Statistical Mechanics Made Simple

Elementary concepts in statistics and probability - The ising model and the lattice gas - Elements of thermodynamics - Statistical mechanics - The world of bosons - All about fermions : theories of metals, superconductors, semiconductors - Kinetic theory - The transfer matrix - Some uses of quantum field theory in statistical physics.

This second edition extends and improves on the first, illustrating through myriad examples, the principles and logic used in extending the simple laws of idealised Newtonian physics and quantum physics into the real world of noise and thermal fluctuations.

This second edition extends and improves on the first, already an acclaimed and original treatment of statistical concepts insofar as they impact theoretical physics and form the basis of modern thermodynamics. This book illustrates through myriad examples the principles and logic used in extending the simple laws of idealized Newtonian physics and quantum physics into the real world of noise and thermal fluctuations. In response to the many helpful comments by users of the first edition, important features have been added in this second, new and revised edition. These additions allow a more coherent picture of thermal physics to emerge. Benefiting from the expertise of the new co-author, the present edition includes a detailed exposition — occupying two separate chapters — of the renormalization group and Monte-Carlo numerical techniques, and of their applications to the study of phase transitions. Additional figures have been included throughout, as have new problems. A new Appendix presents fully worked-out solutions to representative problems; these illustrate various methodologies that are peculiar to physics at finite temperatures, that is, to statistical physics. This new edition incorporates important aspects of many-body theory and of phase transitions. It should better serve the contemporary student, while offering to the instructor a wider selection of topics from which to craft lectures on topics ranging from thermodynamics and random matrices to thermodynamic Green functions and critical exponents, from the propagation of sound in solids and fluids to the nature of quasiparticles in quantum liquids and in transfer matrices.

This book is an elaboration of the author's lecture notes in a graduate course in statistical physics and thermodynamics, augmented by some material suitable for self-teaching as well as for undergraduate study. The first 4 or 5 chapters are suitable for an undergraduate course for engineers and physicists in Thermodynamics and Statistical Physics and include detailed study of the various ensembles and their connections to applied thermodynamics. The Debye law of specific heats and reasons for deviations from the Debye formulas are covered, as are the Einstein theories of Brownian motion, black-body radiation and specific heat of solids. Van der Waals gases and the reason for the apparent failure of his Law of Corresponding States are discussed. The last 5 chapters treat topics of recent interest to researchers, including: the Ising and Potts models, spin waves in ferromagnetic and anti-ferromagnetic media, sound propagation in non-ideal gases and the decay of sound waves, introduction to the understanding of glasses and spin glasses, superfluidity and superconductivity. The selection of material is wide-ranging and the mathematics for handling it completely self-contained, ranging from counting (probability theory) to quantum field theory as used in the study of fermions, bosons and as an adjunct in the solutions of the equations of classical diffusion-reaction theory. In addition to the standard material found in most recent books on statistical physics the constellation of topics covered in this text includes numerous original items: • Generalization of “negative temperature” to interacting spins • Derivation of Gibbs' factor from first principles • Exact free energy of interacting particles in 1D (e.g., classical and quantum Tonk's gas) • Introduction to virial expansions, Equations of State, Correlation Functions and “critical exponents” • Superfluidity in ideal and non-ideal fluids (both Bogolubov and Feynman theories) • Superconductivity: thermodynamical approach and the BCS theory • Derivation of “Central Limit Theorem” and its applications • Boltzmann's “H-Theorem” and the nonlinear Boltzmann equation • Exact solution of nonlinear Boltzmann Equation for electrons in time-dependent electric field and the derivation of Joule heating, transport parameters in crossed electric and magnetic fields, etc. • Frequency spectrum and decay of sound waves in gases • Exact evaluation of free energy and thermodynamic properties of the two-dimensional Ising model in regular and fully frustrated (spin-glass like) lattices • The “zipper” model of crystal fracture or polymer coagulation — calculation of Tc • Potts model in 2D: duality and Tc • “Doi's theory” of diffusion-limited chemical reactions with some exact results — including the evaluation of statistical fluctuations in radioactive decay • Thermodynamic Green Functions and their applications to fermions and bosons with an example drawn from random matrix theory and much more.

This new version of a classic updates much of the material in earlier editions, including the first chapter, on the history of the field. Important modifications reflect major discoveries of the past decades. A historical perspective is maintained throughout. The reader is drawn into the process of discovery: starting with a phenomenon, finding plausible explanations and competing theories — and finally, the solution. The theory of magnetism is practically a metaphor for theoretical physics. The very first quantum many-body theory (Bethe's ansatz) was devised for magnetic chains, just as mean-field theory was invented a century ago by Weiss to explain Curie's Law. The first two chapters of this book are immensely readable, taking us from prehistory to the “spin valves” of the most recent past. Topics in subsequent chapters include: angular momenta and spin (Chapter 3), quantum theory of simple systems, followed by increasingly technical insights into ordered and random systems, thermal fluctuations, phase transitions, chaos and the like. Contemporary developments in nanotechnology now seek to take advantage of the electron's spin as well as of its charge. The time is not far off when nano-circuits made entirely of silicon exhibit such many-body properties as superconductivity or ferromagnetism — without any superconducting materials or magnetic ions being present. The reader of this book will be prepared for such exotic twenty-first century applications. Daniel C Mattis, BS, MS, PhD, Fellow of the American Physical Society (APS), is a frequent lecturer at research institutions and the author of several textbooks and numerous research articles. His expertise includes many-body theory, electrical conductivity, quantum theory of magnetism and most recently, nanotechnology. Prof. Mattis is on the editorial panel for high-temperature superconductivity of the International Journal of Modern Physics B and Modern Physics Letters B, both published by World Scientific. Currently serving as Professor in the Physics department at the University of Utah in Salt Lake City, Utah, USA, at various times he has been visiting Professor at Yale University (New Haven), State University of New York (Buffalo), Temple University (Philadelphia), and served as “Wei-Lun Visiting Professor” at the Chinese University of Hong Kong. A founding member of the “Few-Body Physics” section of the APS, he has also served as Chair of the standing committee of the APS for the “International Freedom of Scientists.”

STATISTICAL MECHANICS JOSEPH EDWARD MAYER... Associate Professor of Chemistry, Columbia University AND MARIA GOEPPERT MAYER Lecturer in Chemistry, Columbia University NEW YORK JOHN WILEY SONS, INC. LONDON CHAPMAN HALL, LIMITED 1940 PREFACE The rapid increase, in the past few decades, of knowledge concerning the structure of molecules has made the science of statistical mechanics a practical tool for interpreting and correlating experimental data. It is therefore desirable to present this subject in a simple manner in order to make it easily available to scientists whose familiarity with theoretical physics is limited. This book, which grew out of lectures and seminars given to graduate students in chemistry and physics, aims to fulfill this purpose. The development of quantum mechanics has altered both the axio matic foundation and the details of the methods of statistical mechanics. Although the results of a large number of statistical calculations are un affected by the introduction of quantum mechanics, the chemists interest happens to be largely in fields where quantum effects are im portant. Consequently, in our presentation, the laws of statistical mechanics are founded on the concepts of both quantum and classical mechanics. The equivalence of the two methods has been stressed, but the quantum-mechanical language has been favored. We believe that this introduction of quantum statistics at the beginning simplifies rather than puts a burden upon the initial concepts. It is to be emphasized that the simpler ideas of quantum mechanics, which are all that is used, are as widely known as the more abstract theorems of classical mechanics which they replace. Simplicity of presentationrather than brevity and elegance has been our endeavor. However, we have not consciously sacrificed rigor. Care has been taken to make the book suitable for reference by sum marizing and tabulating final equations as well as by an attempt to make individual chapters complete in themselves without too much reference to previous subjects. All the theorems and results of mechanics and quantum mechanics which are used later have been summarized, largely without proof, in Chapter 2. The last section, 2k, on Einstein-Bose and Fermi-Dirac systems, ties up closely with Chapters 5 and 16 only. Chapters 3 and 4 contain the derivation of the fundamental statistical laws on which the book is based. Chapter 10 is prerequisite for Chap ters l 1 tol4. Otherwise, individual subj ects may be taken up in different order. vii viii PREFACE In Chapters 7 to 9 considerable space is devoted to the calculation of thermodynamic functions for perfect gases, which was considered justi fied by the value of the results for the chemist. These chapters may be omitted by readers uninterested in the subject. Chapters 13 and 14 on the imperfect gas and condensation theory, respectively, are somewhat more complicated than the remainder, but are included because of our special interest in the subject. The aim of the book is to give the reader a clear understanding of principles and to prepare him thoroughly for the use of the science and the study of recent papers. Many of the simpler applications are dis cussed in some detail, but in general language without comparison with experiment. The more complicated subjects have been omitted, as have been those for which at present only partial solutions are obtained. This choicehas excluded many of the contemporary developments, especially the interesting work of J. G. Kirkwood, L. Onsager, H. Eyring, and W, F. Giauque. In conclusion we express our gratitude to Professors Max Born, Karl F. Hcrzfeld, and Edward Teller, who have read and criticized several parts of the manuscript. We also thank Dr. Elliot Montroll, who aided in reading proof and who made many helpful suggestions. JOSEPH EDWARD MAYER MARIA GOEPPERT MAYER NEW YORK CITY March 31, 1940 Dedicated to our teachers Gilbert N...

Building on the material learned by students in their first few years of study, Topics in Statistical Mechanics (Second Edition) presents an advanced level course on statistical and thermal physics. It begins with a review of the formal structure of statistical mechanics and thermodynamics considered from a unified viewpoint. There is a brief revision of non-interacting systems, including quantum gases and a discussion of negative temperatures. Following this, emphasis is on interacting systems. First, weakly interacting systems are considered, where the interest is in seeing how small interactions cause small deviations from the non-interacting case. Second, systems are examined where interactions lead to drastic changes, namely phase transitions. A number of specific examples is given, and these are unified within the Landau theory of phase transitions. The final chapter of the book looks at non-equilibrium systems, in particular the way they evolve towards equilibrium. This is framed within the context of linear response theory. Here fluctuations play a vital role, as is formalised in the fluctuation-dissipation theorem.The second edition has been revised particularly to help students use this book for self-study. In addition, the section on non-ideal gases has been expanded, with a treatment of the hard-sphere gas, and an accessible discussion of interacting quantum gases. In many cases there are details of Mathematica calculations, including Mathematica Notebooks, and expression of some results in terms of Special Functions.

This book contains a modern selection of about 200 solved problems and examples arranged in a didactic way for hands-on experience with course work in a standard advanced undergraduate/first-year graduate class in thermodynamics and statistical physics. The principles of thermodynamics and equilibrium statistical physics are few and simple, but their application often proves more involved than it may seem at first sight. This book is a comprehensive complement to any textbook in the field, emphasizing the analogies between the different systems, and paves the way for an in-depth study of solid state physics, soft matter physics, and field theory.