constant --- natural units --- Heisenberg cut --- de Sitter space --- dark energy --- dark matter --- cosmological constant --- twin universes --- conformal cyclic cosmology --- quantum walk --- scattering theory --- energy --- survival probability --- attractor eigenspace --- category --- algebra --- state --- category algebra --- state on category --- noncommutative probability --- quantum probability --- GNS representation --- quantum measurement --- C*-algebra --- algebraic quantum field theory --- local net --- extension of local net --- completely positive instrument --- macroscopic distinguishability --- Grassmann manifold --- flag manifold --- pre-homogeneous vector space --- invariants --- category theory --- nonstandard analysis --- coarse geometry --- quantum field --- combinatorial optimization --- Ising spin glass --- coupled oscillator --- eigenmode --- clustering --- localization --- dissipation --- off-shell science --- non-equilibrium open system --- quantum master equation --- quantum density matrix --- projection operator --- renormalization --- discrete-time quantum walk --- scattering quantum random walk --- Grover walk --- pathfinding --- network

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Over recent decades, the increase in computational resources, coupled with the popularity of competitive quantum mechanics alternatives (particularly DFT), has promoted the widespread penetration of quantum mechanics calculations into a variety of fields targeting the reactivity of molecules. This book presents a selection of original research papers and review articles illustrating diverse applications of quantum mechanics in the study of problems involving molecules and their reactivity.

Pyrophosphate --- electronic structure --- mechanical properties --- optical properties --- first-principles calculations --- chemical reactivity theory --- HSAB principle --- information theory --- quantum mechanics --- regional complementarity rule --- virial theorem --- free radical scavengers --- antioxidants --- fluoxetine --- depressive disorder --- major --- oxidative stress --- DFT calculations --- reactive oxygen species --- porphyrins, density functional theory --- DFT --- surfaces --- self-assembly --- scanning tunneling microscopy --- dispersion --- nanostructures --- solid state --- condensed phase --- [NiFeSe] hydrogenase --- quantum mechanics (QM)/molecular mechanics (MM), geometry optimizations --- vibrational frequency analyses --- Fourier transform infrared (FTIR) frequencies --- Quercetin molecule --- conformational mobility --- hydroxyl group --- transition state --- concerted rotation of the hydroxyl groups --- quantum-chemical calculations --- quantum technology --- chemical kinetics --- reaction rate --- RRKM theory --- master equation --- coordination complexes --- donor–acceptor systems --- partial electronic flows --- phase–current relations --- subsystem phases --- n/a --- donor-acceptor systems --- phase-current relations

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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.

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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

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