Centered around the natural phenomena of relaxations and fluctuations, this monograph provides readers with a solid foundation in the linear and nonlinear Fokker-Planck equations that describe the evolution of distribution functions.

It emphasizes principles and notions of the theory (e.g. self-organization, stochastic feedback, free energy, and Markov processes), while also illustrating the wide applicability (e.g. collective behavior, multistability, front dynamics, and quantum particle distribution).

The focus is on relaxation processes in homogeneous many-body systems describable by nonlinear Fokker-Planck equations. Also treated are Langevin equations and correlation functions.

Since these phenomena are exhibited by a diverse spectrum of systems, examples and applications span the fields of physics, biology and neurophysics, mathematics, psychology, and biomechanics.

This book provides a summary of the research conducted at UCLA, Stanford University, and UCSD over the last ?ve years in the area of nonlinear dyn- ics and chaos as applied to digital communications. At ?rst blush, the term "chaotic communications" seems like an oxymoron; how could something as precise and deterministic as digital communications be chaotic? But as this book will demonstrate, the application of chaos and nonlinear dynamicstocommunicationsprovidesmanypromisingnewdirectionsinareas of coding, nonlinear optical communications, and ultra-wideband commu- cations. The eleven chapters of the book summarize many of the promising new approaches that have been developed, and point the way to new research directions in this ?eld. Digital communications techniques have been continuously developed and re?ned for the past ?fty years to the point where today they form the heart of a multi-hundred billion dollar per year industry employing hundreds of thousands of people on a worldwide basis. There is a continuing need for transmission and reception of digital signals at higher and higher data rates. There are a variety of physical limits that place an upper limit on these data rates, and so the question naturally arises: are there alternative communi- tion techniques that can overcome some of these limitations? Most digital communications today is carried out using electronic devices that are essentially "linear," and linear system theory has been used to c- tinually re?ne their performance. In many cases, inherently nonlinear devices are linearized in order to achieve a certain level of linear system performance.

This work systematically investigates a large number of oscillatory network configurations that are able to describe many real systems such as electric power grids, lasers or even the heart muscle, to name but a few. The book is conceived as an introduction to the field for graduate students in physics and applied mathematics as well as being a compendium for researchers from any field of application interested in quantitative models.

Until the 1980s, a tacit agreement among many physical oceanographers was that nothing deserving attention could be found in the upper few meters of the ocean. The lack of adequete knowledge about the near-surface layer of the ocean was mainly due to the fact that the widely used oceanographic instruments (such as bathythermographs, CTDs, current meters, etc.) were practically useless in the upper few meters of the ocean. Interest in the ne- surface layer of the ocean rapidly increased along with the development of remote sensing techniques. The interpretation of ocean surface signals sensed from satellites demanded thorough knowledge of upper ocean processes and their connection to the ocean interior. Despite its accessibility to the investigator, the near-surface layer of the ocean is not a simple subject of experimental study. Random, sometimes huge, vertical motions of the ocean surface due to surface waves are a serious complication for collecting quality data close to the ocean surface. The supposedly minor problem of avoiding disturbances from ships' wakes has frustrated several generations of oceanographers attempting to take reliable data from the upper few meters of the ocean. Important practical applications nevertheless demanded action, and as a result several pioneering works in the 1970s and 1980s laid the foundation for the new subject of oceanography - the near-surface layer of the ocean.

Clouds affect the climate of the Earth, and they are an important factor in the weather. Therefore, their radiative properties must be understood in great detail. This book summarizes current knowledge on cloud optical properties, for example their ability to absorb, transmit, and reflect light, which depends on the clouds geometrical and microphysical characteristics such as sizes of droplets and crystals, their shapes, and structures. In addition, problems related to the image transfer through clouds and cloud remote sensing are addressed in this book in great detail.

This book can serve as a major introductory text in cloud optics for students; it can also be an important source of information on theoretical cloud optics for cloud physicists, meteorologists and optical engineers.

All basic ideas of optics as related to scattering of light in clouds (e.g. Mie theory and radiative transfer) are considered in a self consistent way. Consequently, the book can also be a useful textbook to newcomers to the field."

This text maps out the modern theory of non-linear oscillations. The material is presented in a non-traditional manner and emphasises the new results of the theory - obtained partially by the author, who is one of the leading experts in the area. Among the topics are: synchronization and chaotization of self-oscillatory systems and the influence of weak random vibration on modification of characteristics and behaviour of the non-linear systems.

This book on "Polymer Fracture" might as well have been called "Kinetic Theory of Polymer Fracture." The term "kinetic theory," however, needs some de finition or, at least, some explanation. A kinetic theory deals with and particu larly considers the effect of the existence and discrete size, of the motion and of the physical properties of molecules on the macroscopic behavior of an ensemble, gaseous or other. A kinetic theory of strength does have to consider additional aspects such as elastic and anelastic deformations, chemical and physical reactions, and the sequence and distribution of different disintegration steps. In the last fifteen years considerable progress has been made in the latter do mains. The deformation and rupture of molecular chains, crystals, and morphologi cal structures have been intensively investigated. The understanding of the effect of those processes on the strength of polymeric materials has especially been furthered by the development and application of spectroscopical methods (ESR, IR) and of the tools offracture mechanics. It is the aim of this book to relate the conventional and successful statistical, parametrical, and continuum mechanical treatment of fracture phenomena to new results on the behavior of highly stressed molecular chains."

The methods of statistical physics have become increasingly important in recent years for the treatment of a variety of diverse physical problems. Of principal interest is the microscopic description of the dynamics of dissipative systems. Although a unified theoretical description has at present not yet been achieved, we have assumed the task of writing a textbook which summarizes those of the most important methods which are self-contained and complete in themselves. We cannot, of course, claim to have treated the field exhaustively. A microscopic description of physical phenomena must necessarily be based upon quantum theory, and we have therefore carried out the treatment of dynamic processes strictly within a quantum-theoretical framework. For this reason alone it was necessary to omit a number of extremely important theories which have up to now been formulated only in terms of classical statistics. The goal of this book is, on the one hand, to give an introduction to the general principles of the quantum statistics of dynamical processes, and, on the other, to provide readers who are interested in the treatment of particular phenomena with methods for solving specific problems. The theory is for the most part formulated within the calculational frame work of Liouville space, which, together with projector formalism, has become an expedient mathematical tool in statistical physics."

This work arises from our teaching this subject during many years. The vast majority of these exercises are the exams we gave to our students in this period. We carefully selected the subjects of the exercises to cover all the material which is most needed and which is treated in the most well known texts on these subjects. Each exercise is carefully solved in full details, explaining the theory behind the solution with particular care for those issues that, from our experience, are found most difficult from the average student. Indeed, several exercises are designed to throw light on aspects of the theory that, for one reason or another, are usually neglected with the result to make the students feel uneasy about them. In fact most students get acquainted just with the more common manipulations, which are illustrated by many examples in textbooks. Our exercises never require extensive calculations but tend to be somewhat unusual and force the solver to think about the problem starting from the principles, rather than by analogy with some previously solved exercise."

This volume contains the papers presented at the IUTAM Symposium on Geometry and Statistics of Turbulence, held in November 1999, at the Shonan International Village Center, Hayama (Kanagawa-ken), Japan. The Symposium was proposed in 1996, aiming at organizing concen trated discussions on current understanding of fluid turbulence with empha sis on the statistics and the underlying geometric structures. The decision of the General Assembly of International Union of Theoretical and Applied Mechanics (IUTAM) to accept the proposal was greeted with enthusiasm. Turbulence is often characterized as having the properties of mixing, inter mittency, non-Gaussian statistics, and so on. Interest is growing recently in how these properties are related to formation and evolution of struc tures. Note that the intermittency is meant for passive scalars as well as for turbulence velocity or rate of dissipation. There were eighty-eight participants in the Symposium. They came from thirteen countries, and fifty-seven papers were presented. The presenta tions comprised a wide variety of fundamental subjects of mathematics, statistical analyses, physical models as well as engineering applications. Among the subjects discussed are (a) Degree of self-similarity in cascade, (b) Fine-scale structures and degree of Markovian property in turbulence, (c) Dynamics of vorticity and rates of strain, (d) Statistics associated with vortex structures, (e) Topology, structures and statistics of passive scalar advection, (f) Partial differential equations governing PDFs of velocity in crements, (g) Thermal turbulences, (h) Channel and pipe flow turbulences, and others."

One of the great intellectual challenges for the next few decades is the question of brain organization. What is the basic mechanism for storage of memory? What are the processes that serve as the interphase between the basically chemical processes of the body and the very specific and nonstatistical operations in the brain? Above all, how is concept formation achieved in the human brain? I wonder whether the spirit of the physics that will be involved in these studies will not be akin to that which moved the founders of the "rational foundation of thermodynamics". C. N. Yang! 10 The human brain is said to have roughly 10 neurons connected through about 14 10 synapses. Each neuron is itself a complex device which compares and integrates incoming electrical signals and relays a nonlinear response to other neurons. The brain certainly exceeds in complexity any system which physicists have studied in the past. Nevertheless, there do exist many analogies of the brain to simpler physical systems. We have witnessed during the last decade some surprising contributions of physics to the study of the brain. The most significant parallel between biological brains and many physical systems is that both are made of many tightly interacting components.

This book presents a modern and self-contained treatment of the Liapunov method for stability analysis, in the framework of mathematical nonlinear control theory. A Particular focus is on the problem of the existence of Liapunov functions (converse Liapunov theorems) and their regularity, whose interest is especially motivated by applications to automatic control. Many recent results in this area have been collected and presented in a systematic way. Some of them are given in extended, unified versions and with new, simpler proofs. In the 2nd edition of this successful book several new sections were added and old sections have been improved, e.g., about the Zubovs method, Liapunov functions for discontinuous systems and cascaded systems. Many new examples, explanations and figures were added making this book accessible and well readable for engineers as well as mathematicians.

Product design is characterized by a steady increase in complexity. The main focus of this book is a structural approach on complexity management. This means, system structures are considered in order to address the challenge of complexity in all aspects of product design. Structures arise from the complex dependencies of system elements. Thus, the identification of system structures provides access to the understanding of system behavior in practical applications. The book presents a methodology that enables the analysis, control and optimization of complex structures, and the applicability of domain-spanning problems. The methodology allows significant improvements on handling system complexity by creating improved system understanding on the one hand and optimizing product design that is robust for system adaptations on the other hand. Developers can thereby enhance project coordination and improve communication between team members and as a result shorten development time. The practical application of the methodology is described by means of two detailed examples.

Besides turbulence, there is hardly any other scientific topic which has been considered a prominent scientific challenge for such a long time. The special interest in turbulence is not only based on it being a difficult scientific problem but also on its meaning in the technical world and our daily life. This carefully edited book comprises recent basic research as well as research related to the applications of turbulence. Therefore, both leading engineers and physicists working in the field of turbulence were invited to the iTi Conference on Turbulence held in Bad Zwischenahn, Gemany 21st - 24th of September 2003. Topics discussed include, for example, scaling laws and intermittency, thermal convection, boundary layers at large Reynolds numbers, isotropic turbulence, stochastic processes, passive and active scalars, coherent structures, numerical simulations, and related subjects.

Mobile robotics has until now focused on issues like design of controllers and robot hardware. It is now ready to embrace theoretical methods from dynamical systems theory, statistics and system identification to produce a formalized approach based on quantitative analyses and computer models of the interaction between robot, task and environment.

This book is a step towards a theoretical understanding of the operation of autonomous mobile robots. It presents cutting-edge research on the application of chaos theory, parametric and non-parametric statistics and dynamical systems theory in this field. Practical examples and case studies show how robot behaviour can be logged, analysed, interpreted and modelled, aiding design of controllers, analysis of agent behaviour and verification of results.

As the first book to apply advanced scientific methods to mobile robots it will interest researchers, lecturers and post-graduate students in robotics, artificial intelligence and cognitive science.

This concise but comprehensive textbook reviews the most popular neural-network methods and their associated techniques. Each chapter provides state-of-the-art descriptions of important major research results of the respective neural-network methods. A range of relevant computational intelligence topics, such as fuzzy logic and evolutionary algorithms - powerful tools for neural-network learning - are introduced. The systematic survey of neural-network models and exhaustive references list will point readers toward topics for future research. The algorithms outlined also make this textbook a valuable reference for scientists and practitioners working in pattern recognition, signal processing, speech and image processing, data analysis and artificial intelligence.

Hydrogen can behave as an alkaline metal or a halogen and can react with nearly all elements of the periodic table. This explains the large number of metal hydrides. Since T. Graham's first observation of the absorption of hydrogen in palladium in 1866 the behaviour of hydrogen in metals has been studied very extensively. The interest was motivated by the possible application of metal-hydrogen systems in new technologies (e.g., moderator material in nuclear fission reactors, reversible storage material for thermal energy and large amounts of hydrogen) and by the fact that metal hydrides show very exciting physical properties (e.g., superconductivity, quantum diffusion, order-disorder transitions, phase diagrams, etc.). Many of these properties have been determined for the stable hydrogen isotopes Hand D in various metals. In comparison, very little is known about the behaviour of the ra dioactive isotope tritium in metals. This book is a first attempt to summarize part of the knowledge of tritium gained in the last few years. In addition to the task of presenting the properties of tritium in metals, I have tried to compare these data with those of protium and deuterium. Furthermore, helium-3 is connected inse parably with tritium via the tritium decay. Therefore one chapter of this book is solely devoted to the curious properties of helium in metals caused mainly by its negligible solubility."

This book was originally conceived as a continuation in theme of the collec- tive monograph Limits of Predictability (Yu. A. Kravtsov, Ed. , Springer Series in Synergetics, Vol. 60, Springer-Verlag, Heidelberg, 1993). The main thrust of that book was to examine the various effects and factors (system non- stationarity, measurement noise, predictive model accuracy, and so on) that may limit, in a fundamental fashion, our ability to mathematically predict physical and man-made phenomena and events. Particularly interesting was the diversity of fields from which the papers and examples were drawn, in- cluding climatology, physics, biophysics, cybernetics, synergetics, sociology, and ethnogenesis. Twelve prominant Russian scientists, and one American (Prof. A. J. Lichtman) discussed their philosophical and scientific standpoints on the problem of the limits of predictability in their various fields. During the preparation of that book, the editor (Yu. A. K) had the great pleasure of interacting with world-renowned Russian scientists such as oceanologist A. S. Monin, geophysicist V. I. Keilis-Borok, sociologist I. V. Bestuzhev-Lada, histo- rian L. N. Gumilev, to name a few. Dr. Angela M. Lahee, managing editor of the Synergetics Series at Springer, was enormously helpful in the publishing of that book. In 1992, Prof. H. Haken along with Dr. Lahee kindly supported the idea of publishing a second volume on the theme of nonlinear system predictability, this time with a more international flavor.

Foundational issues in statistical mechanics and the more general question of how probability is to be understood in the context of physical theories are both areas that have been neglected by philosophers of physics. This book fills an important gap in the literature by providing a most systematic study of how to interpret probabilistic assertions in the context of statistical mechanics. The book explores both subjectivist and objectivist accounts of probability, and takes full measure of work in the foundations of probability theory, in statistical mechanics, and in mathematical theory. It will be of particular interest to philosophers of science, physicists and mathematicians interested in foundational issues, and also to historians of science.

The Compendium of Theoretical Physics contains the canonical curriculum of theoretical physics. From classical mechanics over electrodynamics, quantum mechanics and statistical physics/thermodynamics, all topics are treated axiomatic-deductively and confimed by exercises, solutions and short summaries.

A recent paper on subfactors of von Neumann factors has stimulated much research in von Neumann algebras. It was discovered soon after the appearance of this paper that certain algebras which are used there for the analysis of subfactors could also be used to define a new polynomial invariant for links. Recent efforts to understand the fundamental nature of the new link invariants has led to connections with invariant theory, statistical mechanics and quantum theory. In turn, the link invariants, the notion of a quantum group, and the quantum Yang-Baxter equation have had a great impact on the study of subfactors. Our subject is certain algebraic and von Neumann algebraic topics closely related to the original paper. However, in order to promote, in a modest way, the contact between diverse fields of mathematics, we have tried to make this work accessible to the broadest audience. Consequently, this book contains much elementary expository material.

This volume contains contributions based on the lectures delivered at the Third In ternational Workshop on Nonlinear Dynamics and Quantum Phenomena in Opti cal Systems, which was held in Blanes, Girona, Spain, 1-3 October 1990. Blanes is a charming small town located on the well-known Costa Brava. With the con venient facilities of the Centre for Advanced Studies (CEAB), Blanes followed of the previous meetings of this series, which were held at Palma the tradition de Mallorca and Santander. We aimed to provide an opportunity for scientists active in the broad field of quantum optics to meet in an informal atmosphere, thus promoting the ex change of ideas and allowing a search for interconnections between seemingly unrelated topics and a cross fertilization of the different subfields of quantum optics. We encouraged contributions dealing with the newest and most important developments in quantum optics. The main topics included instabilities, chaos, spatiotemporal dynamics, and pattern formation in lasers and nonlinear optical devices; phase dynamics; generation and detection of squeezed and other non classical states of light; coherent interaction of light with atomic systems; and multiphoton processes and above-threshold ionization. The meeting brought together a group of 72 optical scientists from Belgium, France, Germany, Israel, Italy, Spain, the United Kingdom, USA, and USSR. In the technical program, nine invited papers were included, presented by eight distinguished specialists."

Open nonlinear systems are capable of self-organization in space and time. This realization constitutes a major breakthrough of modern science, and is currently at the origin of explosive developments in chemistry, physics and biology. Observations and numerical computations of nonlinear systems surprise us by their inexhaustible and sometimes nonintuitive variety of structures with different shapes and functions. But as well as variety one finds on closer inspection that nonlinear phenomena share universal aspects of pattern formation in time and space. These similarities make it possible to bridge the gap between inanimate and living matter at various levels of complexity, in both theory and experiment. This book is an account of different approaches to the study of this pattern formation. The universality of kinetic, thermodynamic and dimensional approaches is documented through their application to purely mathematical, physical and chemical systems, as well as to systems in nature: biochemical, cellular, multicellular, physiological, neurophysiological, ecological and economic systems. Hints given throughout the book allow the reader to discover how to make use of the principles and methods in different fields of research, including those not treated explicitly in the book.

th This volume contains the proceedings of the X Congress of the Interna- tional Association of Mathematical Physics, held at the University of Leipzig from 30 July until 9 August 1991. There were more than 400 participants, from 29 countries, making it a truly international gathering. The congress had the support of the Deutsche Forschungsgemeinschaft, the European Economic Community, the International Association of Math- ematical Physics, the International Mathematical Union and the Interna- tional Union of Pure and Applied Physics. There were also sponsors from in- dustry and commerce: ATC Mann, Deutsche Bank AG, Miele & Cie GmbH, NEC Deutschland GmbH, Rank Xerox, Siemens AG and Stiftungsfonds IBM Deutschland. On behalf of the congress participants and the members of the International Association of Mathematical Physics, I would like to thank all these organisations for their very generous support. The congress took place under the auspices of the Ministerp6isident des Freistaates Sachsen, K. Biedenkopf. The conference began with an address by A. Uhlmann, Chairman of the Local Organizing Committee. This was followed by speeches of welcome from F. Magirius, City President of Leipzig; C. Weiss, Rector of the Uni- versity of Leipzig; and A. Jaffe, President of the International Association of Mathematical Physics.

An international workshop on Elementary Excitations and Fluctuations in Magnetic Systems was held in Turin for five days beginning 25 May, 1987. The workshop followed much the same format as the one with the same title held in San Miniato in 1984 (proceedings: Springer Series in-Solid-State Sciences, Vol. 54), that most participants contributed talks and provided papers for the proceedings. While many of the participants had attended the first workshop, 15 of the 40 invited review papers were presented by scientists who had not. The majority of the talks reported theoretical work concerned with the introduction of new techniques. However, experimental work was also well represented, not least because many of the reported theoretical studies were motivated by experimental findings, and a highlight of the workshop was an extremely stimulating session devoted to recent neutron scattering measure- ments, on various systems, that exploited polarization analysis. The fine venue of the workshop, Villa Gualino, with its excellent facili- ties and spacious accommodation, helped to produce a delightful relaxed and friendly atmosphere. For the use of Villa Gualino and significant financial support we are indebted to our host organization, the Institute for Scien- tific Interchange (lSI). Additional financial support came from the Consiglio Nazionale delle Ricerche (CNR), Centro Interuniversitario di Struttura della Material del Ministero della Pubblica Istruzione (CISM-MPI) and Gruppo Nazionale di Struttura della Materia (GNSM-CNR).