In a number of famous works, M. Kac showed that various methods of probability theory can be fruitfully applied to important problems of analysis. The interconnection between probability and analysis also plays a central role in the present book. However, our approach is mainly based on the application of analysis methods (the method of operator identities, integral equations theory, dual systems, integrable equations) to probability theory (Levy processes, M. Kac's problems, the principle of imperceptibility of the boundary, signal theory). The essential part of the book is dedicated to problems of statistical physics (classical and quantum cases). We consider the corresponding statistical problems (Gibbs-type formulas, non-extensive statistical mechanics, Boltzmann equation) from the game point of view (the game between energy and entropy). One chapter is dedicated to the construction of special examples instead of existence theorems (D. Larson's theorem, Ringrose's hypothesis, the Kadison-Singer and Gohberg-Krein questions). We also investigate the Bezoutiant operator. In this context, we do not make the assumption that the Bezoutiant operator is normally solvable, allowing us to investigate the special classes of the entire functions.

This textbook covers a broad spectrum of developments in QFT, emphasizing those aspects that are now well consolidated and for which satisfactory theoretical descriptions have been provided. The book is unique in that it offers a new approach to the subject and explores many topics merely touched upon, if covered at all, in standard reference works. A detailed and largely non-technical introductory chapter traces the development of QFT from its inception in 1926. The elegant functional differential approach put forward by Schwinger, referred to as the quantum dynamical (action) principle, and its underlying theory are used systematically in order to generate the so-called vacuum-to-vacuum transition amplitude of both abelian and non-abelian gauge theories, in addition to Feynman's well-known functional integral approach, referred to as the path-integral approach. Given the wealth of information also to be found in the abelian case, equal importance is put on both abelian and non-abelian gauge theories. Particular emphasis is placed on the concept of a quantum field and its particle content to provide an appropriate description of physical processes at high energies, where relativity becomes indispensable. Moreover, quantum mechanics implies that a wave function renormalization arises in the QFT field independent of any perturbation theory - a point not sufficiently emphasized in the literature. The book provides an overview of all the fields encountered in present high-energy physics, together with the details of the underlying derivations. Further, it presents "deep inelastic" experiments as a fundamental application of quantum chromodynamics. Though the author makes a point of deriving points in detail, the book still requires good background knowledge of quantum mechanics, including the Dirac Theory, as well as elements of the Klein-Gordon equation. The present volume sets the language, the notation and provides additional background for reading Quantum Field Theory II - Introduction to Quantum Gravity, Supersymmetry and String Theory, by the same author. Students in this field might benefit from first reading the book Quantum Theory: A Wide Spectrum (Springer, 2006), by the same author.

This monograph presents, from the viewpoint of continuum mechanics, a newly emerging field of irreversible thermodynamics, in which linear irreversible thermodynamics are extended to the nonlinear regime and macroscopic phenomena far removed from equilibrium are studied in a manner consistent with the laws of thermodynamics. The tool to develop this thermodynamic theory of irreversible processes are the generalized thermodynamics, which also extends the classical hydrodynamics of Navier, Stokes and Fourier to nonlinear irreversible processes. On the basis of mathematically rigorous representations of the first and the second law of thermodynamics, phenomenological theory (continuum mechanics) deductions are made from the thermodynamic laws of R. Clausius and Lord Kelvin and by this continuum mechanics theories are formulated for macroscopic irreversible processes occurring far removed from equilibrium. Non-equilibrium thermodynamics are developed for thermodynamic functions.

The macroscopic irreversible processes studied include global irreversible processes as well as local hydrodynamic processes at an arbitrary degree of removal from equilibrium. Applications of the theories cover global irreversible processes, simple flows of non-Newtonian and non-Fourier fluids, shock waves of monatomic and diatomic gases, rarefied gas dynamics, ultrasonic wave absorption and dispersion of monatomic and diatomic gases, electrochemical processes, neural networks of chemical reactors, microflows, etc. Variational principles in irreversible thermodynamics and contact manifolds in thermodynamics are also discussed.'

This monograph, will be of interest to condensed matter physicists, chemicalphysicists, biophysicists, mechanical and aerospace engineers, and specialists and graduate students in the fields of irreversible thermodynamics and non-equilibrium statistical mechanics.

This set of lecture notes gives a first coherent account on a novel aspect of the living world that can be called biological information. The book presents both a pedagogical and state-of-the art roadmap of this rapidly evolving field and covers the whole range from information which is encoded in the molecular genetic code to the description of large-scale evolution of complex species networks. The book will prove useful for all those who work at the interface of biology, physics and information science.

social network analysis has been an established eld since the 1950s; in computer and information sciences, in biology, and of course in mathematics (graph theory) networks are central representations of objects and methods (De Nooy, forthc- ing). More detailed bibliometric studies have examined the individual, cognitive, and institutional composition of complex network theory (Morris and Yen 2004), and social network theory (Otte and Rousseau 2002). Among the more impor- .. tant pieces of literature are Borner et al. (2007), Bornholdt and Schuster (2003), Buchanan (2002), Dorogovtsev and Mendes (2003), Otte and Rousseau (2002), Newman (2003), and Watts (1999, 2004). Of these, Borner .. et al. (2007) stand out because they have most recently re-examined network science, considering it as a possible innovation in information science. All the reviews mentioned include efforts to build bridges between different scienti c disciplines and specialties. In this book we draw particular attention to the link between evolutionary economics and statistical physics. Despite this impressive development, claims that an entirely new science has been created (Barabasi ' 2002) have nevertheless been the subject of criticism. - depth analyses of a subset of "complex networks" contributions (1991-2003) have shown that the notion of "complex networks" was already prevalent in a number of different elds before it became practically a "brand name" or the popular label for a new specialty area in physics, or a new cross-disciplinary paradigm.

Bridging the gap between statistical theory and physical experiment, this is a thorough introduction to the statistical methods used in the experimental physical sciences and to the numerical methods used to implement them. An accompanying CD-ROM provides detailed code for implementing many of these algorithms. The treatment emphasises concise but rigorous mathematics but always retains its focus on applications. Readers are assumed to have a sound basic knowledge of differential and integral calculus and some knowledge of vectors and matrices. After an introduction to probability, random variables, computer generation of random numbers and important distributions, the book turns to statistical samples, the maximum likelihood method, and the testing of statistical hypotheses. The discussion concludes with several important statistical methods: least squares, analysis of variance, polynomial regression, and analysis of time series. Appendices provide the necessary methods of matrix algebra, combinatorics, and many sets of useful algorithms and formulae.

Only recently did mankind realise that it resides in a world of networks. The Internet and World Wide Web are changing our life. Our physical existence is based on various biological networks. The term "network" turns out to be a central notion in our time, and the consequent explosion of interest in networks is a social and cultural phenomenon. The principles of the complex organization and evolution of networks, natural and artificial, are the topic of this book, which is written by physicists and is addressed to all involved researchers and students. The aim of the text is to understand networks. The ideas are presented in a clear and a pedagogical way, with minimal mathematics, so even students without a deep knowledge of mathematics and statistical physics will be able to rely on this as a reference. Special attention is given to real networks, both natural and artificial. Collected empirical data and numerous real applications of existing theories are discussed in detail, as well as the topical problems of communication networks.

Starting from basic principles, the book covers a wide variety of topics, ranging from Heisenberg, Schroedinger, second quantization, density matrix and path integral formulations of quantum mechanics, to applications that are (or will be) corner stones of present and future technologies. The emphasis is on spin waves, quantum information, recent tests of quantum physics and decoherence. The book provides a large amount of information without unbalancing the flow of the main ideas by laborious detail.

This volume explores the complex problems that arise in the modeling and simulation of crowd dynamics in order to present the state-of-the-art of this emerging field and contribute to future research activities. Experts in various areas apply their unique perspectives to specific aspects of crowd dynamics, covering the topic from multiple angles. These include a demonstration of how virtual reality may solve dilemmas in collecting empirical data; a detailed study on pedestrian movement in smoke-filled environments; a presentation of one-dimensional conservation laws with point constraints on the flux; a collection of new ideas on the modeling of crowd dynamics at the microscopic scale; and others. Applied mathematicians interested in crowd dynamics, pedestrian movement, traffic flow modeling, urban planning, and other topics will find this volume a valuable resource. Additionally, researchers in social psychology, architecture, and engineering may find this information relevant to their work.

This book provides a unique insight into the latest breakthroughs in a consistent manner, at a level accessible to undergraduates, yet with enough attention to the theory and computation to satisfy the professional researcher Statistical physics addresses the study and understanding of systems with many degrees of freedom. As such it has a rich and varied history, with applications to thermodynamics, magnetic phase transitions, and order/disorder transformations, to name just a few. However, the tools of statistical physics can be profitably used to investigate any system with a large number of components. Thus, recent years have seen these methods applied in many unexpected directions, three of which are the main focus of this volume. These applications have been remarkably successful and have enriched the financial, biological, and engineering literature. Although reported in the physics literature, the results tend to be scattered and the underlying unity of the field overlooked.

This book illustrates how models of complex systems are built up and provides indispensable mathematical tools for studying their dynamics. This second edition includes more recent research results and many new and improved worked out examples and exercises.

The primary goal of the book is to present the ideas and research findings of active researchers such as physicists, economists, mathematicians and financial engineers working in the field of "Econophysics," who have undertaken the task of modeling and analyzing systemic risk, network dynamics and other topics.

Of primary interest in these studies is the aspect of systemic risk, which has long been identified as a potential scenario in which financial institutions trigger a dangerous contagion mechanism, spreading from the financial economy to the real economy.

This type of risk, long confined to the monetary market, has spread considerably in the recent past, culminating in the subprime crisis of 2008. As such, understanding and controlling systemic risk has become an extremely important societal and economic challenge. The Econophys-Kolkata VI conference proceedings are dedicated to addressing a number of key issues involved. Several leading researchers in these fields report on their recent work and also review contemporary literature on the subject.

With many areas of science reaching across their boundaries and becoming more and more interdisciplinary, students and researchers in these fields are confronted with techniques and tools not covered by their particular education. Especially in the life- and neurosciences quantitative models based on nonlinear dynamics and complex systems are becoming as frequently implemented as traditional statistical analysis. Unfamiliarity with the terminology and rigorous mathematics may discourage many scientists to adopt these methods for their own work, even though such reluctance in most cases is not justified.

This book bridges this gap by introducing the procedures and methods used for analyzing nonlinear dynamical systems. In Part I, the concepts of fixed points, phase space, stability and transitions, among others, are discussed in great detail and implemented on the basis of example elementary systems. Part II is devoted to specific, non-trivial applications: coordination of human limb movement (Haken-Kelso-Bunz model), self-organization and pattern formation in complex systems (Synergetics), and models of dynamical properties of neurons (Hodgkin-Huxley, Fitzhugh-Nagumo and Hindmarsh-Rose). Part III may serve as a refresher and companion of some mathematical basics that have been forgotten or were not covered in basic math courses. Finally, the appendix contains an explicit derivation and basic numerical methods together with some programming examples as well as solutions to the exercises provided at the end of certain chapters. Throughout this book all derivations are as detailed and explicit as possible, and everybody with some knowledge of calculus should be able to extract meaningful guidance follow and apply the methods of nonlinear dynamics to their own work.

"This book is a masterful treatment, one might even say a gift, to the interdisciplinary scientist of the future."

"With the authoritative voice of a genuine practitioner, Fuchs is a master teacher of how to handle complex dynamical systems."

"What I find beautiful in this book is its clarity, the clear definition of terms, every step explained simply and systematically."

(J.A.Scott Kelso, excerpts from the foreword)

Why writing a book about a specialized task of the large topic of complex systems? And who will read it? The answer is simple: The fascination for a didactically valuable point of view, the elegance of a closed concept and the lack of a comprehensive disquisition.

The fascinating part is that field equations can have localized solutions exhibiting the typical characteristics of particles. Regarding the field equations this book focuses on, the field phenomenon of localized solutions can be described in the context of a particle formalism, which leads to a set of ordinary differential equations covering the time evolution of the position and the velocity of each particle. Moreover, starting from these particle dynamics and making the transition to many body systems, one considers typical phenomena of many body systems as shock waves and phase transitions, which themselves can be described as field phenomena. Such transitions between different level of modelling are well known from conservative systems, where localized solutions of quantum field theory lead to the mechanisms of elementary particle interaction and from this to field equations describing the properties of matter. However, in dissipative systems such transitions have not been considered yet, which is adjusted by the presented book. The elegance of a closed concept starts with the observation of self-organized current filaments in a semiconductor gas discharge system. These filaments move on random paths and exhibit certain particle features like scattering or the formation of bound states. Neither the reasons for the propagation of the filaments nor the laws of the interaction between the filaments can be registered by direct observations. Therefore a model is established, which is phenomenological in the first instance due to the complexity of the experimental system. This model allows to understand the existence of localized structures, their mechanisms of movement, and their interaction, at least, on a qualitative level. But this model is also the starting point for developing a data analysis method that enables the detection of movement and interaction mechanisms of the investigated localized solutions. The topic is rounded of by applying the data analysis to real experimental data and comparing the experimental observations to the predictions of the model.

A comprehensive publication covering the interesting topic of localized solutions in reaction diffusion systems in its width and its relation to the well known phenomena of spirals and patterns does not yet exist, and this is the third reason for writing this book. Although the book focuses on a specific experimental system the model equations are as simple as possible so that the discussed methods should be adaptable to a large class of systems showing particle-like structures.

Therefore, this book should attract not only the experienced scientist, who is interested in self-organization phenomena, but also the student, who would like to understand the investigation of a complex system on the basis of a continuous description.

The book reports on the latest advances in and applications of chaos theory and intelligent control. Written by eminent scientists and active researchers and using a clear, matter-of-fact style, it covers advanced theories, methods, and applications in a variety of research areas, and explains key concepts in modeling, analysis, and control of chaotic and hyperchaotic systems. Topics include fractional chaotic systems, chaos control, chaos synchronization, memristors, jerk circuits, chaotic systems with hidden attractors, mechanical and biological chaos, and circuit realization of chaotic systems. The book further covers fuzzy logic controllers, evolutionary algorithms, swarm intelligence, and petri nets among other topics. Not only does it provide the readers with chaos fundamentals and intelligent control-based algorithms; it also discusses key applications of chaos as well as multidisciplinary solutions developed via intelligent control. The book is a timely and comprehensive reference guide for graduate students, researchers, and practitioners in the areas of chaos theory and intelligent control.

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. Particular attention is given to the theory of optical bistability formulated in terms of the positive P-representation, and the theory of small bistable systems. This is a textbook at an advanced graduate level. It is intended as a bridge between an introductory discussion of the master equation method and problems of current research.

This book draws on the latest research to discuss the history and development of high-entropy alloys and ceramics in bulk, film, and fiber form. High-entropy materials have recently been developed using the entropy of mixing and entropy of configuration of materials, and have proven to exhibit unique properties superior to those of conventional materials. The field of high-entropy alloys was born in 2004, and has since been developed for both scientific and engineering applications. Although there is extensive literature, this field is rapidly transforming. This book highlights the cutting edge of high-entropy materials, including their fundamentals and applications. Above all, it reflects two major milestones in their development: the equi-atomic ratio single-phase high-entropy alloys; and the non-equi-atomic ratio dual-phase high-entropy alloys.

Computer science and physics have been closely linked since the birth of modern computing. In recent years, an interdisciplinary area has blossomed at the junction of these fields, connecting insights from statistical physics with basic computational challenges. Researchers have successfully applied techniques from the study of phase transitions to analyze NP-complete problems such as satisfiability and graph coloring. This is leading to a new understanding of the structure of these problems, and of how algorithms perform on them.
Computational Complexity and Statistical Physics will serve as a standard reference and pedagogical aid to statistical physics methods in computer science, with a particular focus on phase transitions in combinatorial problems. Addressed to a broad range of readers, the book includes substantial background material along with current research by leading computer scientists, mathematicians, and physicists. It will prepare students and researchers from all of these fields to contribute to this exciting area.

Within the framework of Jaynes' "Predictive Statistical Mechanics," this book presents a detailed derivation of an ensemble formalism for open systems arbitrarily away from equilibrium. This involves a large systematization and extension of the fundamental works and ideas of the outstanding pioneers Gibbs and Boltzmann, and of Bogoliubov, Kirkwood, Green, Mori, Zwanzig, Prigogine and Zubarev, among others.
Chapters 1 to 5 include a description of the philosophy, foundations, and construction (methodology) of the formalism, including the derivation of a nonequilibrium grand-canonical ensemble for far-from-equilibrium systems as well as the derivation of a quantum nonlinear kinetic theory and a response function theory together with a theory of scattering. In chapter 6 applications of the theory are cataloged, making comparisons with experimental data (a basic step for the validation of any theory). Chapter 7 is devoted to the description of irreversible thermodynamics, providing a far-reaching generalization of Informational-Statistical Thermodynamics. The last chapter gives an overall picture of the formalism, and questions and criticisms related to it are discussed.
Audience: This book is directed at an audience of researchers in the field of Statistical Mechanics and Thermodynamics of open nonequilibrium systems. In addition, it is relevant for the study of far-from-equilibrium processes in condensed matter, particularly semiconductor physics, as well as molecular Hydrodynamics, Rheology, many-body systems with complex behavior and areas of engineering, etc. The book can also be used as a complement to advanced graduate courses in Statistical Mechanics.

Complexity science has been a source of new insight in physical and social systems and has demonstrated that unpredictability and surprise are fundamental aspects of the world around us. This book is the outcome of a discussion meeting of leading scholars and critical thinkers with expertise in complex systems sciences and leaders from a variety of organizations, sponsored by the Prigogine Center at The University of Texas at Austin and the Plexus Institute, to explore strategies for understanding uncertainty and surprise. Besides contributions to the conference, it includes a key digest by the editors as well as a commentary by the late nobel laureate Ilya Prigogine, "Surprises in half of a century." The book is intended for researchers and scientists in complexity science, as well as for a broad interdisciplinary audience of both practitioners and scholars. It will well serve those interested in the research issues and in the application of complexity science to physical and social systems.

This book introduces readers to essential tools for the measurement and analysis of information loss in signal processing systems. Employing a new information-theoretic systems theory, the book analyzes various systems in the signal processing engineer's toolbox: polynomials, quantizers, rectifiers, linear filters with and without quantization effects, principal components analysis, multirate systems, etc. The user benefit of signal processing is further highlighted with the concept of relevant information loss. Signal or data processing operates on the physical representation of information so that users can easily access and extract that information. However, a fundamental theorem in information theory-data processing inequality-states that deterministic processing always involves information loss. These measures form the basis of a new information-theoretic systems theory, which complements the currently prevailing approaches based on second-order statistics, such as the mean-squared error or error energy. This theory not only provides a deeper understanding but also extends the design space for the applied engineer with a wide range of methods rooted in information theory, adding to existing methods based on energy or quadratic representations.

Kinetic Theory of granular Gases provides an introduction to the rapidly developing theory of dissipative gas dynamics as it has been developed mainly during the past decade. The book is aimed at readers from the advanced undergraduate level onwards and leads up to the present state of research. The text is self-contained, in the sense that no mathematical or physical knowledge is required that goes beyond standard undergraduate physics courses. The material is adequate for a one-semester course and contains chapter summaries as well as exercises with detailed solutions. Special emphasis is put on a microscopically consistent description of pairwise particle collisions which leads to an impact-velocity dependent coefficient of restitution. The description of the many-particle system, based on the Boltzmann equation, starts with the derivation of the velocity distribution function, followed by the investigation of self-diffusion and Brownian motion. Using hydrodynamical methods, transport processes and self-organized structure formulation are studies. An appendix gives a brief introduction to event-driven molecular dynamics. A second appendix describes a novel mathematical technique for the derivation of the kinetic properties which allows for the application of computer algebra. The book is accompanied by a web page where the molecular dynamics program as well as the computer-algebra programs are provided.

This thesis discusses in detail the measurement of the polarizations of all S-wave vector quarkonium states in LHC proton-proton collisions with the CMS detector. Heavy quarkonium states constitute an ideal laboratory to study non-perturbative effects of quantum chromodynamics and to understand how quarks bind into hadrons. The experimental results are interpreted through an original phenomenological approach, which leads to a coherent picture of quarkonium production cross sections and polarizations within a simple model, dominated by one single color-octet production mechanism. These findings provide new insights into the dynamics of heavy quarkonium production at the LHC, an important step towards a satisfactory understanding of hadron formation within the standard model of particle physics.

This book presents the latest advances in ultrafast science, including both ultrafast optical technology and the study of ultrafast phenomena. It covers picosecond, femtosecond, and attosecond processes relevant to applications in physics, chemistry, biology, and engineering. Ultrafast technology has a profound impact in a wide range of applications, amongst them biomedical imaging, chemical dynamics, frequency standards, material processing, and ultrahigh-speed communications. This book summarizes the results presented at the 19th International Conference on Ultrafast Phenomena and provides an up-to-date view of this important and rapidly advancing field.

Design happens everywhere, whether in animate objects (e.g., dendritic lung structures, bacterial colonies, and corals), inanimate patterns (river basins, beach slope, and dendritic crystals), social dynamics (pedestrian traffic flows), or engineered systems (heat dissipation in electronic circuitry). This design in nature often takes on remarkably similar patterns, which can be explained under one unifying Constructal Law. This book explores the unifying power of the Constructal Law and its applications in all domains of design generation and evolution, ranging from biology and geophysics to globalization, energy, sustainability, and security.
The Constructal Law accounts for the universal tendency of flow systems to morph into evolving configurations that provide greater and easier access over time. The Constructal Law resolves the many and contradictory ad hoc statements of optimality, end design, and destiny in nature, such as minimum and maximum entropy production and minimum and maximum flow resistance, and also explains the designs that are observed and copied in biomimetics.
"Constructal Law and the Unifying Principle of Design" covers the fundamentals of Constructal Theory and Design, as well as presenting a variety ofstate-of-the-art applications. Experts from the biological, physical and social sciences demonstrate the unification of all design phenomena in nature, and apply this knowledge to novel designs in modern engineering, such as vascularization for self-healing and self-cooling materials for aircraft, and tree fins and cavities for heat transfer enhancement.