This book presents the theory and practical applications of the Master equation approach, which provides a powerful general framework for model building in a variety of disciplines. The aim of the book is to not only highlight different mathematical solution methods, but also reveal their potential by means of practical examples. Part I of the book, which can be used as a toolbox, introduces selected statistical fundamentals and solution methods for the Master equation. In Part II and Part III, the Master equation approach is applied to important applications in the natural and social sciences. The case studies presented mainly hail from the social sciences, including urban and regional dynamics, population dynamics, dynamic decision theory, opinion formation and traffic dynamics; however, some applications from physics and chemistry are treated as well, underlining the interdisciplinary modelling potential of the Master equation approach. Drawing upon the author’s extensive teaching and research experience and consulting work, the book offers a valuable guide for researchers, graduate students and professionals alike.

This book highlights the theory and practical applications of the chemical master equation (CME) approach for very large biochemical networks, which provides a powerful general framework for model building in a variety of biological networks. The aim of the book is to not only highlight advanced numerical solution methods for the CME, but also reveal their potential by means of practical examples. The case studies presented are mainly from biology; however, the applications from novel methods are discussed comprehensively, underlining the interdisciplinary approach in simulation and the potential of the chemical master equation approach for modelling bionetworks. The book is a valuable guide for researchers, graduate students, and professionals alike.

This book describes the analysis of macroeconomic agent based models using the tools of statistical mechanics.

This is the first of a two-volume presentation on current research problems in quantum optics, and will serve as a standard reference in the field for many years to come. The book provides an introduction to the methods of quantum statistical mechanics used in quantum optics and their application to the quantum theories of the single-mode laser and optical bistability. The generalized representations of Drummond and Gardiner are discussed together with the more standard methods for deriving Fokker-Planck equations.

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 it originated 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.

This book treats the central physical concepts and mathematical techniques used to investigate the dynamics of open quantum systems. To provide a self-contained presentation the text begins with a survey of classical probability theory and with an introduction into the foundations of quantum mechanics with particular emphasis on its statistical interpretation. The fundamentals of density matrix theory, quantum Markov processes and dynamical semigroups are developed. The most important master equations used in quantum optics and in the theory of quantum Brownian motion are applied to the study of many examples. Special attention is paid to the theory of environment induced decoherence, its role in the dynamical description of the measurement process and to the experimental observation of decohering Schrodinger cat states. The book includes the modern formulation of open quantum systems in terms of stochastic processes in Hilbert space. Stochastic wave function methods and Monte Carlo algorithms are designed and applied to important examples from quantum optics and atomic physics, such as Levy statistics in the laser cooling of atoms, and the damped Jaynes-Cummings model. The basic features of the non-Markovian quantum behaviour of open systems are examined on the basis of projection operator techniques. In addition, the book expounds the relativistic theory of quantum measurements and discusses several examples from a unified perspective, e.g. non-local measurements and quantum teleportation. Influence functional and super-operator techniques are employed to study the density matrix theory in quantum electrodynamics and applications to the destruction of quantum coherence are presented. The text addresses graduate students and lecturers in physics and applied mathematics, as well as researchers with interests in fundamental questions in quantum mechanics and its applications. Many analytical methods and computer simulation techniques are developed and illustrated with the help of numerous specific examples. Only a basic understanding of quantum mechanics and of elementary concepts of probability theory is assumed.

"Highly recommended. . . . This is an important book in putting the burgeoning field of sociodynamics on a solid footing."—Journal of Artificial Societies and Social Simulation This text deals with general modelling concepts in the social sciences, their applications, and their mathematical methods. The author's well-organized approach offers a clear, coherent introduction to terminology, approaches, and goals in modelling. Appropriate for advanced undergraduates and graduate students, it requires a solid background in algebra and calculus. The three-part treatment begins by addressing general modelling concepts, the second part provides applications, and the third discusses mathematical method. Topics include population dynamics, group interaction, political transitions, evolutionary economics, and urbanization. Guiding students through a series of practical applications that illustrate the methods' potential scope, the text concludes with a detailed look at mathematical methods.

This two-volume book offers a comprehensive treatment of the probabilistic approach to mean field game models and their applications. The book is self-contained in nature and includes original material and applications with explicit examples throughout, including numerical solutions. Volume II tackles the analysis of mean field games in which the players are affected by a common source of noise. The first part of the volume introduces and studies the concepts of weak and strong equilibria, and establishes general solvability results. The second part is devoted to the study of the master equation, a partial differential equation satisfied by the value function of the game over the space of probability measures. Existence of viscosity and classical solutions are proven and used to study asymptotics of games with finitely many players. Together, both Volume I and Volume II will greatly benefit mathematical graduate students and researchers interested in mean field games. The authors provide a detailed road map through the book allowing different access points for different readers and building up the level of technical detail. The accessible approach and overview will allow interested researchers in the applied sciences to obtain a clear overview of the state of the art in mean field games.

This monograph provides graduate students and also professional researchers aiming to understand the dynamics of open quantum systems with a valuable and self-contained toolbox. Special focus is laid on the link between microscopic models and the resulting open-system dynamics. This includes how to derive the celebrated Lindblad master equation without applying the rotating wave approximation. As typical representatives for non-equilibrium configurations it treats systems coupled to multiple reservoirs (including the description of quantum transport), driven systems and feedback-controlled quantum systems. Each method is illustrated with easy-to-follow examples from recent research. Exercises and short summaries at the end of every chapter enable the reader to approach the frontiers of current research quickly and make the book useful for quick reference.

This book presents powerful tools for integrating interrelated composites--such as capabilities, policies, treatments, indices, and systems--into structural equation modeling (SEM). Jörg Henseler introduces the types of research questions that can be addressed with composite-based SEM and explores the differences between composite- and factor-based SEM, variance- and covariance-based SEM, and emergent and latent variables. Using rich illustrations and walked-through data sets, the book covers how to specify, identify, estimate, and assess composite models using partial least squares path modeling, maximum likelihood, and other estimators, as well as how to interpret findings and report the results. Advanced topics include confirmatory composite analysis, mediation analysis, second-order constructs, interaction effects, and importance–performance analysis. Most chapters conclude with software tutorials for ADANCO and the R package cSEM. The companion website includes data files and syntax for the book's examples, along with presentation slides.