The core of the material on large scale dynamics of interacting particles grew out of courses I taught at the Katholieke Universiteit Leuven, Rutgers Universi ty, and the Ludwig-Maximilians-Universitat Munchen and out of lectures I gave at the workshop "Hydrodynamical Behavior of Microscopic Systems" at the Universita dell'Aquila. I had the good luck of being helped through difficult ground by many friends. Amongst them I am deeply indebted to Joel L. Lebowitz. He got me started. Relatively little would have been achieved without his never-ending curiosity and insistence on clarity. Furthermore, I gratefully acknowledge the cooperation of Michael Aizenman, Henk van Beijeren, Carlo Boldrighini, Jean Bricmont, Paola Calderoni, Brian Davies, Anna DeMasi, Roland Dobrushin, Detlef Durr, Gregory Eyink, Mark Fannes, Pablo Ferrari, Alberto Frigerio, Joseph Fritz, Antonio Galves, Shelly Goldstein, Vittorio Gorini, Reinhard Illner, Claude Kipnis, Joachim Krug, Oscar Lanford, Reinhard Lang, Joel Lebowitz, Christian Maes, Stefano Olla, George Papanicolaou, Errico Presutti, Mario Pulvirenti, Fraydoun Rezakhanlou, Hermann Rost, Yasha Sinai, Yuri Suhov, Domo Szasz, Ragu Varadhan, Andre Verbeure, David Wick, and Horng-Tzer Yau. The list is somewhat lengthy, perhaps, but besides thanks I want to make clear that what I will describe is the outcome of a common scientific enterprise. I thank Henk van Beijeren and Detlef Durr for careful reading of and com ments on a previous version. Paola Calderoni and Detlef Durr supplied me with the proof in Part I, Chapter 8. 4 which is most appreciated. Munchen, May 1991 Herbert Spohn Contents Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Scales . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ."

Locality is a fundamental restriction in nature. On the other hand, adaptive complex systems, life in particular, exhibit a sense of permanence and time lessness amidst relentless constant changes in surrounding environments that make the global properties of the physical world the most important problems in understanding their nature and structure. Thus, much of the differential and integral Calculus deals with the problem of passing from local information (as expressed, for example, by a differential equation, or the contour of a region) to global features of a system's behavior (an equation of growth, or an area). Fundamental laws in the exact sciences seek to express the observable global behavior of physical objects through equations about local interaction of their components, on the assumption that the continuum is the most accurate model of physical reality. Paradoxically, much of modern physics calls for a fundamen tal discrete component in our understanding of the physical world. Useful computational models must be eventually constructed in hardware, and as such can only be based on local interaction of simple processing elements."

The field of neural information processing has two main objects: investigation into the functioning of biological neural networks and use of artificial neural networks to sol ve real world problems. Even before the reincarnation of the field of artificial neural networks in mid nineteen eighties, researchers have attempted to explore the engineering of human brain function. After the reincarnation, we have seen an emergence of a large number of neural network models and their successful applications to solve real world problems. This volume presents a collection of recent research and developments in the field of neural information processing. The book is organized in three Parts, i.e., (1) architectures, (2) learning algorithms, and (3) applications. Artificial neural networks consist of simple processing elements called neurons, which are connected by weights. The number of neurons and how they are connected to each other defines the architecture of a particular neural network. Part 1 of the book has nine chapters, demonstrating some of recent neural network architectures derived either to mimic aspects of human brain function or applied in some real world problems. Muresan provides a simple neural network model, based on spiking neurons that make use of shunting inhibition, which is capable of resisting small scale changes of stimulus. Hoshino and Zheng simulate a neural network of the auditory cortex to investigate neural basis for encoding and perception of vowel sounds.

Nonlinear Modeling: Advanced Black-Box Techniques discusses methods on Neural nets and related model structures for nonlinear system identification; Enhanced multi-stream Kalman filter training for recurrent networks; The support vector method of function estimation; Parametric density estimation for the classification of acoustic feature vectors in speech recognition; Wavelet-based modeling of nonlinear systems; Nonlinear identification based on fuzzy models; Statistical learning in control and matrix theory; Nonlinear time-series analysis. It also contains the results of the K.U. Leuven time series prediction competition, held within the framework of an international workshop at the K.U. Leuven, Belgium in July 1998.

The Fifteenth International Workshop on Maximum Entropy and Bayesian Meth- ods was held July 31-August 4, 1995 in Santa Fe, New Mexico, USA. St. John's College, located in the foothills of the Sangre de Cristo Mountains, provided a congenial setting for the Workshop. The relaxed atmosphere of the College, which was thoroughly enjoyed by all the attendees, stimulated free-flowing and thought- ful discussions. Conversations continued at the social events, which included a reception at the Santa Fe Institute, a New Mexican dinner at Richard Silver's home, and an excursion to Los Alamos that ended with a mixed grill at FUller Lodge, the main hall of the former Los Alamos Ranch School. This volume represents the Proceedings of the Workshop. Articles on the tra- ditional theme of the Workshop, application of the maximum-entropy principle and Bayesian methods for statistical inference in diverse areas of scientific re- search, are contained in these Proceedings. As is tradition, the Workshop opened with a tutorial on Bayesian methods, lucidly presented by Peter Cheeseman and Wray Buntine (NASA AMES, Moffett Field). The lecture notes for their tutorial are available on the World Wide Web at http://bayes .lanl. gov / "'maxent/. In addition, several new thrusts for the Workshop are described below.

Nervous System Actions and Interactions: Concepts in Neurophysiology approaches the nervous system from a functional, rather than structural, point of view.

While all of the central topics of functional neuroscience are covered, these topics are organized from a neurophysiological perspective yielding chapters on subjects such as information storage and effector actions. Each chapter is organized around general concepts that then are further developed in the text. The authors attempt to establish a dialogue with the reader by means of proposed experiments and open ended questions that are designed to both reinforce and question the text. This volume is intended to be a book of ideas for the novice or seasoned researcher in neuroscience.

This book gives the first detailed coherent treatment of a relatively young branch of statistical physics - nonlinear nonequilibrium and fluctuation-dissipative thermo dynamics. This area of research has taken shape fairly recently: its development began in 1959. The earlier theory -linear nonequilibrium thermodynamics - is in principle a simple special case of the new theory. Despite the fact that the title of this book includes the word "nonlinear", it also covers the results of linear nonequilibrium thermodynamics. The presentation of the linear and nonlinear theories is done within a common theoretical framework that is not subject to the linearity condition. The author hopes that the reader will perceive the intrinsic unity of this discipline, and the uniformity and generality of its constituent parts. This theory has a wide variety of applications in various domains of physics and physical chemistry, enabling one to calculate thermal fluctuations in various nonlinear systems. The book is divided into two volumes. Fluctuation-dissipation theorems (or relations) of various types (linear, quadratic and cubic, classical and quantum) are considered in the first volume. Here one encounters the Markov and non-Markov fluctuation-dissipation theorems (FDTs), theorems of the first, second and third kinds. Nonlinear FDTs are less well known than their linear counterparts.

Ordinary thermodynamics provides reliable results when the thermodynamic fields are smooth, in the sense that there are no steep gradients and no rapid changes. In fluids and gases this is the domain of the equations of Navier-Stokes and Fourier. Extended thermodynamics becomes relevant for rapidly varying and strongly inhomogeneous processes. Thus the propagation of high frequency waves, and the shape of shock waves, and the regression of small-scale fluctuation are governed by extended thermodynamics. The field equations of ordinary thermodynamics are parabolic while extended thermodynamics is governed by hyperbolic systems. The main ingredients of extended thermodynamics are * field equations of balance type, * constitutive quantities depending on the present local state and * entropy as a concave function of the state variables. This set of assumptions leads to first order quasi-linear symmetric hyperbolic systems of field equations; it guarantees the well-posedness of initial value problems and finite speeds of propaga tion. Several tenets of irreversible thermodynamics had to be changed in subtle ways to make extended thermodynamics work. Thus, the entropy is allowed to depend on nonequilibrium vari ables, the entropy flux is a general constitutive quantity, and the equations for stress and heat flux contain inertial terms. New insight is therefore provided into the principle of material frame indifference. With these modifications an elegant formal structure can be set up in which, just as in classical thermostatics, all restrictive conditions--derived from the entropy principle-take the form of integrability conditions.

This is the Proceedings of the Taniguchi International Symposium on "Relaxation of Elementary Excitations" which was held October 12-16,1979, at Susono-shi (at the foot of f1t. Fuji) in Japan. The pleasant atmosphere of the Symposium is evidenced in the picture of the participants shown on the next page. The purpose of the symposium was to provide an opportunity for a limited number of active researchers to meet and to discuss relaxation processes and related phenomena not only of excitons and phonons in solids but also electronic and vibrational excitations in molecules and biological systems. First, the lattice relaxation, i.e., multi-phonon process, associated with electronic excitation, which plays important roles in self-trapping of an exciton and a particle (electron and hole) and also in degradation of semi conductor lasers, is discussed. Second, this lattice relaxation is studied as the intermediate state interaction in the second-order optical responses, i.e., in connection with the competitive behavior of Raman scattering and luminescence. Third, relaxation mechanisms and relaxation constants are by spectroscopic methods as well as by genuine nonlinear optical determined phenomena. Conversely the relaxation is decisive in coherent nonlinear optical phenomena such as laser, superradiance, and optical bistability. Fourth, the role played by relaxation processes is discussed for optical phenomena in macromolecules and biological system such as photosynthesis."

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).

It is universally recognized that the end of the current and the beginning of the next century will be characterized by a radical change in the existing trends in the economic development of all countries and a transition to new principles of economic management on the basis of a resource and energy conservation policy. Thus there is an urgent necessity to study methods, technical aids and economic consequences of this change, and particularly, to determine the possible amounts of energy resources which could be conserved (energy "reserves") in different spheres of the national economy. An increased interest towards energy conservation in industry, one of the largest energy consumers, is quite natural and is manifested by the large num ber of publications on this topic. But the majority of publications are devoted to the solution of narrowly defined problems, determination of energy reserves in specific processes and plants, efficiency estimation of individual energy conserva tion measures, etc. However, it is necessary to develop a general methodological approach to the solution of such problems and create a scientific and methodical base for realizing an energy conservation policy. Such an effort is made in this book, which is concerned with methods for studying energy use efficiency in technological processes and estimation of the theoretical and actual energy reserves in a given process, technology, or industrial sector on the basis of their complete energy balances."

In the past three decades there has been enormous progress in identifying the essential role that nonlinearity plays in physical systems, including supporting soliton-like solutions and self-trapped sxcitations such as polarons. during the same period, similarly impressive progress has occurred in understanding the effects of disorder in linear quantum problems, especially regarding Anderson localization arising from impurities, random spatial structures, stochastic applied fields, and so forth. These striking consequences of disorder, noise and nonlinearity frequently occur together in physical systems. Yet there have been only limited attempts to develop systematic techniques which can include all of these ingredients, which may reinforce, complement or frustrate each other. This book contains a range of articles which provide important steps toward the goal of systematic understanding and classification of phenomenology. Experts from Australia, Europe, Japan, USA, and the USSR describe both mathematical and numerical techniques - especially from soliton and statistical physics disciplines - and applicaations to a number of important physical systems and devices, including optical and electronic transmission lines, liquid crystals, biophysics and magnetism.

This volume contains survey and research articles by some of the leading researchers in mathematical systems theory - a vibrant research area in its own right. Many authors have taken special care that their articles are self-contained and accessible also to non-specialists.

In 1979, a historical meeting took place at the Institute for Theoretical Physics in Kiev, USSR, where 48 American Scientists, specialists in nonlinear and turbulent processes, met for two weeks with their soviet counterparts. This meeting pro vided the unique opportunity for USA and USSR participants to directly interact personally and scientifically with each other. This interaction was of great impor not only for the individuals involved but also for the science of nonlinear tance phenomena in general. At the end of the meeting, it was agreed that this exchange should continue, and it was decided to have the next meeting in the USA in 1981. Unfortunately, due to the political situation at that time, the second meeting in the USA never materialized. However, in 1983, the Soviet scientists organized in Kiev a second Workshop. This second meeting was again quite successful. Similar meetings, with growing success were organized at Kiev in 1987, and 1989. It should be noted that 405 participants from 22 countries participated at the fourth Kiev workshop on Nonlinear and Turbulent Processes. The Chainnan of this workshop was V. Zakharov, who has also been a co-chainnan of all the previous workshops."

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."

For any research field to have a lasting impact, there must be a firm theoretical foundation. Neural networks research is no exception. Some of the founda tional concepts, established several decades ago, led to the early promise of developing machines exhibiting intelligence. The motivation for studying such machines comes from the fact that the brain is far more efficient in visual processing and speech recognition than existing computers. Undoubtedly, neu robiological systems employ very different computational principles. The study of artificial neural networks aims at understanding these computational prin ciples and applying them in the solutions of engineering problems. Due to the recent advances in both device technology and computational science, we are currently witnessing an explosive growth in the studies of neural networks and their applications. It may take many years before we have a complete understanding about the mechanisms of neural systems. Before this ultimate goal can be achieved, an swers are needed to important fundamental questions such as (a) what can neu ral networks do that traditional computing techniques cannot, (b) how does the complexity of the network for an application relate to the complexity of that problem, and (c) how much training data are required for the resulting network to learn properly? Everyone working in the field has attempted to answer these questions, but general solutions remain elusive. However, encouraging progress in studying specific neural models has been made by researchers from various disciplines."

We have classified the articles presented here in two Sections according to their general content. In Part I we have included papers which deal with statistical mechanics, math ematical aspects of dynamical systems and sthochastic effects in nonequilibrium systems. Part II is devoted mainly to instabilities and self-organization in extended nonequilibrium systems. The study of partial differential equations by numerical and analytic methods plays a great role here and many works are related to this subject. Most recent developments in this fascinating and rapidly growing area are discussed. PART I STATISTICAL MECHANICS AND RELATED TOPICS NONEQUILIBRIUM POTENTIALS FOR PERIOD DOUBLING R. Graham and A. Hamm Fachbereich Physik, Universitiit Gesamthochschule Essen D4300 Essen 1 Germany ABSTRACT. In this lecture we consider the influence of weak stochastic perturbations on period doubling using nonequilibrium potentials, a concept which is explained in section 1 and formulated for the case of maps in section 2. In section 3 nonequilibrium potentials are considered for the family of quadratic maps (a) at the Feigenbaum 'attractor' with Gaussian noise, (b) for more general non Gaussian noise, and (c) for the case of a strange repeller. Our discussion will be informal. A more detailed account of this and related material can be found in our papers [1-3] and in the reviews [4, 5], where further references to related work are also given. 1.

In this paper we shall discuss the construction of formal short-wave asymp totic solutions of problems of mathematical physics. The topic is very broad. It can somewhat conveniently be divided into three parts: 1. Finding the short-wave asymptotics of a rather narrow class of problems, which admit a solution in an explicit form, via formulas that represent this solution. 2. Finding formal asymptotic solutions of equations that describe wave processes by basing them on some ansatz or other. We explain what 2 means. Giving an ansatz is knowing how to give a formula for the desired asymptotic solution in the form of a series or some expression containing a series, where the analytic nature of the terms of these series is indicated up to functions and coefficients that are undetermined at the first stage of consideration. The second stage is to determine these functions and coefficients using a direct substitution of the ansatz in the equation, the boundary conditions and the initial conditions. Sometimes it is necessary to use different ansiitze in different domains, and in the overlapping parts of these domains the formal asymptotic solutions must be asymptotically equivalent (the method of matched asymptotic expansions). The basis for success in the search for formal asymptotic solutions is a suitable choice of ansiitze. The study of the asymptotics of explicit solutions of special model problems allows us to "surmise" what the correct ansiitze are for the general solution."

Human Face Recognition Using Third-Order Synthetic Neural Networks explores the viability of the application of High-order synthetic neural network technology to transformation-invariant recognition of complex visual patterns. High-order networks require little training data (hence, short training times) and have been used to perform transformation-invariant recognition of relatively simple visual patterns, achieving very high recognition rates. The successful results of these methods provided inspiration to address more practical problems which have grayscale as opposed to binary patterns (e.g., alphanumeric characters, aircraft silhouettes) and are also more complex in nature as opposed to purely edge-extracted images - human face recognition is such a problem. Human Face Recognition Using Third-Order Synthetic Neural Networks serves as an excellent reference for researchers and professionals working on applying neural network technology to the recognition of complex visual patterns.

The fourth Nishinomiya-Yukawa Memorial Symposium, devoted to the topic of dynamics and patterns in complex fluids, was held on October 26 and 27, 1989, in Nishinomiya City, Japan, where ten invited speakers gave their lectures. A one-day meeting, comprising short talks and poster sessions, was then held on the same topic on October 28 at the Research Institute for Fundamental Physics, Kyoto University. The present volume contains the 10 invited papers and 38 contributed papers presented at these two meetings. The symposium was sponsored by Nishinomiya City, where Prof. Hideki Yukawa once lived and where he wrote the celebrated paper describing the work that was later honored by a Nobel prize. The topic of the fourth symposium was chosen from one of the most vigorously evolving and highly interdisciplinary fields in condensed matter physics. The field of complex fluids is very diverse and still in its infancy and, as a result, the definition of a complex fluid varies greatly from one researcher to the next. One of the objectives of the symposium was to clarify its definition by explicitly posing a number of potentially rich problems waiting to be explored. Indeed, experimentalists are disclosing a variety of intriguing dynamical phenomena in complex systems such as polymers, liquid crystals, gels, colloids, and surfactant systems. We, the organizers, hope that the symposium will contribute to the increasing importance of the field in the coming years.

A good deal of the material presented in this book has been prepared by top experts in the field lecturing in January 1987 at the Winter School on Solitons in Tiruchirapalli, India. The lectures begin at an elementary level but go on to include even the most recent developments in the field. The book makes a handy introduction to the various facets of the soliton concept, and will be useful both to newcomers to the field and to researchers who are interested in developments in new branches of physics and mathematics

Bifurcation and Chaos presents a collection of especially written articles describing the theory and application of nonlinear dynamics to a wide variety of problems encountered in physics and engineering. Each chapter is self-contained and includes an elementary introduction, an exposition of the present state of the art, and details of recent theoretical, computational and experimental results. Included among the practical systems analysed are: hysteretic circuits, Josephson circuits, magnetic systems, railway dynamics, rotor dynamics and nonlinear dynamics of speech. This book contains important information and ideas for all mathematicians, physicists and engineers whose work in R & D or academia involves the practical consequences of chaotic dynamics.

Research in the past thirty years on the foundations of thermodynamics has led not only to a better understanding of the early developments of the subject but also to formulations of the First and Second Laws that permit both a rigorous analysis of the consequences of these laws and a substantial broadening of the class of systems to which the laws can fruitfully be applied. Moreover, modem formulations of the laws of thermodynamics have now achieved logically parallel forms at a level accessible to under graduate students in science and engineering who have completed the standard calculus sequence and who wish to understand the role which mathematics can play in scientific inquiry. My goal in writing this book is to make some of the modem develop ments in thermodyamics available to readers with the background and orientation just mentioned and to present this material in the form of a text suitable for a one-semester junior-level course. Most of this presentation is taken from notes that I assembled while teaching such a course on two occasions. I found that, aside from a brief review of line integrals and exact differentials in two dimensions and a short discussion of infima and suprema of sets of real numbers, juniors (and even some mature sophomores) had sufficient mathematical background to handle the subject matter. Many of the students whom I taught had very limited experience with formal and rigorous mathematical exposition."

This volume is the proceedings of the Hiroshima Symposium on Elementary Excitations in Quantum Fluids, which was held on August 17 and 18, 1987, in Hiroshima, Japan, and was attended by thirty-two scientists from seven countries. Quantum fluids have been the subject of intense study as a consequence of their superfluid properties at very low temperatures. Elementary excitations in them are an important concept about which many important discoveries have been made in recent years. This symposium was arranged by a group of physicists from Hiroshima University to provide an opportunity to discuss these recent developments. It was conceived as a satellite conference of the 18th International Conference on Low Temperature Physics (LT 18), which was held in Kyoto, August 20-26, 1987. Emphasis was placed on the dynamic structures and correlations of ele mentary excitations, which resulted in invited speakers being selected from this field. However, enthusiastic contributors reported notable new results on various other aspects of the elementary excitations, which made the sympo sium lively and successful. It is our great satisfaction to present this volume, which includes papers of good quality and originality. We thank all the parti cipants for their cooperation throughout this symposium, and for preparing their manuscripts within a reasonable time."

In Nonlinear Dynamics and Pattern Formation in Semiconductors and Devices the contributions of the International Conference on Nonlinear Dynamics and Pattern Formation in the Natural Environment (ICPF '94) in Noordwijkerhout, held by many internationally reknown experts, are compiled. To connect the field of semiconductor physics with the theory of nonequilibrium dissipative systems, the emphasis lies on the study of localized structures, their stability and bifurcation behaviour. A point of special interest is the evolution of dynamic structures and the investigation of more complex structures arising from interactions between these structures. Possible applications of nonlinear effects and self-organization phenomena with respect to signal processing are discussed.