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.

The Third European Turbulence Conference was held at the Royal Institute of Technology, Sweden, from the 3rd to the 6th of July 1990 under the auspices of the European Mechanics Committee. This series of conferences is primarily devoted to fundamental aspects of turbulence and aimed at bringing together engineers, physicists and mathematicians. The scientific committee - serving also as the European Turbulence Con ference Committee - consisted of the following members: G. Comte-Bellot (Lyon), H.H. Fernholz (Berlin), H. Fiedler (Berlin), U. Frisch (Nice), J.c.R. Hunt (Cambridge), J. Jimenez (Madrid), A.V. Johansson (Stockholm), E. Krause (Aachen), G. Ooms (Amsterdam), and 1. Rhyming (Lausanne). The local organizing committee consisted of A.V. Johansson and P.H. Alfredsson. The conference programme comprised six invited lectures, given by D. Bechert, Y. Gagne, J. Jimenez, M.T. Landahl, C.H. Priddin and G.M. Za slavsky, and 96 contributions given as either oral or poster presentations.There were 172 participants from 17 countries. The members of this scientific com mittee acted as referees for the selection and scrutiny of the papers published in these proceedings. The main topics of this meeting were turbulence structure; transition and dynamical systems; turbulent combustion and mixing; turbulence affected by body forces; turbulence modelling; drag reduction and turbulence control; and novel experimental techniques. During the first evening of the conference the new MTL low-speed, low-turbulence wind-tunnel at KTH was inaugurated. The wind-tunnel outline, also acting as decoration for the "official conference cakes," is shown below."

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."

The aim of this book is to give a unified and critical account of the fundamental aspects of liquid crystals. Preference is given to discussing the assumptions made in developing theories and analyzing experimental data rather than to attempting to compile all the latest results. The book has four parts. Part I is quite descriptive in character and gives a general overview of the various liquid crystalline phases. Part II deals with the macroscopic continuum theory of liquid crystals and gives a systematic development of the theory from a tensorial point of view thus emphasizing the relevant symmetries. Part III concentrates on experiments that provide microscopic information on the orientational behaviour of the molecules. Finally Part IV discusses the theory of the various phases and their attendant phase transitions from both a Landau and a molecular-statistical point of view. Simplifying the various models as far as possible, it critically examines the merits of a molecular-statistical approach.

Macroscopic physics provides us with a great variety of pattern-forming systems displaying propagation phenomena, from reactive fronts in combustion, to wavy structures in convection and to shear flow instabilities in hydrodynamics. These proceedings record progress in this rapidly expanding field. The contributions have the following major themes: - The problems of velocity selection and front morphology of propagating interfaces in multiphase media, with emphasis on recent theoretical and experimental results on dendritic crystal growth, Saffman-Taylor fingering, directional solidification and chemical waves. - The "unfolding" of large-scale, low-frequency behavior in weakly confined homogeneous systems driven far from equilibrium, and more specifically, the envelope approach to the mathematical description of textures in different cases: steady cells, propagating waves, structural defects, and phase instabilities. - The implications of the presence of global downstream transport in open flows for the nature, convective or absolute, of shear flow instabilities, with applications to real boundary layer flows or shear layers, as reported in contributions covering experimental situations of fundamental and/or engineering interest.

Large-scale winds and currents tend to balance Coriolis and pressure gradient forces. The time evolution of these winds and currents is the subject of the quasi-geostrophic theory.
Chapter 1 presents concepts and equations of classical inertial fluid mechanics.
Chapter 2 deals with the equations of thermodynamics that close the governing equations of the fluids. Then, the motion is reformulated in a uniformly rotating reference frame.
Chapter 3 deals with the shallow-water model and the homogeneous model of wind-driven circulation. The chapter also describes a classical application of the Ekman layer to the atmosphere.
Chapter 4 considers the two-layer model, as an introduction to baroclinic flows, together with the concept of available potential energy.
Chapter 5 takes into account continuously stratified flows in the ocean and in the atmosphere.

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.

The Symposium on "The Influence of Polymer Additives on Veloc- ity and Temperature Fields" was proposed to the General Assem- bly of the International Union of Theoretical and Applied Me- chanics (IUTAM) by the "Gesellschaft fur Angewandte Mathematik und Mechanik" (GAMM). The Symposium was held under the auspices of IUTAM in association with the "Deutsche Rheologische Ge- sellschaft" (DRG) with responsibility for the organization ly- ing with B. Gampert (Universitat-GH-Essen). The main aim of this IUTAM Symposium was to consider the funda- mental aspects of the phenomena that occur when small amounts of polymers are added to turbulent flows (turbulent drag re- duction) and laminar porous media flows. In particular atten- tion was devoted to - the influence of molecular parameters of the polymers and solution properties, especially the elongational viscosity, on turbulent flow and laminar porous media flow; the influence of polymers on the turbulence structure in polymer drag reduction.

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.

The development of the modern theory of metals and alloys has coincided with great advances in quantum-mechanical many-body theory, in electronic structure calculations, in theories of lattice dynamics and of the configura tional thermodynamics of crystals, in liquid-state theory, and in the theory of phase transformations. For a long time all these different fields expanded quite independently, but now their overlap has become sufficiently large that they are beginning to form the basis of a comprehensive first-principles the ory of the cohesive, structural, and thermodynamical properties of metals and alloys in the crystalline as well as in the liquid state. Today, we can set out from the quantum-mechanical many-body Hamiltonian of the system of electrons and ions, and, following the path laid out by generations of the oreticians, we can progress far enough to calculate a pressure-temperature phase diagram of a metal or a composition-temperature phase diagram of a binary alloy by methods which are essentially rigorous and from first prin ciples. This book was written with the intention of confronting the materials scientist, the metallurgist, the physical chemist, but also the experimen tal and theoretical condensed-matter physicist, with this new and exciting possibility. Of course there are limitations to such a vast undertaking as this. The selection of the theories and techniques to be discussed, as well as the way in which they are presented, are necessarily biased by personal inclination and personal expertise."

This is a unique approach to noise theory and its application to physical measurements that will find its place among the graduate course books. In a very systematic way, the foundations are laid and applied in a way that the book will also be useful to those not focusing on optics. Exercises and solutions help students to deepen their knowledge.

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.

During the last decade there has been a renewed interest in research on supramolecular assemblies in solutions, such as micelles and microemulsions, not only because of their extensive applications in industries dealing with catalysts, detergency, biotechnology, and enhanced oil recovery, but also due to the development of new and more powerful experimental and theoretical tools for probing the microscopic behavior of these systems. Prominent among the array of the newly available experimental techniques are photon correlation spectroscopy, small-angle neutron and X-ray scattering, and neutron spin-echo and nuclear magnetic resonance spectroscopies. On the theoretical side, the traditionally emphasized thermodynamic approach to the study of the phase behavior of self-assembled systems in solutions is gradually being replaced by statistical mechanical studies of semi-micro scopic and microscopic models of the assemblies. Since the statistical mechanical approach demands as its starting point the microscopic struc tural information of the self-assembled system, the experimental determina tion of the structures of micelles and microemulsions becomes of paramount interest. In this regard the scattering techniques mentioned above have played an important role in recent years and will continue to do so in the future. In applying the scattering techniques to the supramolecular species in solution, one cannot often regard the solution to be ideal. This is because the inter-aggregate interaction is often long-ranged since it is coulombic in nature and the interparticle correlations are thus appreciable."

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.

The idea of a workshop devoted to the static and dynamic properties of liquids arose from two smaller meetings on the same topic held at the Interfaculty Reactor Institute, Delft, The Netherlands in 1985 and 1987, and from the LAM conference (International Conference on Liquid and Amorphous Metals) held at Garmisch Partenkirchen, FRG, in 1986. The object of the Workshop was to bring together experimentalists, theoreticians and people working on computer simulations in or der to learn about developments in the microscopic behaviour of liquids. Approx imately 35 scientists from Austria, France, Israel, Italy, Sweden, The Netherlands, the United Kingdom, the Federal Republic of Germany and Yugoslavia gathered at the Hotel "Libertas", Dubrovnik, in Yugoslavia, June 27-30, 1988 to hear ten in vited talks and nineteen contributed papers on a wide range of subjects, including generalized hydrodynamics, mode coupling theory, liquids at surfaces, extraction of effective potentials, and the glass transition. The most numerous were the papers dealing with computer simulation, and then those on supercooled liquids and the glass transition. The number of experimental results was relatively small, which points to difficulties in this kind of research (evaluation of experimental data). The workshop closed with an expanded round-table discussion of the main themes of the meeting, with an emphasis on the likely trend of investigations in the future. This discussion is summarized at the end of this volume.