From the preface: "The book is written for scientists, practicing engineers and students interested in the analysis of system dynamics. Experience has shown that the volume is of special value to analysts and designers of control systems in many disciplines of engineering. The first two chapters can be of great use as a textbook for subjects from the field of dynamics and control systems in university undergraduate courses, while the third chapter is intended for more detailed graduate study. Having this in mind, every section of the book ends in many solved numerical examples. This can be of great use in the continuing education and home-study of all those who are concerned with this fast-developing field."

Material particles, electrons, atoms, molecules, interact with one another by means of electromagnetic forces. That is, these forces are the cause of their being combined into condensed (liquid or solid) states. In these condensed states, the motion of the particles relative to one another proceeds in orderly fashion; their individual properties as well as the electric and magnetic dipole moments and the radiation and absorption spectra, ordinarily vary little by comparison with their properties in the free state. Exceptiotls are the special so-called collective states of condensed media that are formed under phase transitions of the second kind. The collective states of matter are characterized to a high degree by the micro-ordering that arises as a result of the interaction between the particles and which is broken down by chaotic thermal motion under heating. Examples of such pheonomena are the superfluidity of liquid helium, and the superconductivity and ferromagnetism of metals, which exist only at temperatures below the critical temperature. At low temperature states the particles do not exhibit their individual characteristics and conduct themselves as a single whole in many respects. They flow along capillaries in ordered fashion and create an undamped current in a conductor or a macroscopic magnetic moment. In this regard the material acquires special properties that are not usually inherent to it.

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.

This volume collects papers dedicated toWalterNoll on his sixtieth birthday, January 7, 1985. They first appeared in Volumes 86-97 (1984-1987) of the Archive for Rational Mechanics and Analysis. At the request ofthe Editors the list of authors to be invited was drawn up by B.D. Coleman, M. Feinberg, and J. Serrin. WalterNoll's influence upon research into the foundations of mechanics and thermodynamics is plain, everywhere acknowledged. Less obvious is the wide effect his writings have exerted upon those who apply mechanics to special problems, but it is witnessed by the now frequent use of terms, concepts, and styles of argument he introduced, use sometimes by young engineers who have learnt them in some recent textbook and hence take them for granted, oftenwith no idea whence they come. Examples are "objectivity", "material frame- indifference", "constitutive equation", "reduced form" of the last-named, "sim- plematerial", "simplesolid", "simplefluid", "isotropygroup",andtheassociated notations and lines of reasoning.

It was when I saw the countless number of icebergs floating in the North Atlantic Ocean in August 1964 that I decided to commence research into the solidification mechanism of metals, and already I have been continu- ing this research for two decades. In 1970 my former professor Susumu Miyata received a letter from Pre- sident Jiro Komatsu of the Tokyo Keigokin Seisaskusho Co., Ltd. The letter enclosed a copy of an article written by Mr. Imao Sasaki, chief of the casting research section at Daihatsu Motor Co., Ltd., entitled "Aruminyumu gyoko riron no shimpo" (Progress in aluminum solidi- fication theory) from a journal called "Kinzoku Zairyo" (Metals in En- gineering). This article introduced my research in considerable detail and well-written style. After stating that I "rebuffed conventional theories and enlightened casting engineers", Sasaki wrote that "These research results deserve great attention from casting engineers for the progress they have brought about in basic theories on improved soundness in casting quality." The letter from President Komatsu was a request to arrange a meeting with me. Saying that "I've heard you mention your theory occasionally, but I'd like to hear a full discussion of it for once", Professor Miyata accompanied me to the Tokyo Keigokin Seisakusho Co., Ltd. at Gyoda in Saitama Pre- fecture.

Thermodynamics Problem Solving in Physical Chemistry: Study Guide and Map is an innovative and unique workbook that guides physical chemistry students through the decision-making process to assess a problem situation, create appropriate solutions, and gain confidence through practice solving physical chemistry problems. The workbook includes six major sections with 20 - 30 solved problems in each section that span from easy, single objective questions to difficult, multistep analysis problems. Each section of the workbook contains key points that highlight major features of the topic to remind students of what they need to apply to solve problems in the topic area. Key Features: Provides instructor access to a visual map depicting how all equations used in thermodynamics are connected and how they are derived from the three major energy laws. Acts as a guide in deriving the correct solution to a problem. Illustrates the questions students should ask themselves about the critical features of the concepts to solve problems in physical chemistry Can be used as a stand-alone product for review of Thermodynamics questions for major tests.

The present simulation method has been developed at the Institute for Power Technology and Steam Generation (IVD) of the University of Stuttgart. It is being successfully employed in the analysis of processes involving large state changes such as start-ups, shut downs, malfunctions and failures in steam power generating unit, which is a large scale system consisting of several subsystems with distributed parameters, to which the steam generator also belongs. This research resulted from the increasing use of the once-through boiler, while simultaneously raising the steam parameters into the region of the supercritical state, using sliding pressure operation, combined processes with gas and steam turbines etc. The objective of this system simulation is to reduce losses of heat and condensate and to minimise unavoidable thermal stresses. The project was financed between 1979 and 1983 by the German Research Society (DFG) as part of the special research section Nr. 157 'Thermal power plants'. The Westfalen Power Company Inc. (VEW) sponsored the start-up code 'DYSTAR'. We would like to express our thanks for this support. The following members of the IVD were involved in this research project: Dr.-Ing. J. Kley Dipl.-Ing. G. Riemenschneider Dr.-Ing. A. Rolf Dipl.-Ing. U. Mayer Dipl.-lng. E. Dr.-lng. M. Klug Pfleger Dr.-Ing. G. Berndt Presently it is intended to use this non-linear, time-variant model of a power generating unit with a variable process and system struc ture as the basis for simple code versions, which one can employ e.g."

In a certain sense this book has been twenty-five years in the writing, since I first struggled with the foundations of the subject as a graduate student. It has taken that long to develop a deep appreciation of what Gibbs was attempting to convey to us near the end of his life and to understand fully the same ideas as resurrected by E.T. Jaynes much later. Many classes of students were destined to help me sharpen these thoughts before I finally felt confident that, for me at least, the foundations of the subject had been clarified sufficiently. More than anything, this work strives to address the following questions: What is statistical mechanics? Why is this approach so extraordinarily effective in describing bulk matter in terms of its constituents? The response given here is in the form of a very definite point of view-the principle of maximum entropy (PME). There have been earlier attempts to approach the subject in this way, to be sure, reflected in the books by Tribus [Thermostat ics and Thermodynamics, Van Nostrand, 1961], Baierlein [Atoms and Information Theory, Freeman, 1971], and Hobson [Concepts in Statistical Mechanics, Gordon and Breach, 1971].

This book is the Proceedings of the First International Symposium for Science on Form. The Symposium was held on November 26 through 30, 1985 at the University of Tsukuba, Japan. It was organized by The Society for Science on Form, J::.!pan, and sponsored by the Foundation for Advancement of International Science (F AIS). The purpose of the Symposium was to discuss interdisciplinal science aspects of form. "Form", to exhibit its tremendous characters, depends on the material and the changes. But, it is the form that appears evident at once and endures. Form is absorbed from every field as media of information. Thirty years and more ago, interdisciplinal problems between earthethics and science were submitted to a symposium on Form in Nature and Art. The relation between form and function had been emphasized philosophically and psychologically. In this quarter century, information theory had exactly decided figures, electronic computer had easily calculated graphics, and laser hologram had completely contained the objective image and reconstructed it.

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

The material included in this book was first presented in a series of lectures de livered at the University of Minnesota in June 1983 in connection with the con ference "Thermodynamics and Phase Transitions." This conference was one of the principal events in the first year of operation of the Institute for Mathematics and its Applications (lMA) at the University of Minnesota. The Institute was founded under the auspices of the National Science Foun dation of the United States and the University of Minnesota and is devoted to strengthening and fostering the relation of mathematics with its various applica tions to problems of the real world. The present volume constitutes an important element in the continuing pub lication program of the Ipstitute. Previous publications in this program have ap peared as lecture notes in the well-known Springer series, and future ones will be part of a new series "IMA Volumes in Applied Mathematics." Preface Until recently it was believed that thermodynamics could be given a rigorous foundation only in certain restricted circumstances, particularly those involving reversible and quasi-static processes. More general situations, commonly arising in continuum theories, have therefore been treated on the assumption that inter nal energy, entropy and absolute temperature are a priori given quantities, or have been dealt with on a more or less ad hoc basis, with emphasis for example on various types of variational formulations and maximization rules."

Computational fluid flow is not an easy subject. Not only is the mathematical representation of physico-chemical hydrodynamics complex, but the accurate numerical solution of the resulting equations has challenged many numerate scientists and engineers over the past two decades. The modelling of physical phenomena and testing of new numerical schemes has been aided in the last 10 years or so by a number of basic fluid flow programs (MAC, TEACH, 2-E-FIX, GENMIX, etc). However, in 1981 a program (perhaps more precisely, a software product) called PHOENICS was released that was then (and still remains) arguably, the most powerful computational tool in the whole area of endeavour surrounding fluid dynamics. The aim of PHOENICS is to provide a framework for the modelling of complex processes involving fluid flow, heat transfer and chemical reactions. PHOENICS has now been is use for four years by a wide range of users across the world. It was thus perceived as useful to provide a forum for PHOENICS users to share their experiences in trying to address a wide range of problems. So it was that the First International PHOENICS Users Conference was conceived and planned for September 1985. The location, at the Dartford Campus of Thames Polytechnic, in the event, proved to be an ideal site, encouraging substantial interaction between the participants.

In the seven years since this volume first appeared. there has been an enormous expansion of the range of problems to which Monte Carlo computer simulation methods have been applied. This fact has already led to the addition of a companion volume ("Applications of the Monte Carlo Method in Statistical Physics", Topics in Current Physics. Vol . 36), edited in 1984, to this book. But the field continues to develop further; rapid progress is being made with respect to the implementation of Monte Carlo algorithms, the construction of special-purpose computers dedicated to exe cute Monte Carlo programs, and new methods to analyze the "data" generated by these programs. Brief descriptions of these and other developments, together with numerous addi tional references, are included in a new chapter , "Recent Trends in Monte Carlo Simulations" , which has been written for this second edition. Typographical correc tions have been made and fuller references given where appropriate, but otherwise the layout and contents of the other chapters are left unchanged. Thus this book, together with its companion volume mentioned above, gives a fairly complete and up to-date review of the field. It is hoped that the reduced price of this paperback edition will make it accessible to a wide range of scientists and students in the fields to which it is relevant: theoretical phYSics and physical chemistry , con densed-matter physics and materials science, computational physics and applied mathematics, etc.

It is a great pleasure to have the opportunity to honor our distinguished colleague, Professor Leo Brewer, on the occasion of his sixty-fifth birth day, with this special volume of High Temperature Science. Leo and his wife, Rose, are personal friends of several generations of students and postdoctoral researchers at the University of California at Berkeley. Their concern and understanding has been important to many of us over the past forty years. Each paper in this volume has at least one author who was a gradu ate student or a postdoctoral researcher in Leo's laboratory at Berkeley. The variety of topics is indicative of the wide-ranging science done by Brewer-ites and by Leo Brewer himself. He has personally participated in the resolution of many of the classical problems of high-temperature science-from the heat of sublimation of graphite to the dissociation en ergy of nitrogen to the prediction of binary and ternary phase diagrams. He and his students have made major contributions to atomic and molec ular spectroscopy. He has made significant contributions to the develop ment of efficient systems for energy conversion and to ceramics. In addi tion to his research activities, Leo Brewer has been a long-time participant in the dynamic undergraduate teaching program of the Berkeley Chemistry Department. He has provided crucial insight for stu dents involved in those career-shaping experiences that one endures while acquiring the basics of inorganic, organic, and physical chemistry with that interwoven common bond of thermodynamics."

Groundwater has long been one of the world's most important resources. It accounts for approximately 96% of all fresh water in the United States and supplies more than 50% of the population with potable water. Historically, this water source has generally been regarded as pristine. However, in recent years, contamination of ground water by industrial products has become a problem of growing concern. During the past four decades, the variety and quantity of organic chemicals produced in the U. S. has steadily increased. Currently, more than 40,000 different organic compounds are being manufactured, trans ported, used and eventually disposed of in the environment (Wilson, et !l (1981". Production and consumption of petroleum products has also risen in this same time period. Many of these industrial compounds are highly toxic and slightly water soluble. Thus, they pose a poten tial threat to large volumes of groundwater if they are somehow intro duced into the subsurface. Increased production of chemicals implies the increased risk of accidental spills or leakage to the soil, and indeed, the literature abounds with contamination case histories. 2 Incidences of petroleum contamination of groundwater have been documented by many authors. For example, see: Schwi11e (1967); Toms (1971); Guenther (1972); McKee, et!l (1912); Williams and Wilder (1971); Van100cke, et ~]-

Kompakt und verstandlich fuhrt dieses Lehrbuch in die Grundlagen der theoretischen Physik ein. Dabei werden die ublichen Themen der Grundvorlesungen Mechanik, Elektrodynamik, Relativitatstheorie, Quantenmechanik , Thermodynamik und Statistik in einem Band zusammengefasst, um den Zusammenhang zwischen den einzelnen Teilgebieten besonders zu betonen. Ein Kapitel mit mathematischen Grundlagen der Physik erleichtert den Einstieg. Zahlreiche UEbungsaufgaben dienen der Vertiefung des Stoffes.

The problem of deriving irreversible thermodynamics from the re versible microscopic dynamics has been on the agenda of theoreti cal physics for a century and has produced more papers than can be digested by any single scientist. Why add to this too long list with yet another work? The goal is definitely not to give a gen eral review of previous work in this field. My ambition is rather to present an approach differing in some key aspects from the stan dard treatments, and to develop it as far as possible using rather simple mathematical tools (mainly inequalities of various kinds). However, in the course of this work I have used a large number of results and ideas from the existing literature, and the reference list contains contributions from many different lines of research. As a consequence the reader may find the arguments a bit difficult to follow without some previous exposure to this set of problems."

The equations which we are going to study in these notes were first presented in 1963 by E. N. Lorenz. They define a three-dimensional system of ordinary differential equations that depends on three real positive parameters. As we vary the parameters, we change the behaviour of the flow determined by the equations. For some parameter values, numerically computed solutions of the equations oscillate, apparently forever, in the pseudo-random way we now call "chaotic"; this is the main reason for the immense amount of interest generated by the equations in the eighteen years since Lorenz first presented them. In addition, there are some parameter values for which we see "preturbulence," a phenomenon in which trajectories oscillate chaotically for long periods of time before finally settling down to stable stationary or stable periodic behaviour, others in which we see "intermittent chaos," where trajectories alternate be tween chaotic and apparently stable periodic behaviours, and yet others in which we see "noisy periodicity," where trajectories appear chaotic though they stay very close to a non-stable periodic orbit. Though the Lorenz equations were not much studied in the years be tween 1963 and 1975, the number of man, woman, and computer hours spent on them in recent years - since they came to the general attention of mathematicians and other researchers - must be truly immense."