This volume contains most of the invited papers presented at the International Work shop on Synergetics, Schloss E1mau, Bavaria, May 2 to.May 7, 1977. This workshop fol lowed an International Symposium on SynergetiGS at Schloss E1mau, 1972, and an Inter national SUl11l1erschoo1 at Erice, Sicily, 1974. Synergetics is a rather new field of interdisciplinary research which studies the self-organized behavior of systems leading to the formation of structures and func tionings. Indeed the whole universe seems to be organized, with pronounced structures starting from spiral galaxies down to living cells. Furthermore, very many of the most interesting phenomena occur in systems which are far from thermal equilibrium. Synergetics in its present form focusses its attention on those phenomena where dra matic changes occur on a macroscopic scale. Here indeed Synergetics was able to re veal profound analogies between systems in different disciplines ranging from physics to sociology. This volume contains contributions from various fields but the reader will easily discover their cOl11J1on goal. Not only in the natural sciences but also in ecology, sociology, and economy, man is confronted with the problems of complex sys tems. The principles and analogies unearthed by Synergetics will certainly be very he1pfu to cope with such difficult problems. I use this opportunity to thank the Vo1kswagenwerk Foundation for its support of the project Synergetics and in particular for sponsoring the International Workshop on Synergetics."

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.

In this monograph, a statistical description of natural phenomena is used to develop an information processing system capable of modeling non-linear relationships between sensory data. The system, based on self-organized, optimal preservation of empirical information, applies these relationships for prediction and adaptive control.
This monograph is written for students, scientists and engineers in academia and industry who are interested in experimental work related to the adaptive modeling of natural laws, the development of sensory-neural networks, intelligent control, synergetics and informatics. No specific knowledge of advanced mathematics is presupposed. Examples taken from physics, engineering, medicine and economics demonstrate the applicability of such intelligent systems.

The basic subjects and main topics covered by this book are: (1) Physics of Black Holes (classical and quantum); (2) Thermodynamics, entropy and internal dynamics; (3) Creation of particles and evaporation; (4) Mini black holes; (5) Quantum mechanics of black holes in curved spacetime; (6) The role of spin and torsion in the black hole physics; (7) Equilibrium geometry and membrane paradigm; (8) Black hole in string and superstring theory; (9) Strings, quantum gravity and black holes; (10) The problem of singularity; (11) Astrophysics of black holes; (12) Observational evidence of black holes. The book reveals the deep connection between gravitational, quantum and statistical physics and also the importance of black hole behaviour in the very early universe. An important new point discussed concerns the introduction of spin in the physics of black holes, showing its central role when correctly put into the Einstein equations through the geometric concept of torsion, with the new concept of a time-temperature uncertainty relation, minimal time, minimal entropy, quantization of entropy and the connection of black hole with wormholes. Besides theoretical aspects, the reader will also find observational evidence for black holes in active galactic nuclei, in binary X-ray sources and in supernova remnants. The book will thus interest physicists, astronomers, and astrophysicists at different levels of their career who specialize in classical properties, quantum processes, statistical thermodynamics, numerical collapse, observational evidence, general relativity and other related problems.

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.

Substances possessing heterogeneous microstructure on the nanometer and micron scales are scientifically fascinating and technologically useful. Examples of such substances include liquid crystals, microemulsions, biological matter, polymer mixtures and composites, vycor glasses, and zeolites. In this volume, an interdisciplinary group of researchers report their developments in this field. Topics include statistical mechanical free energy theories which predict the appearance of various microstructures, the topological and geometrical methods needed for a mathematical description of the subparts and dividing surfaces of heterogeneous materials, and modern computer-aided mathematical models and graphics for effective exposition of the salient features of microstructured materials.

Bibliography . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 325 Critical point dominance in quantum field models. . . . . . . . . . . . . . . . . . . . 326 q>,' quantum field model in the single-phase regions: Differentiability of the mass and bounds on critical exponents. . . . 341 Remark on the existence of q>:. . . * . . . . * . . . . * . . . . . . . . * . * . . . . . . . . . . * . 345 On the approach to the critical point . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 348 Critical exponents and elementary particles. . . . . . . . . . . . . . . . . . . . . . . . . . 362 V Particle Structure Introduction. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 371 Bibliography . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 371 The entropy principle for vertex functions in quantum field models. . . . . 372 Three-particle structure of q>4 interactions and the scaling limit . . . . . . . . . 397 Two and three body equations in quantum field models. . . . . . . . . . . . . . . 409 Particles and scaling for lattice fields and Ising models. . . . . . . . . . . . . . . . 437 The resummation of one particle lines. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 450 VI Bounds on Coupling Constants Introduction. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 479 Bibliography . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 479 Absolute bounds on vertices and couplings. . . . . . . . . . . . . . . . . . . . . . . . . . 480 The coupling constant in a q>4 field theory. . . . . . . . . . . . . . . . . . . . . . . . . . . 491 VII Confinement and Instantons Introduction. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 497 Bibliography . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 497 Instantons in a U(I) lattice gauge theory: A coulomb dipole gas. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 498 Charges, vortices and confinement. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 516 ix VIII Reflection Positivity Introduction. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 531 Bibliography . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 531 A note on reflection positivity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 532 x Introduction This volume contains a selection of expository articles on quantum field theory and statistical mechanics by James Glimm and Arthur Jaffe. They include a solution of the original interacting quantum field equations and a description of the physics which these equations contain. Quantum fields were proposed in the late 1920s as the natural framework which combines quantum theory with relativ ity. They have survived ever since.

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.

Controlling Chaos achieves three goals: the suppression, synchronisation and generation of chaos, each of which is the focus of a separate part of the book. The text deals with the well-known Lorenz, Roessler and Henon attractors and the Chua circuit and with less celebrated novel systems. Modelling of chaos is accomplished using difference equations and ordinary and time-delayed differential equations. The methods directed at controlling chaos benefit from the influence of advanced nonlinear control theory: inverse optimal control is used for stabilization; exact linearization for synchronization; and impulsive control for chaotification. Notably, a fusion of chaos and fuzzy systems theories is employed. Time-delayed systems are also studied. The results presented are general for a broad class of chaotic systems. This monograph is self-contained with introductory material providing a review of the history of chaos control and the necessary mathematical preliminaries for working with dynamical systems.

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.

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 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."

This book contains the lectures given at the Conference on Dynamics and Randomness held at the Centro de Modelamiento Matematico of the Universidad de Chile from December 11th to 15th, 2000. This meeting brought together mathematicians, theoretical physicists and theoretical computer scientists, and graduate students interested in fields re lated to probability theory, ergodic theory, symbolic and topological dynam ics. We would like to express our gratitude to all the participants of the con ference and to the people who contributed to its organization. In particular, to Pierre Collet, Bernard Host and Mike Keane for their scientific advise. VVe want to thank especially the authors of each chapter for their well prepared manuscripts and the stimulating conferences they gave at Santiago. We are also indebted to our sponsors and supporting institutions, whose interest and help was essential to organize this meeting: ECOS-CONICYT, FONDAP Program in Applied Mathematics, French Cooperation, Fundacion Andes, Presidential Fellowship and Universidad de Chile. We are grateful to Ms. Gladys Cavallone for their excellent work during the preparation of the meeting as well as for the considerable task of unifying the typography of the different chapters of this book.

In this volume we continue the logical development of the work begun in Volume I, and the equilibrium theory now becomes a very special case of the exposition presented here. Once a departure is made from equilibrium, however, the problems become deeper and more subtle-and unlike the equilibrium theory, many aspects of nonequilibrium phenomena remain poorly understood. For over a century a great deal of effort has been expended on the attempt to develop a comprehensive and sensible description of nonequilibrium phenomena and irreversible processes. What has emerged is a hodgepodge of ad hoc constructs that do little to provide either a firm foundation, or a systematic means for proceeding to higher levels of understanding with respect to ever more complicated examples of nonequilibria. Although one should rightfully consider this situation shameful, the amount of effort invested testifies to the degree of difficulty of the problems. In Volume I it was emphasized strongly that the traditional exposition of equilibrium theory lacked a certain cogency which tended to impede progress with extending those considerations to more complex nonequilibrium problems. The reasons for this were adduced to be an unfortunate reliance on ergodicity and the notions of kinetic theory, but in the long run little harm was done regarding the treatment of equilibrium problems. On the nonequilibrium level the potential for disaster increases enormously, as becomes evident already in Chapter 1.

7 Les Houches Number theory, or arithmetic, sometimes referred to as the queen of mathematics, is often considered as the purest branch of mathematics. It also has the false repu tation of being without any application to other areas of knowledge. Nevertheless, throughout their history, physical and natural sciences have experienced numerous unexpected relationships to number theory. The book entitled Number Theory in Science and Communication, by M.R. Schroeder (Springer Series in Information Sciences, Vol. 7, 1984) provides plenty of examples of cross-fertilization between number theory and a large variety of scientific topics. The most recent developments of theoretical physics have involved more and more questions related to number theory, and in an increasingly direct way. This new trend is especially visible in two broad families of physical problems. The first class, dynamical systems and quasiperiodicity, includes classical and quantum chaos, the stability of orbits in dynamical systems, K.A.M. theory, and problems with "small denominators", as well as the study of incommensurate structures, aperiodic tilings, and quasicrystals. The second class, which includes the string theory of fundamental interactions, completely integrable models, and conformally invariant two-dimensional field theories, seems to involve modular forms and p adic numbers in a remarkable way.

Correlation Effects in Low-Dimensional Electron Systems describes recent developments in theoretical condensed-matter physics, emphasizing exact solutions in one dimension including conformal-field theoretical approaches, the application of quantum groups, and numerical diagonalization techniques. Various key properties are presented for two-dimensional, highly correlated electron systems.

This text takes readers in a clear and progressive format from simple to recent and advanced topics in pure and applied probability such as contraction and annealed properties of non-linear semi-groups, functional entropy inequalities, empirical process convergence, increasing propagations of chaos, central limit, and Berry Esseen type theorems as well as large deviation principles for strong topologies on path-distribution spaces. Topics also include a body of powerful branching and interacting particle methods.

One service mathematics has rendered the 'Et BIOi. .... si j'avait su comment en revenir. human race. It has put common sense back je n'y serais point aile.' Jules Verne where it belongs. on the topmost shelf next to the dusty canister labelled 'discarded non The series is divergent; therefore we may be sense'. able to do something with it. Eric T. Bell O. Heaviside Math@matics is a tool for thought. A highly necessary tool in a world where both feedback and non Iinearities abound. Similarly, all kinds of parts of mathematics serve as tools for other parts and for other sciences. Applying a simple rewriting rule to the quote on the right above one finds such statements as: 'One service topology has rendered mathematical physics .. .'; 'One service logic has rendered com puter science .. .'; 'One service category theory has rendered mathematics .. .'. All arguably true. And all statements obtainable this way form part of the raison d'etre of this series."

Once upon a time, science was not divided into disciplines as we know it today. There was no distinction between so-called social and natural sciences, not to mention the fragmentation of the latter into physics, chemistry, biology, geology, etc. According to legend, the scientists those days would do their research in whatever environment they happened to find comfortable, which more often than not was in bathtubs or giant hot tubs - remember Archimedes! Then, somehow, these days we find ourselves compartmentalized into different departments in our universities, or divisions in our research institutes. (We suspect, for one thing, that is to ensure that we will get our paychecks delivered on time at the end of each month. ) Anyway, as anyone who has worked in the real world knows: when one is confronted with a completely new problem or phenomenon, it is usually impossible to neatly assign the problem to physics, chemistry, or, for that matter, computer science. One needs to recall and fuse together the knowledge one learned before and, if that alone is insufficient, to consult experts in other areas. This points to the shortcomings of the compartmentalization of knowledge in our educational systems. In recent years, something has changed. Under the banner of Complex Systems, some brave souls are not afraid to tackle problems that are considered intractable by others, and dare to venture out of their trained disciplines or departments to which they are attached.

Recent years have seen a growing interest in and activity at the interface between physics and biology, with the realization that both subjects have a great deal to learn from and to teach to one another. A particularly promising aspect of this interface concerns the area of cooperative phenomena and phase transitions. The present book addresses both the structure and motion of biological materials and the increasingly complex behaviour that arises out of interactions in large systems, giving rise to self organization, adaptation, selection and evolution: concepts of interest not only to biology and living systems but also within condensed matter physics. The approach adopted by Physics of Biomaterials: Fluctuations, Self Assembly and Evolution is tutorial, but the book is fully up to date with the latest research. Written at a level appropriate to graduate researchers, preferably with a background either in condensed matter physics or theoretical or physically-oriented experimental biology.

What is thermodynamics? What does statistical physics teach us? In the pages of this slim book, we confront the answers. The reader will discover that where thermodynami cs provi des a 1 arge scal e, macroscopi c theory of the ef fects of temperature on physical systems, statistical mechanics provides the microscopic analysis of these effects which, invariably, are the results of thermal disorder. A number of systems in nature undergo dramatic changes in aspect and in their properties when subjected to changes in ambient temperature or pres sure, or when electric or magnetic fields are applied. The ancients already knew that a liquid, a solid, or a gas can represent different states of the same matter. But what is meant by "state"? It is here that the systematic study of magnetic materials has provided one of the best ways of examining this question, which is one of the principal concerns of statistical physics (alias "statistical mechanics") and of modern thermodynamics."

The concept of this book was developed during the Winter Seminar held in the Austrian mountains at the Alpengasthof Zeinisjoch, Tirol-Vorarlberg, from February 27 to March 3, 1988. Leading experts and advanced students in math ematics, physics, chemistry and computer science met to present and discuss their most recent results in an informal seminar. These were the circumstances that led to the idea of compiling some of the essential contributions presented at this seminar together with others describing basic features of "optimal struc tures in heterogeneous reaction systems". The aim of this book is to present the scientific results of the intensive work carried out in each of the specific fields of research. Each contribution therefore presents the current state of the art together with a deeper treatment enabling a more comprehensive understanding of that particular field of work. The common ideas which unite all the different contributions are already ex pressed in the title of this book. The nature of heterogeneous reaction systems is quite varied. An example is provided by the chemical systems such as noble metal particles which may act as heterogeneous catalysts for gaseous chemical compounds. Under these circumstances the metal particles and/or their sur faces may undergo phase transitions during reaction. Imbihl and Plath report on special catalytic systems of this kind, which are of industrial importance.

Many novel cooperative phenomena found in a variety of systems studied by scientists can be treated using the uniting principles of synergetics. Examples are frustrated and random systems, polymers, spin glasses, neural networks, chemical and biological systems, and fluids. In this book attention is focused on two main problems. First, how local, topological constraints (frustrations) can cause macroscopic cooperative behavior: related ideas initially developed for spin glasses are shown to play key roles also for optimization and the modeling of neural networks. Second, the dynamical constraints that arise from the nonlinear dynamics of the systems: the discussion covers turbulence in fluids, pattern formation, and conventional 1/f noise. The volume will be of interest to anyone wishing to understand the current development of work on complex systems, which is presently one of the most challenging subjects in statistical and condensed matter physics.

WAVE TURBULENCE is a state of a system of many simultaneously excited and interacting waves characterized by an energy distribution which is not in any sense close to thermodynamic equilibrium. Such situations in a choppy sea, in a hot plasma, in dielectrics under arise, for example, a powerful laser beam, in magnets placed in a strong microwave field, etc. Among the great variety of physical situations in which wave turbulence arises, it is possible to select two large limiting groups which allow a detailed analysis. The first is fully developed wave turbulence arising when energy pumping and dissipation have essentially different space scales. In this case there is a wide power spectrum of turbulence. This type of turbulence is described in detail e. g. in Zakharov et al. 1 In the second limiting case the scales in which energy pumping and dissipation occur are the same. As a rule, in this case a narrow, almost singular spectrum of turbulence appears which is concentrated near surfaces, curves or even points in k-space. One of the most important, widely investigated and instructive examples of this kind of turbulence is parametric wave turbulence appearing as a result of the evolution of a parametric instability of waves in media under strong external periodic modulation (laser beam, microwave electromagnetic field, etc. ). The present book deals with parametric wave turbulence.

This book offers a discussion of Niels Bohr's conception of "complementarity," arguably his greatest contribution to physics and philosophy. By tracing Bohr's work from his 1913 atomic theory to the introduction and then refinement of the idea of complementarity, and by explicating different meanings of "complementarity" in Bohr and the relationships between it and Bohr's other concepts, the book aims to offer a contained and accessible, and yet sufficiently comprehensive account of Bohr's work on complementarity and its significance.