com PHASETRAMSITIOMS A Brief Account with Modern Applications www.com This page intentionally left blank www.com PHASETRAMSITIOMS A Brief Account with Modern Applications Moshe Gitterrnan Vivian (Hairn) Halpern Bar-Ilan University, Israel r pWorld Scientific N E W JERSEY LONDON SINGAPORE BElJlNG SHANGHAI HONG KONG TAIPEI CHENNAI www.com Published by World Scientific Publishing Co. 5 Toh Tuck Link, Singapore 596224 USA office: 27 Warren Street, Suite 401–402, Hackensack, NJ 07601 UK office: 57 Shelton Street, Covent Garden, London WC2H 9HE British Library Cataloguing-in-Publication Data A catalogue record for this book is available from the British Library. PHASE TRANSITIONS A Brief Account with Modern Applications Copyright © 2004 by World Scientific Publishing Co. All rights reserved.
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June 25, 2004 14:20 WSPC/Book Trim Size for 9in x 6in fm www.com Contents Preface ix 1. Phases and Phase Transitions 1 1.1 Classification of Phase Transitions .2 Appearance of a Second Order Phase Transition. The Ising Model 13 2. Mean Field Theory 25 3.1 Landau Mean Field Theory .2 First Order Phase Transitions in Landau Theory .3 Landau Theory Supplemented with Fluctuations.
36 v June 25, 2004 14:20 WSPC/Book Trim Size for 9in x 6in fm www.com vi Phase Transition 4.1 Relations Between Thermodynamic Critical Indices. The Renormalization Group 49 5.1 Fixed Points of a Map .2 Basic Idea of the Renormalization Group .3 RG: 1D Ising Model .4 RG: 2D Ising Model for the Square Lattice (1) .5 RG: 2D Ising Model for the Square Lattice (2). Phase Transitions in Quantum Systems 63 6.1 Symmetry of the Wave Function .2 Exchange Interactions of Fermions .3 Quantum Statistical Physics .5 Bose–Einstein Condensation of Atoms .7 High Temperature (High-Tc ) Superconductors .1 Heisenberg Ferromagnet and Related Models .2 Many-Spin Interactions .3 Gaussian and Spherical Models .6 Interactions Between Vortices .7 Vortices in Superfluids and Superconductors. 96 June 25, 2004 14:20 WSPC/Book Trim Size for 9in x 6in fm www.com Contents vii 8.
Random and Small World Systems 99 8.2 Ising Model with Random Interactions .4 Small World Systems .6 Phase Transitions in Small World Systems. Self-Organized Criticality 113 9.1 Power Law Distributions .3 Distribution of Links in Networks .4 Dynamics of Networks .5 Mean Field Analysis of Networks .6 Hubs in Scale-Free Networks. 128 Bibliography 129 Index 133 www.com This page intentionally left blank June 25, 2004 14:20 WSPC/Book Trim Size for 9in x 6in fm www.com Preface This book is based on a short graduate course given by one of us (M.G) at New York University and at Bar-Ilan University, Israel. The decision to publish these lectures as a book was made, after some doubts, for the following reason.
The theory of phase transi- tions, with excellent agreement between theory and experiment, was developed some forty years ago culminating in Wilson’s Nobel prize and the Wolf prize awarded to Kadanoff, Fisher and Wilson. In spite of this, new books on phase transitions appear each year, and each of them starts with the justification of the need for an additional book. Following this tradition we would like to underline two main features that distinguish this book from its predecessors. Firstly, in addition to the five pillars of the modern theory of phase transitions (Ising model, mean field, scaling, renormalization group and universality) described in Chapters 2–5 and in Chapter 7, we have tried to describe somewhat more extensively those problems which are of major interest in modern statistical mechanics.
Thus, in Chapter 6 we consider the superfluidity of helium and its connec- tion with the Bose–Einstein condensation of alkali atoms, and also the general theory of superconductivity and its relation to the high temperature superconductors, while in Chapter 7 we treat the x–y model associated with the theory of vortices in superconductors. The short description of percolation and of spin glasses in Chapter 8 is complemented by the presentation of the small world phenomena, which also involve short and long range order. Finally, we consider in Chapter 9 the applications of critical phenomena to self-organized ix June 25, 2004 14:20 WSPC/Book Trim Size for 9in x 6in fm www.com x Phase Transition criticality in scale-free non-equilibrium systems. While each of these topics has been treated individually and in much greater detail in different books, we feel that there is a lot to be gained by presenting them all together in a more elementary treatment which emphasizes the connection between them.
In line with this attempt to combine the traditional, well-established issues with the recently published and not yet so widely known and more tentative topics, our fairly short list of references consists of two clearly distinguishable parts, one related to the classical theory of the sixties and seventies and the other to the developments in the past few years. In the index, we only list the pages where a topic is discussed in some detail, and if the discussion extends over more than one page then only the first page is listed. We hope that simplicity and brevity are the second characteris- tic property of this book. We tried to avoid those problems which require a deep knowledge of specialized topics in physics and math- ematics, and where this was unavoidable we brought the necessary details in the text.
It is desirable these days that every scientist or engineer should be able to follow the new wide-ranging applications of statistical mechanics in science, economics and sociology. Accord- ingly, we hope that this short exposition of the modern theory of phase transitions could usefully be a part of a course on statistical physics for chemists, biologists or engineers who have a basic knowl- edge of mathematics, statistical mechanics and quantum mechanics. Our book provides a basis for understanding current publications on these topics in scientific periodicals. In addition, although students of physics who intend to do their own research will need more basic material than is presented here, this book should provide them with a useful introduction to the subject and overview of it.
Mosh Gitterman & Vivian (Haim) Halpern January 2004 June 25, 2004 14:17 WSPC/Book Trim Size for 9in x 6in chap01 www.com Chapter 1 Phases and Phase Transitions In discussing phase transitions, the first thing that we have to do is to define a phase. This is a concept from thermodynamics and statistical mechanics, where a phase is defined as a homogeneous system. As a simple example, let us consider instant coffee. This consists of coffee powder dissolved in water, and after stirring it we have a homogeneous mixture, i.
If we add to a cup of coffee a spoonful of sugar and stir it well, we still have a single phase — sweet coffee. However, if we add ten spoonfuls of sugar, then the contents of the cup will no longer be homogeneous, but rather a mixture of two homogeneous systems or phases, sweet liquid coffee on top and coffee-flavored wet sugar at the bottom. In the above example, we obtained two different phases by chang- ing the composition of the system. However, the more usual type of phase transition, and the one that we will consider mostly in this book, is when a single system changes its phase as a result of a change in the external conditions, such as temperature, pressure, or an external magnetic or electric field.
The most familiar example from everyday life is water. At room temperature and normal atmo- spheric pressure this is a liquid, but if its temperature is reduced to below 0◦ C it will change into ice, a solid, while if its temperature is raised to above 100◦ C it will change into steam, a gas. As one varies both the temperature and pressure, one finds a line of points in the pressure–temperature diagram, Fig.1, along which two phases can exist in equilibrium, and this is called the coexistence curve. 1 June 25, 2004 14:17 WSPC/Book Trim Size for 9in x 6in chap01 www.com 2 Phase Transitions P B Liquid 1 2 Solid Vapor A T Fig.1 The phase diagram for water.
We now consider in more detail the change of phase when water boils, in order to show how to characterize the different phases, instead of just using the terms solid, liquid or gas. Let us examine the density ρ(T ) of the system as a function of the temperature T. The type of phase transition that occurs depends on the experimen- tal conditions. If the temperature is raised at a constant pressure of 1 atmosphere (thermodynamic path 2 in Fig.1), then initially the density is close to 1 g/cm3 , and when the system reaches the phase transition line (at the temperature of 100◦ C) a second (vapor) phase appears with a much lower density, of order 0.001 g/cm3 , and the two phases coexist.
After crossing this line, the system fully transforms into the vapor phase. This type of phase transition, with a disconti- nuity in the density, is called a first order phase transition, because the density is the first derivative of the thermodynamic potential. However, if both the temperature and pressure are changed so that the system remains on the coexistence curve AB (thermodynamic path 1 in Fig.1), one has a two-phase system all along the path until the critical point B (Tc = 374◦ C, pc = 220 atm.) is reached, when the system transforms into a single (“fluid”) phase. The criti- cal point is the end-point of the coexistence curve, and one expects some anomalous behavior at such a point.
This type of phase transi- tion is called a second order one, because at the critical point B the density is continuous and only a second derivative of the thermody- namic potential, the thermal expansion coefficient, behaves anoma- lously. Anomalies in thermodynamical quantities are the hallmarks of a phase transition. June 25, 2004 14:17 WSPC/Book Trim Size for 9in x 6in chap01 www.com Phases and Phase Transitions 3 Phase transitions, of which the above is just an everyday exam- ple, occur in a wide variety of conditions and systems, including some in fields such as economics and sociology in which they have only recently been recognized as such. The paradigm for such tran- sitions, because of its conceptual simplicity, is the paramagnetic– ferromagnetic transition in magnetic systems.
These systems consist of magnetic moments which at high temperatures point in random directions, so that the system has no net magnetic moment. As the system is cooled, a critical temperature is reached at which the moments start to align themselves parallel to each other, so that the system acquires a net magnetic moment (at least in the presence of a weak magnetic field which defines a preferred direction). This can be called an order–disorder phase transition, since below this crit- ical temperature the moments are ordered while above it they are disordered, i., the phase transition is accompanied by symmetry breaking. Another example of such a phase transition is provided by binary systems consisting of equal numbers of two types of particle, A and B.
For instance, in a binary metal alloy with attractive forces between atoms of different type, the atoms are situated at the sites of a crystal lattice, and at high temperatures the A and B atoms will be randomly distributed among these sites.