A Modern Course in Statistical Physics 4th Edition

Book Preface

A Modern Course in Statistical Physics has gone through several editions. The first edition was published in 1980 by University of Texas Press. It was well re-ceived because it contained a presentation of statistical physics that synthesized the best of the american and european “schools” of statistical physics at that time. In 1997, the rights to A Modern Course in Statistical Physics were transferred to John Wiley & Sons and the second edition was published. The second edition was a much expanded version of the first edition, and as we subsequently realized, was too long to be used easily as a textbook although it served as a great reference on statistical physics. In 2004, Wiley-VCH Verlag assumed rights to the second edi-tion, and in 2007 we decided to produce a shortened edition (the third) that was explicitly written as a textbook. The third edition appeared in 2009.
Statistical physics is a fast moving subject and many new developments have occurred in the last ten years. Therefore, in order to keep the book “modern”, we decided that it was time to adjust the focus of the book to include more applica-tions in biology, chemistry and condensed matter physics. The core material of the book has not changed, so previous editions are still extremely useful. Howev-er, the new fourth edition, which is slightly longer than the third edition, changes some of its focus to resonate with modern research topics.
The first edition acknowledged the support and encouragement of Ilya Pri-gogine, who directed the Center for Statistical Mechanics at the U.T. Austin from 1968 to 2003. He had an incredible depth of knowledge in many fields of science and helped make U.T. Austin an exciting place to be. The second edition was ded-icated to Ilya Prigogine “for his encouragement and support, and because he has changed our view of the world.” The second edition also acknowledged another great scientist, Nico van Kampen, whose beautiful lectures on stochastic process-es, and critically humorous view of everything, were an inspiration and spurred my interest statistical physics. Although both of these great people are now gone, I thank them both.
The world exists and is stable because of a few symmetries at the microscopic level. Statistical physics explains how thermodynamics, and the incredible com-plexity of the world around us, emerges from those symmetries. This book at-tempts to tell the story of how that happens.

Austin, Texas January 2016

Thermodynamics, which is a macroscopic theory of matter, emerges from the symmetries of nature at the microscopic level and provides a universal theory of matter at the macroscopic level. Quantities that cannot be destroyed at the mi-croscopic level, due to symmetries and their resulting conservation laws, give rise to the state variables upon which the theory of thermodynamics is built.
Statistical physics provides the microscopic foundations of thermodynamics. At the microscopic level, many-body systems have a huge number of states avail-able to them and are continually sampling large subsets of these states. The task of statistical physics is to determine the macroscopic (measurable) behavior of many-body systems, given some knowledge of properties of the underlying mi-croscopic states, and to recover the thermodynamic behavior of such systems.
The field of statistical physics has expanded dramatically during the last half-century. New results in quantum fluids, nonlinear chemical physics, critical phe-nomena, transport theory, and biophysics have revolutionized the subject, and yet these results are rarely presented in a form that students who have little back-ground in statistical physics can appreciate or understand. This book attempts to incorporate many of these subjects into a basic course on statistical physics. It in-cludes, in a unified and integrated manner, the foundations of statistical physics and develops from them most of the tools needed to understand the concepts underlying modern research in the above fields.
There is a tendency in many books to focus on equilibrium statistical mechan-ics and derive thermodynamics as a consequence. As a result, students do not get the experience of traversing the vast world of thermodynamics and do not under-stand how to apply it to systems which are too complicated to be analyzed using the methods of statistical mechanics. We will begin in Chapter 2, by deriving the equations of state for some simple systems starting from our knowledge of the microscopic states of those systems (the microcanonical ensemble). This will give some intuition about the complexity of microscopic behavior underlying the very simple equations of state that emerge in those systems.
In Chapter 3, we provide a thorough grounding in thermodynamics. We review the foundations of thermodynamics and thermodynamic stability theory and de-vote a large part of the chapter to a variety of applications which do not involve phase transitions, such as heat engines, the cooling of gases, mixing, osmosis, chemical thermodynamics, and batteries. Chapter 4 is devoted to the thermo-dynamics of phase transitions and the use of thermodynamic stability theory in analyzing these phase transitions. We discuss first-order phase transitions in liq-uid–vapor–solid transitions, with particular emphasis on the liquid–vapor transi-tion and its critical point and critical exponents. We also introduce the Ginzburg–Landau theory of continuous phase transitions and discuss a variety of transitions which involve broken symmetries. And we introduce the critical exponents which characterize the behavior of key thermodynamic quantities as a system approach-es its critical point.
In Chapter 5, we derive the probability density operator for systems in thermal contact with the outside world but isolated chemically (the canonical ensemble). We use the canonical ensemble to derive the thermodynamic properties of a va-riety of model systems, including semiclassical gases, harmonic lattices and spin systems. We also introduce the concept of scaling of free energies as we approach the critical point and we derive values for critical exponents using Wilson renor-malization theory for some particular spin lattices.
In Chapter 6, we derive the probability density operator for open systems (the grand canonical ensemble), and use it to discuss adsorption processes, properties of interacting classical gases, ideal quantum gases, Bose–Einstein condensation, Bogoliubov mean field theory, diamagnetism, and super-conductors.
The discrete nature of matter introduces fluctuations about the average (ther-modynamic) behavior of systems. These fluctuations can be measured and give valuable information about decay processes and the hydrodynamic behavior of many-body systems. Therefore, in Chapter 7 we introduce the theory of Browni-an motion which is the paradigm theory describing the effect of underlying fluc-tuations on macroscopic quantities. The relation between fluctuations and decay processes is the content of the so-called fluctuation–dissipation theorem which is derived in this chapter. We also derive Onsager’s relations between transport coefficients, and we introduce the mathematics needed to introduce the effect of causality on correlation functions. We conclude this chapter with a discussion of thermal noise and Landauer conductivity in ballistic electron waveguides.
Chapter 8 is devoted to hydrodynamic processes for systems near equilibrium. We derive the Navier–Stokes equations from the symmetry properties of a fluid of point particles, and we use the derived expression for entropy production to ob-tain the transport coefficients for the system. We also use the solutions of the lin-earized Navier–Stokes equations to predict the outcome of light-scattering exper-iments. We next derive a general expression for the entropy production in binary mixtures and use this theory to describe thermal and chemical transport process-es in mixtures, and in electrical circuits. We conclude Chapter 8 with a derivation of hydrodynamic equations for superfluids and consider the types of sound that can exist in such fluids.
In Chapter 9, we derive microscopic expressions for the coefficients of diffusion, shear viscosity, and thermal conductivity, starting both from mean free path ar-guments and from the Boltzmann and Lorentz–Boltzmann equations. We obtain explicit microscopic expressions for the transport coefficients of a hard-sphere gas Finally, in Chapter 10 we conclude with the fascinating subject of nonequilibri-um phase transitions. We show how nonlinearities in the rate equations for chem-ical reaction–diffusion systems lead to nonequilibrium phase transitions which give rise to chemical clocks, nonlinear chemical waves, and spatially periodic chemical structures, while nonlinearities in the Rayleigh–Bénard hydrodynamic system lead to spatially periodic convection cells.
The book contains Appendices with background material on a variety of top-ics. Appendix A, gives a review of basic concepts from probability theory and the theory of stochastic processes. Appendix B reviews the theory of exact differen-tials which is the mathematics underlying thermodynamics. In Appendix C, we review ergodic theory. Ergodicity is a fundamental ingredient for the microscop-ic foundations of thermodynamics. In Appendix D, we derive the second quan-tized formalism of quantum mechanics and show how it can be used in statistical mechanics. Appendix E reviews basic classical scattering theory. Finally, in Ap-pendix F, we give some useful math formulas and data. Appendix F also contains solutions to some of the problems that appear at the end of each chapter.
The material covered in this textbook is designed to provide a solid grounding in the statistical physics underlying most modern physics research topics.