Modern structural applications of crystallography make extensive use of statistical methods, in particular the probability density function (pdf) of the magnitude of the structure factor. Similarly, direct methods of phase determination have been responsible for much of the success of crystallography - methods based on properties of joint pdfs. This monograph, from two authorities in the field of structure factor statics, presents a survey of techniques and theories in this field of research in a self-contained and consistent way, with an emphasis on the probabilistic principles involved.

This monograph gives a systematic presentation of ideas, methods and results of the modern statistical theory of open systems -- systems capable of exchanging matter, energy and information with the surrounding world. The resulting self-organization can lead to more sophisticated and advanced structures. Central to this work are the statistical criteria of self-organization. The feasibility of a unified description of kinetic, hydrodynamic and diffusion processes in passive and active macroscopic systems without resorting to the methods of perturbation theory is demonstrated. On this basis, a general definition of thermal flux is given in terms of the entropy gradient. Moreover, a consistent method for calculating both kinetic and hydrodynamic fluctuations is proposed. This approach is then used to construct a theory of classical and anomalous Brownian motion in nonlinear media. This theory makes it possible to treat in an original way the phenomenon of turbulence, and to propose a unified kinetic description of laminar and turbulent motion. The proposed methods are also applied to the statistical description of quantum macroscopic open systems. This provides answers as to whether or not the quantum mechanical description is complete, and whether or not there are hidden parameters in quantum mechanics. The book has no analogy in the existing literature. It is both a monograph and a textbook, and is based largely on the author's original research. The book will be useful to postgraduate students and researchers in chemistry, physics, mathematics, economics, sociology, and engineering.

Computer Simulation and Computer Algebra. Starting from simple examples in classical mechanics, these introductory lectures proceed to simulations in statistical physics (using FORTRAN) and then explain in detail the use of computer algebra (by means of Reduce). This third edition takes into account the most recent version of Reduce (3.4.1) and updates the description of large-scale simulations to subjects such as the 170000 X 170000 Ising model. Furthermore, an introduction to both vector and parallel computing is given.

This book is a collection of thirty invited papers, covering the important parts of a rapidly developing area like "computational statistics." All contributions supply information about a specialized topic in a tutorial and comprehensive style. Newest results and developments are discussed. Starting with the foundations of computational statistics, i.e. numerical reliability of software packages or construction principles for pseudorandom number generators, the volume includes design considerations on statistical programming languages and the basic issues of resampling techniques. Also covered are areas like design of experiments, graphical techniques, modelling and testing problems, a review of clustering algorithms, and concise discussions of regression trees or cognitive aspects of authoring systems.

This book is based on my research work I did between 1986 and 1992 at the Institut fur Theoretische Physik und Synergetik, Universitiit Stuttgart. It might be of interest to all students and scientists who are interested in modern mathematical physics. It deals with one of the essential topics of modern physical research, namely t. he problem of universal aspects in statistical physics, and in doing so multi-component systems, like thermodynamic systems, laser systems and high-dimensional quan tum systems, are considered. Synergetic aspects of multi-component systems will be discussed. Remarks to biological systems will be made, in particular remarks to the problem of EEG analysis. Non-linear aspects will be discussed. The problem of a macroscopic access to multi-component systems was an essential part of my research work. In this context it has to be emphasized that in this book the concept of hyper-surface equations will be introduced, which is a totally analytical strategy to determine distribution function parameters in an exact way by using only given measurement quantities. This concept can be used to determine both distribution functions of macroscopic systems and path integrals of quantum systems. In this context the property self-similarity will playa crucial role. A special extreme princi ple, the maximum information entropy principle, can be taken as a basis to introduce this concept. This principle will be discussed. In order to determine the dynamic behavior of multi-component systems, special evolution equations are needed."

The book explores several open questions in the philosophy and the foundations of statistical mechanics. Each chapter is written by a leading expert in philosophy of physics and/or mathematical physics. Here is a list of questions that are addressed in the book:

Symmetries and singularity structures play important roles in the study of nonlinear dynamical systems. It was Sophus Lie who originally stressed the importance of symmetries and invariance in the study of nonlinear differential equations. How- ever, the full potentialities of symmetries had been realized only after the advent of solitons in 1965. It is now a well-accepted fact that associated with the infinite number of integrals of motion of a given soliton system, an infinite number of gep. eralized Lie BAcklund symmetries exist. The associated bi-Hamiltonian struc- ture, Kac-Moody, Vrrasoro algebras, and so on, have been increasingly attracting the attention of scientists working in this area. Similarly, in recent times the role of symmetries in analyzing both the classical and quantum integrable and nonintegrable finite dimensional systems has been remarkable. On the other hand, the works of Fuchs, Kovalevskaya, Painleve and coworkers on the singularity structures associated with the solutions of nonlinear differen- tial equations have helped to classify first and second order nonlinear ordinary differential equations. The recent work of Ablowitz, Ramani and Segur, con- jecturing a connection between soliton systems and Painleve equations that are free from movable critical points, has motivated considerably the search for the connection between integrable dynamical systems with finite degrees of freedom and the Painleve property. Weiss, Tabor and Carnevale have extended these ideas to partial differential equations.

Proceedings of the NATO Advanced Study Institute on Propagation of Correlations in Constrained Systems, Cargese, Corsica, France, July 2-14, 1990

The statistical mechanical theory of liquids and solutions is a fundamental area of physical sciences with important implications in other fields of science and for many industrial applications. This book introduces equilibrium statistical mechanics in general, and statistical mechanics of liquids and solutions in particular. A major theme is the intimate relationship between forces in a fluid and the fluid structure - a relationship that is paramount for the understanding of the subject of interactions in dense fluids. Using this microscopic, molecular approach, the text emphasizes clarity of physical explanations for phenomena and mechanisms relevant to fluids, addressing the structure and behavior of liquids and solutions under various conditions. A notable feature is the author's treatment of forces between particles that include nanoparticles, macroparticles, and surfaces. The book provides an expanded, in-depth treatment of simple liquids and electrolytes in the bulk and in confinement. Provides an introduction to statistical mechanics of liquids and solutions with special attention to structure and interactions. Offers an extensive presentation starting with the basics of statistical mechanics to modern aspects of the theory of liquids and solutions, including intermolecular interactions in fluids. Treats both homogeneous bulk fluids and inhomogeneous fluids near surfaces and in confinement. Takes a microscopic, molecular approach that combines physical transparency, theoretical sharpness and a pedagogical and accessible style. Gives explicit and clear textual explanations and physical interpretations for any mathematical relationships and derivations. Goes deeper than the available texts on interactions in fluids, by taking the discussion beyond simple approximations and mean field approaches. The book will be an invaluable resource for advanced undergraduate, graduate, and postgraduate students in physics, chemistry, soft matter science, surface and colloid science and related fields, as well as professionals and instructors in those areas of science.

Since 1972 the Schools on Nonlinear Physics in Gorky have been a meeting place for Soviet scientists working in this field. Instead of producing for the first time English proceedings it has been decided to present a good cross section of nonlinear physics in the USSR. Thus the participants at the last School were invited to provide English reviews and research papers for these two volumes (which in the years to come will be followed by the proceedings of forthcoming schools). "The second volume" deals with dynamical chaos in classical and quantum systems, with evolution in chemical systems and self-organisation in biology, and with applications of nonlinear dynamics to condensed matter, sea waves, and astrophysics.

This proceedings volume gathers together original articles and survey works that originate from presentations given at the conference Transient Transcendence in Transylvania, held in Brasov, Romania, from May 13th to 17th, 2019. The conference gathered international experts from various fields of mathematics and computer science, with diverse interests and viewpoints on transcendence. The covered topics are related to algebraic and transcendental aspects of special functions and special numbers arising in algebra, combinatorics, geometry and number theory. Besides contributions on key topics from invited speakers, this volume also brings selected papers from attendees.

The book is meant to be primarily a textbook. Therefore most of the examples of special thermodynamic systems are standards in different fields. The number of the examples, however, is restricted. For more applications we refer the reader to the rich literature on thermodynamics. The author hopes that the book, which is concerned above all with basic connections, will be interesting not only for students but also for academic teachers and other scientists who like the structural analysis of fundamentals in physics. According to the character of a textbook, this book is not intended to demonstrate new results. Nevertheless, the way of the logical deductions, and of the presentations used in this book, as well as the choice of illustrating examples are not only influenced by literature but also by discussions with colleagues and friends. In this respect I should like to mention the Professors A. Stahl, J. Meixner, R. Bausch, H.-K. Janssen, R. Bessenrodt, Dr. E. Scholl, and Dr. C. Escher in Germany, as well as Professor C. A. Mead in Minneapolis, and Professor R. St. Berry in Chicago. Particular thank is directed to Professor V. Dohm for critically reading certain parts of the manuscript and making valuable proposals for improvements.

Deals with the computer simulation of complex physical sys- tems encounteredin condensed-matter physics and statistical mechanics as well as in related fields such as metallurgy, polymer research,lattice gauge theory and quantummechanics.

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.

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. The mathematical description for quantum theory starts with a Hilbert space H of state vectors. Quantum fields are linear operators on this space, which satisfy nonlinear wave equations of fundamental physics, including coupled Dirac, Max well and Yang-Mills equations. The field operators are restricted to satisfy a "locality" requirement that they commute (or anti-commute in the case of fer mions) at space-like separated points. This condition is compatible with finite propagation speed, and hence with special relativity. Asymptotically, these fields converge for large time to linear fields describing free particles. Using these ideas a scattering theory had been developed, based on the existence of local quantum fields."