EQUILIBRIUM STATISTICAL PHYSICS PLISCHKE PDF

Some Relevant Textbooks and Monographs: L. Reichl: A modern course in statistical physics. Wiley-Interscience, New York Chandler: Introduction to Modern statistical mechanics. Oxford University Press

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Statistical physics. Critical phenomena Physics. I, Plischke, Michael II. Bergersen, Birger. QC P55 Pte, Led. All rights reserved. This book, or parts thereof, may not be reproduced in any form or by any means, electronic or mechanical, including photocopying, recording or any information storage and retrieval system now known or to be invented, without written permission from the Publisher.

In this case permission to photocopy is not required from the publisher. State Variables and Equations of State. First law 1. Second law. Thermodynamic Potentials Systems at Fixed Temperature: Canonical Ensemble Grand Canonical Ensemble.. Virial Expansion.. BBGKY hierarchy Contents vii 5. Analysis of series 53 Scaling. Detoiled balance and the Metropolis algorithm 7. Finitetemperature BCS theory Sherrington-Kirkpatrick model fyvlc 3 goupdooGooooooDoooooD eGo ooodGo Appendix: Occupation Number Representation Bibliography Index Preface to First Edition During the last decade each of the authors has regularly taught a graduate or senior undergraduate course in statistical mechanics.

During this same period, the renormalization group approach to critical phenomena, pioneered by K. Wilson, greatly altered our approach to condensed matter physics. Since its introduction in the context of phase transitions, the method has found appli- cation in many other areas of physics, such as many-body theory, chaos, the conductivity of disordered materials, and fractal structures.

So pervasive is its influence that we feel that it now essential that graduate students be intro- duced at an early stage in their career to the concepts of scaling, universality, fixed points, and renormalization transformations, which were developed in the context of critical phenomena, but are relevant in many other situations. In this book we describe both the traditional methods of statistical mechan- ics and the newer techniques of the last two decades.

Most graduate students are exposed to only one course in statistical physics. We believe that this course should provide a bridge from the typical under-graduate course usu- ally concerned primarily with noninteracting systems such as ideal gases and paramagnets to the sophisticated concepts necessary to a researcher. We assume that the student has been exposed previously to the material of these two chapters and thus our treatment is rather concise.

We have, however, included a substantial number of exercises that complement the review. In Chapter 3 we begin our discussion of strongly interacting systems with a lengthy exposition of mean field theory. A number of examples are worked out in detail. The more general Landau theary of phase transitions is developed and used to discuss critical points, tricritical points, and first-order phase tran- sitions.

The limitations of mean field and Landau theory are described and xi xii Preface the role of fluctuations is explored in the framework of the Landau-Ginzburg model. Chapter 4 is concerned with the theory of dense gases and liquids. Many of the techniques commonly used in the theory of liquids have a long history and are well described in other texts.

Nevertheless, we feel that they are sufficiently important that we could not omit them. The traditional method of viral expansions is presented and we emphasize the important role played in both theory and experiment by the pair correlation function. We briefly describe some of the useful and still popular integral equation methods based on the Ornstein-Zernike equation used to calculate this function as well as the modern perturbation theories of liquids.

Simulation methods Monte Carlo and molecular dynamics are introduced. In the final section of the chapter we present an interesting application of mean field theory, namely the van der Waals theory of the liquid-vapor interface and a simple model of roughening of this interface due to capillary waves.

Chapters 5 and 6 are devoted to continuous phase transitions and crit- ical phenomena. In Chapter 5 we review the Onsager solution of the two- dimensional Ising model on the square lattice and continue with a description of the series expansion methods, which were historically very important in the theory of critical phenomena. We formulate the scaling theory of phase tran- sitions following the ideas of Kadanoff, introduce the concept of universality of critical behavior, and conclude with a mainly qualitative discussion of the Kosterlitz-Thouless theory of phase transitions in two-dimensional systems with continuous symmetry.

Chapter 6 is entirely concerned with the renormalization group approach to phase transitions. The ideas are introduced by means of technically straight- forward calculations for the one- and two-dimensional Ising models. We discuss the role of the fixed points of renormalization transformations and show how the theory leads to universal critical behavior. The original e-expansion of Wilson and Fisher is also discussed. This section is rather detailed, as we have attempted to make it accessible to students without a background in field theory.

In Chapter 7 we turn to quantum fluids and discuss the ideal Bose gas, the weakly interacting Bose gas, the BCS theory of superconductivity, and the phe- nomenological Landau-Ginzburg theory of superconductivity. Our treatment of these topics except for the ideal Bose gas is very much in the spirit of mean field theory and provides more challenging applications of the formalism developed in Chapter 3.

Preface xiii Chapter 8 is devoted to linear response theory. The fluctuation-dissipation theorem, the Kubo formalism, and the Onsager relations for transport coef- ficients are discussed. This chapter is consistent with our emphasis on equi- librium phenomena — in the linear response approximation the centfal role is played by equilibrium correlation functions.

A number of applications of the formalism, such as the dielectric response of an electron gas, the elementary excitations of a Heisenberg ferromagnet, and the excitation spectrum of an interacting Bose fluid, are discussed in detail. The complementary approach to transport via the linearized Boltzmann equation is also presented.

Chapter 9 provides an introduction to the physics of disordered materials. We discuss the effect of disorder on the quantum states of a system and in- troduce as an example the notion of localization of electronic states by an explicit calculation for a one-dimensional model.

Percolation theory is intro- duced and its analogy to thermal phase transitions is elucidated. The nature of phase transitions in disordered materials is discussed and we conclude with a very brief and qualitative description of the glass and spin-glass transitions. In compensation, we have provided a more extensive list of references to recent articles on these topics than elsewhere in the book.

We have found the material presented here suitable for an introductory graduate course, or with some selectivity, for a senior undergraduate course. A student with a previous course in statistical mechanics, some background in quantum mechanics, and preferably, some exposure to solid state physics should be adequately prepared.

The notation of second quantization is used extensively in the latter part of the book and the formalism is developed in de- tail in the Appendix. The instructor should be forewarned that although some of the problems, particularly in the early chapters, are quite straightforward, those toward the end of the book can be rather challenging.

Much of this book deals with topics on which there is a great deal of recent research. For this reason we have found it necessary to give a large number of references to journal articles. Whenever possible, we have referred to recent review articles rather than to the original sources. The writing of this book has been an ongoing frequently interrupted pro- cess for a number of years.

We have benefited from discussion with, and critical comments from, a number of our colleagues. Our students Dan Ciarniello, Victor Finberg and Barbara Frisken have also helped to decrease the number of errors, ambiguities, and obscurities. The responsibility for the remaining faults rests entirely with the authors. Michael Plischke Birger Bergersen Preface to the Second Edition During the five years that have passed since the first edition of this book was published, we have received numerous helpful suggestions from friends and colleagues both at our own institutions and at others.

As well, the field of statistical mechanics had continued to evolve. In composing this second edition we have attempted to take all of this into account. The purpose of the book remains the same: to provide an introduction to state-of-the-art techniques in statistical physics for graduate students in physics, chemistry and materials science. While the general structure of the second edition is very similar to that of the first edition, there are a number of important additions.

The rather ab- breviated treatment of computer simulations has been expanded considerably and now forms a separate Chapter 7. We have included an introduction to density-functional methods in the chapter on classical liquids. We have added an entirely new Chapter 8 on polymers and membranes. In the discussion of critical phenomena, we have corrected an important omission of the first edition and have added sections on finite-size scaling and phenomenological renormalization group.

Finally, we have considerably expanded the discussion of spin-glasses and have also added a number of new problems. We have also compiled a solution manual which is available from the publisher. It goes without saying that we have corrected those errors of the first edition that we are aware of.

In this task we have been greatly helped by a number of individuals. It com- plements Chapter 2, in which the connection between thermodynamics and statistical mechanical ensembles is established. The outline of the present chapter is as follows. In Section 1. Section 1. The Gibbs-Duhem equation and a number of useful Maxwell relations are derived in Section 1.

Review of Thermodynamics 1. Thermodynamics thus concerns itself with the relation between a small number of variables which are sufficient to describe the bulk behavior of the system in question.

In the case of a gas or liquid the appropriate variables are the pressure P, volume V, and temperature T. In the case of a magnetic solid the appropriate variables are the magnetic field H, the magnetization M, and the temperature T.

In more complicated situations, such as when a liquid is in contact with its vapor, more variables are needed: the pressure P, temperature T, volume of liquid and gas Vz, Ve, interfacial area A, and surface tension o. If the thermodynamic variables are independent of time, the system is said to be in a steady state. If, moreover, there are no macroscopic currents in the system, such as a flow of heat or particles through the material, the system is in equilibrium.

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