This book provides a comprehensive review of the theory of phase transitions and its modern applications, based on the five pillars of the modern theory of phase transitions: the Ising model, mean field, scaling, renormalization group and universality. This expanded second edition includes, along with a description of vortices and high temperature superconductivity, a discussion of phase transitions in chemical reactions and moving systems. The book covers the close connection between phase transitions and small world phenomena as well as scale-free systems such as the stock market and the Internet.

This book presents both the fundamentals and the major research topics in statistical physics of systems out of equilibrium. It summarizes different approaches to describe such systems on the thermodynamic and stochastic levels, and discusses a variety of areas including reactions, anomalous kinetics, and the behavior of self-propelling particles.

Path following in combination with boundary value problem solvers has emerged as a continuing and strong influence in the development of dynamical systems theory and its application. It is widely acknowledged that the software package AUTO - developed by Eusebius J. Doedel about thirty years ago and further expanded and developed ever since - plays a central role in the brief history of numerical continuation. This book has been compiled on the occasion of Sebius Doedel's 60th birthday. Bringing together for the first time a large amount of material in a single, accessible source, it is hoped that the book will become the natural entry point for researchers in diverse disciplines who wish to learn what numerical continuation techniques can achieve. The book opens with a foreword by Herbert B. Keller and lecture notes by Sebius Doedel himself that introduce the basic concepts of numerical bifurcation analysis. The other chapters by leading experts discuss continuation for various types of systems and objects and showcase examples of how numerical bifurcation analysis can be used in concrete applications. Topics that are treated include: interactive continuation tools, higher-dimensional continuation, the computation of invariant manifolds, and continuation techniques for slow-fast systems, for symmetric Hamiltonian systems, for spatially extended systems and for systems with delay. Three chapters review physical applications: the dynamics of a SQUID, global bifurcations in laser systems, and dynamics and bifurcations in electronic circuits.

Flocks of birds, schools of fish and swarms of locusts display amazing forms of collective motion, while huge numbers of glow worms can emit light signals with almost unbelievable synchronization. These and many other collective phenomena in animal societies take place according to laws very similar to those governing the collective behavior in the inanimate nature, such as the magnetization of iron and the light radiation of lasers. During recent years, this has led to the study of swarm behavior as a challenging new field of science, in which ideas from the physical world are applied in order to understand the formation and structure of animal swarms. From these studies, it has become clear that such collective behavior of animals emerges in a self-organized way, without any need of overall coordination. In this book, we present different swarm phenomena of the animal world and compare them to their counterparts in physics, in a conceptual and non-technical way, addressed to a general readership.

This invaluable book provides a broad introduction to a rapidly growing area of nonequilibrium statistical physics. The first part of the book complements the classical book on the Langevin and Fokker-Planck equations (H. Risken, The Fokker-Planck Equation: Methods of Solution and Applications (Springer, 1996)). Some topics and methods of solutions are presented and discussed in details which are not described in Risken's book, such as the method of similarity solution, the method of characteristics, transformation of diffusion processes into the Wiener process in different prescriptions, harmonic noise and relativistic Brownian motion. Connection between the Langevin equation and Tsallis distribution is also discussed.Due to the growing interest in the research on the generalized Langevin equations, several of them are presented. They are described with some details.Recent research on the integro-differential Fokker-Planck equation derived from the continuous time random walk model shows that the topic has several aspects to be explored. This equation is worked analytically for the linear force and the generic waiting time probability distribution function. Moreover, generalized Klein-Kramers equations are also presented and discussed. They have the potential to be applied to natural systems, such as biological systems.

The book summarizes the state-of-the-art of research on control of self-organizing nonlinear systems with contributions from leading international experts in the field. The first focus concerns recent methodological developments including control of networks and of noisy and time-delayed systems. As a second focus, the book features emerging concepts of application including control of quantum systems, soft condensed matter, and biological systems. Special topics reflecting the active research in the field are the analysis and control of chimera states in classical networks and in quantum systems, the mathematical treatment of multiscale systems, the control of colloidal and quantum transport, the control of epidemics and of neural network dynamics.

In this book we have solved the complicated problem of constructing upper bounds for many-time averages for the case of a fairly broad class of model systems with four-fermion interaction. The methods proposed in this book for solving this problem will undoubtedly find application not only for the model systems associated with the theory of superconductivity considered here. The theoretical methods developed in Chapters 1 and 2 are already applicable to a much broader class of model systems from statistical physics and the theory of elementary particles.

Statistics links microscopic and macroscopic phenomena, and requires for this reason a large number of microscopic elements like atoms. The results are values of maximum probability or of averaging. This introduction to statistical physics concentrates on the basic principles, and attempts to explain these in simple terms supplemented by numerous examples. These basic principles include the difference between classical and quantum statistics, a priori probabilities as related to degeneracies, the vital aspect of indistinguishability as compared with distinguishability in classical physics, the differences between conserved and non-conserved elements, the different ways of counting arrangements in the three statistics (Maxwell-Boltzmann, Fermi-Dirac, Bose-Einstein), the difference between maximization of the number of arrangements of elements, and averaging in the Darwin-Fowler method. Significant applications to solids, radiation and electrons in metals are treated in separate chapters, as well as Bose-Einstein condensation. This revised second edition contains an additional chapter on the Boltzmann transport equation along with appropriate applications. Also, more examples have been added throughout, as well as further references to literature.

The aim of this book is to give a physical treatment of the kinetic theory of gases and magnetoplasmas, covering the standard material in as simple a way as possible, using mean-free-path arguments when possible and identifying problem areas where received theory has either failed or has fallen short of expectations. Examples are provided by strong shock waves, ultrasonic waves (high Knudsen numbers), and transport across strong magnetic fields. Examples of problem areas provided by strong shock waves, ultrasonic waves (high Knudsen numbers), and transport across strong magnetic fields. One of the paradoxes arising in kinetic theory concerns the fluid pressure. Collisions are necessary for a fluid force to result, yet standard kinetic theory does not entail this, being satisfied to bypass Newton's equations by defining pressure as a momentum flux. This omission usually has no adverse consequences, but with increasing Knudsen number, it leads to errors. This text pays particular attention to pressure, explaining the importance of allowing for its collisional nature from the outset in developing kinetic theory.

Examining important results and analytical techniques, this graduate-level textbook is a step-by-step presentation of the structure and function of complex networks. Using a range of examples, from the stability of the internet to efficient methods of immunizing populations, and from epidemic spreading to how one might efficiently search for individuals, this textbook explains the theoretical methods that can be used, and the experimental and analytical results obtained in the study and research of complex networks. Giving detailed derivations of many results in complex networks theory, this is an ideal text to be used by graduate students entering the field. End-of-chapter review questions help students monitor their own understanding of the materials presented.

Gets right to the point with step-by-step guidance on solving physics problems. Covers all topics in standard general physics courses in the same sequence. Keeps learning about physics fun and engaging through the story of dinosaurs being tested on their knowledge for a final challenge (deflecting an asteroid headed to Earth!). Enables the reader to quickly flip through and locate steps needed for a particular problem. Includes tons of easy to follow diagrams and worked solutions.

Cosmology has undergone a revolution in recent years. The exciting interplay between astronomy and fundamental physics has led to dramatic revelations, including the existence of the dark matter and the dark energy that appear to dominate our cosmos. But these discoveries only reveal themselves through small effects in noisy experimental data. Dealing with such observations requires the careful application of probability and statistics.
But it is not only in the arcane world of fundamental physics that probability theory plays such an important role. It has an impact in many aspects of our everyday life, from the law courts to the lottery.
Why then do so few people understand probability? And why do so few people understand why it is so important for science? Why do so many people think that science is about absolute certainty when, at its core, it is actually dominated by uncertainty?
This book attempts to explain the basics of probability theory, and illustrate their application across the entire spectrum of science.

The properties of the harmonic oscillator with random frequency or/and random damping formed the content of the first edition. The second edition includes hundreds of publications on this subject since 2005. The noisy oscillator continues to be the subject of intensive studies in physics, chemistry, biology, and social sciences.The new and the latest type of a stochastic oscillator has also been considered, namely, an oscillator with random mass. Such model describes, among other phenomena, Brownian motion with adhesion, where the molecules of the surrounding medium not only randomly collide, but also stick to the Brownian particle for some (random) time, thereby changing its mass. This edition contains two new chapters, eight new sections and an expanded bibliography. A wide group of researchers, students and teachers will benefit from this book.

This volume presents a collection of original and peer-reviewed articles related with the applications of Statistical Physics dedicated to Professor Dr Leopoldo Garcia-Colin, in commemoration of his 80th birthday in 2010. Professor Garcia-Colin has worked in many different fields of statistical physics, and has applied it to biological physics, solid state physics, relativity and cosmology. These are pioneering works of Prof Garcia-Colin involved in all various fields which have their roots in Mexico. His influence is found in each of these works that cover a wide range of topics including thermodynamics, statistical mechanics and kinetic theory applied to biological systems, cosmology and condensed matter, among others.Papers contributed by important experts in the field, such as J Lebowitz, as well as the latest classical applications of statistical physics can be found in this volume.

This book describes the theory of how processes on the unobservable molecular scale give rise to observable effects such as diffusion and electrical noise on the macroscopic or laboratory scale. It puts the modern theory into historical context, and features new applications, statistical mechanics derivations, and the mathematical background of the topic.

This modern and self-contained book offers a clear and accessible introduction to the important topic of machine learning with neural networks. In addition to describing the mathematical principles of the topic, and its historical evolution, strong connections are drawn with underlying methods from statistical physics and current applications within science and engineering. Closely based around a well-established undergraduate course, this pedagogical text provides a solid understanding of the key aspects of modern machine learning with artificial neural networks, for students in physics, mathematics, and engineering. Numerous exercises expand and reinforce key concepts within the book and allow students to hone their programming skills. Frequent references to current research develop a detailed perspective on the state-of-the-art in machine learning research.

This book is designed for use in an introductory course in thermodynamics. It is aimed at students of Physics, Chemistry, Materials Science, and Engineering. As an undergraduate text, it gives a clear description of the theoretical framework of thermodynamics, while providing specific examples of its use in a wide variety of problems. These examples include topics that are atypical of undergraduate texts, such as biological systems, atmospheric phenomena, and polymers.

The narrative is infused with historical notes on the characters who make up the story of thermodynamics, enlivening the material while keeping the reader engaged.

This book reviews one-dimensional reactions, dynamics, diffusion and adsorption. In studies of complex systems in biology, chemistry and physics, understanding can be gained by analytical and numerical analyses of simple models. This book presents review articles at an advanced research level, describing results for one-dimensional models of dynamical processes such as chemical reactions and catalysis, kinetic ising models, phase separation and cluster growth, monolayer and multilayer adsorption with added relaxation, surface and hard-core particle dynamics, diffusional transport, random systems and experimental results. It also covers experimental results for systems ranging from chemical reactions to adsorption and reactions on polymer chains, steps on crystalline surfaces, and DNA. All chapters are written by leading scientists in the field. They present a self-contained review of this subject that will guide a reader from basic concepts, ideas, methods and models to the forefront of research.

* Explores the exciting and new topic of econophysics * Multidisciplinary approach, that will be of interest to students and researchers from physics, engineering, mathematics, statistics, and other physical sciences * Useful to both students and researchers

This book provides a comprehensive introduction to the theory of phase transitions and critical phenomena. The content covers a period of more than 100 years of theoretical research of condensed matter phases and phase transitions providing a clear interrelationship with experimental problems. It starts from certain basic University knowledge of thermodynamics, statistical physics and quantum mechanics. The text is illustrated with classic examples of phase transitions. Various types of phase transition and (multi)critical points are introduced and explained. The classic aspects of the theory are naturally related with the modern developments. This interrelationship and the field-theoretical renormalization group method are presented in details. The main applications of the renormalization group methods are presented. Special attention is paid to the description of quantum phase transitions. This edition contains a more detailed presentation of the renormalization group method and its applications to particular systems.

Approach your problems from the right end It isn't that they can't see the solution. It is and begin with the answers. Then one day, that they can't see the problem. perhaps you will find the final question. G. K. Chesterton. The Scandal of Father 'The Hermit Clad in Crane Feathers' in R. Brown 'The point of a Pin'. van Gulik's The Chif1ese Maze Murders. Growing specialization and diversification have brought a host of monographs and textbooks on increasingly specialized topics. However, the "tree" of knowledge of mathematics and related fields does not grow only by putting forth new branches. It also happens, quite often in fact, that branches which were thought to be completely disparate are suddenly seen to be related. Further, the kind and level of sophistication of mathematics applied in various sciences has changed drastically in recent years: measure theory is used (non trivially) in regional and theoretical economics; algebraic geometry interacts with physics; the Minkowsky lemma, coding theory and the structure of water meet one another in packing and covering theory; quantum fields, crystal defects and mathematical programming profit from homotopy theory; Lie algebras are relevant to filtering; and prediction and electrical engineering can use Stein spaces. And in addition to this there are such new emerging subdisciplines as "experimental mathematics," "CFD," "completely integrable systems," "chaos, synergetics and large-scale order," which are almost impossible to fit into the existing classification schemes. They draw upon widely different sections of mathematics."

The spectacular success of the scientific enterprise over the last four hundred years has led to the promise of an all encompassing vision of the natural world. In this elegant picture, everything we observe is based upon just a few fundamental processes and entities. The almost infinite variety and complexity of the world is thus the product of emergence. But the concept of emergence is fraught with controversy and confusion. This book ponders the question of how emergence should be understood within the scientific picture, and whether a complete vision of the world can be attained that includes consciousness.

This book provides a comprehensive introduction to the theory of phase transitions and critical phenomena. The content covers a period of more than 100 years of theoretical research of condensed matter phases and phase transitions providing a clear interrelationship with experimental problems. It starts from certain basic University knowledge of thermodynamics, statistical physics and quantum mechanics. The text is illustrated with classic examples of phase transitions. Various types of phase transition and (multi)critical points are introduced and explained. The classic aspects of the theory are naturally related with the modern developments. This interrelationship and the field-theoretical renormalization group method are presented in details. The main applications of the renormalization group methods are presented. Special attention is paid to the description of quantum phase transitions. This edition contains a more detailed presentation of the renormalization group method and its applications to particular systems.

This book presents quantum kinetic theory in a comprehensive way. The focus is on density operator methods and on non-equilibrium Green functions. The theory allows to rigorously treat nonequilibrium dynamics in quantum many-body systems. Of particular interest are ultrafast processes in plasmas, condensed matter and trapped atoms that are stimulated by rapidly developing experiments with short pulse lasers and free electron lasers. To describe these experiments theoretically, the most powerful approach is given by non-Markovian quantum kinetic equations that are discussed in detail, including computational aspects.

How can one construct dynamical systems obeying the first and second laws of thermodynamics: mean energy is conserved and entropy increases with time? This book answers the question for classical probability (Part I) and quantum probability (Part II). A novel feature is the introduction of heat particles which supply thermal noise and represent the kinetic energy of the molecules. When applied to chemical reactions, the theory leads to the usual nonlinear reaction-diffusion equations as well as modifications of them. These can exhibit oscillations, or can converge to equilibrium. In this second edition, the text is simplified in parts and the bibliography has been expanded. The main difference is the addition of two new chapters; in the first, classical fluid dynamics is introduced. A lattice model is developed, which in the continuum limit gives us the Euler equations. The five Navier???Stokes equations are also presented, modified by a diffusion term in the continuity equation. The second addition is in the last chapter, which now includes estimation theory, both classical and quantum, using information geometry.