Stealing the Gold: A Celebration of the Pioneering Physics of Sam Edwards

Stealing the Gold: A Celebration of the Pioneering Physics of Sam Edwards

PAUL M. GOLDBART NIGEL GOLDENFELD DAVID SHERRINGTON

OXFORD UNIVERSITY PRESS

THE I N T E R N A T I O N A L SERIES OF MONOGRAPHS ON PHYSICS

SERIES E D I T O R S J. B I R M A N S.F.EDWARDS R. FRIEND M. REES D. S H E R R I N G T O N G. V E N E Z I A N O

CITY UNIVERSITY OF NEW YORK UNIVERSITY OF C A M B R I D G E UNIVERSITY OF C A M B R I D G E UNIVERSITY OF C A M B R I D G E UNIVERSITY OF OXFORD C E R N , GENEVA

Stealing the Gold A celebration of the pioneering physics of Sam Edwards

PAUL M. G O L D B A R T

and NIGEL GOLDENFELD Department of Physics, University of Illinois at Urbana-Champaign DAVID S H E R R I N G T O N Department of Physics, University of Oxford

C L A R E N D O N PRESS • OXFORD 2004

OXFORD UNIVERSITY PRESS

Great Clarendon Street, Oxford OX2 6DP Oxford University Press is a department of the University of Oxford. It furthers the University's objective of excellence in research, scholarship, and education by publishing worldwide in Oxford New York Auckland Bangkok Buenos Aires Cape Town Chennai Dar es Salaam Delhi Hong Kong Istanbul Karachi Kolkata Kuala Lumpur Madrid Melbourne Mexico City Mumbai Nairobi Sao Paulo Shanghai Taipei Tokyo Toronto Oxford is a registered trade mark of Oxford University Press in the UK and in certain other countries Published in the United States by Oxford University Press Inc., New York © Oxford University Press 2005 The moral rights of the authors have been asserted Database right Oxford University Press (maker) First published 2005 All rights reserved. No part of this publication may be reproduced, stored in a retrieval system, or transmitted, in any form or by any means, without the prior permission in writing of Oxford University Press, or as expressly permitted by law, or under terms agreed with the appropriate reprographics rights organization. Enquiries concerning reproduction outside the scope of the above should be sent to the Rights Department, Oxford University Press, at the address above You must not circulate this book in any other binding or cover and you must impose this same condition on any acquirer A catalogue record for this title is available from the British Library Data available Library of Congress Cataloging in Publication Data (Data available) ISBN 0-19-852853-1 (hbk) 10 9 8 7 6 5 4 3 2 1 Typeset by Newgen Imaging Systems (P) Ltd., Chennai, India Printed in Great Britain on acid-free paper by Biddies Ltd., King's Lynn

CONTENTS

Papers of Sir Sam Edwards reprinted, with permission, in this volume Contributing Authors

xi xiii

Foreword P.-G. de Gennes

1

Preface

3

1 Reprint: A new method for the evaluation of electric conductivity in metals S. F. Edwards

10

2 Impurity diagrammatics and the physics of disordered metals D. Khmelnitskii References

23 28

3 Reprint: The statistical dynamics of homogeneous turbulence S. F. Edwards

30

4 Sam Edwards and the turbulence theory K. R. Sreenivasan and G. L. Eyink

66

4.1 Introduction 4.2 Contributions of Edwards 4.3 The white-noise passive scalar model 4.4 Navier-Stokes turbulence 4.5 Conclusion References

66 67 73 77 79 80

5 Reprint: The statistical mechanics of polymers with excluded volume S. F. Edwards 6 The entry of field theory into polymer science R. C. Ball 6.1 Background 6.2 The new polymer theory, and excluded volume

86 99 99 100

Contents

vi 6.3 Polymer solutions 6.4 Mathematical aspects 6.5 Future directions References

7 Reprint: The theory of polymer solutions at intermediate concentration S. F. Edwards 8 The coarse grained approach in polymer physics Y. Oono and T. Ohta 8.1 Introduction 8.2 Edwards model and cutoff 8.3 Edwards goes beyond dilute solutions 8.4 Semidilute solutions 8.5 Block copolymer melts 8.6 Some reflections on models References 9 Reprint: Statistical mechanics with topological constraints: II S. F. Edwards 10 Notes on 'Statistical mechanics with topological constraints: I & II' E. Witten References 11 Reprint: Theory of spin glasses S. F. Edwards and P. W. Anderson

101 102 104 106

108 125 125 125 127 131 134 139 142

144 159 162 164

12 Remarks on the Edwards Anderson paper P. W. Anderson References

175

13 Edwards Anderson: Opening up the world of complexity D. Sherrington

179

13.1 Introduction 13.2 Edwards-Anderson spin glass and beyond 13.2.1 Edwards-Anderson 13.2.2 Exactly soluble model 13.2.3 Beyond spin glasses 13.2.4 Dynamics

177

179 181 181 181 183 185

Contents

vii

13.2.5 Ed wards-Anderson again 13.2.6 Mathematics 13.3 Concluding Remarks References

187 188 188 189

14 The overlap in glassy systems

192

G. Parisi

14.1 Introduction 14.2 The original definition of the overlap 14.2.1 Only one state 14.2.2 Many states 14.2.3 A soluble model 14.3 The thermodynamic definition of the overlap 14.4 The two susceptibilities 14.5 Virtual probabilities 14.5.1 General considerations 14.5.2 A first attempt 14.5.3 Generalized susceptibilities 14.5.4 Local overlap 14.6 Fluctuation-dissipation relations 14.6.1 The global fluctuation-dissipation relations 14.6.2 The local fluctuation-dissipation relations 14.7 Conclusions References 15 Theory of random solid states M. Mezard 15.1

A few landmarks 15.1.1 Structural glasses 15.1.2 From rubber to spin glass and proteins 15.1.3 Networks of interacting individuals: global equilibrium 15.1.4 Networks of interacting individuals: dynamics 15.2 Tools and concepts 15.2.1 Statistical description 15.2.2 Physics without symmetry: equilibrium 15.2.3 Replicas 15.2.4 Physics without symmetry: dynamics 15.2.5 Simulations 15.3 Directions 15.3.1 Physical glasses 15.3.2 Random systems 15.3.3 The unreasonable inefficiency of mathematics 15.3.4 Consilience References

192 193 193 195 197 198 199 202 202 202 203 205 207 207 208 209 210 212 212 212 214 216 219 221 221 223 224 227 229 229 230 231 232 233 235

Contents

viii 16

17

18

19

20

21

Reprint: The theory of rubber elasticity R. T. Deam and S. F. Edwards

237

Sam Edwards and the statistical mechanics of rubber P. M. Goldbart and N. Goldenfeld

275

17.1 Introduction 17.2 Edwards' formulation of the statistical mechanics of vulcanized macromolecular systems 17.2.1 Idealized model 17.2.2 Quenched disorder: treating the cross-links statistically 17.2.3 Handling the quenched disorder via replicas 17.2.4 Modelling the statistics of the cross-links 17.2.5 Effective pure theory of coupled replicas 17.3 Predictions of the Deam-Edwards theory 17.4 Nature of the vulcanization transition 17.5 The emergent amorphous solid state 17.5.1 Microscopic character 17.5.2 Macroscopic character 17.5.3 Goldstone fluctuations; low dimensions 17.6 Ongoing directions: Dynamics at the liquid to solid transition 17.7 Concluding remarks References

275

Reprint: Dynamics of concentrated polymer systems Part 2.—Molecular motion under flow M. Doi and S. F. Edwards The Doi Edwards theory W. W. Graessley and T. C. B. McLeish References Reprint: The surface statistics of a granular aggregate S. F. Edwards and D. R. Wilkinson

280 280 282 282 284 286 287 288 291 291 293 294 295 296 297

300 318 326

328

The surface statistics of a growing aggregate M. Kardar

344

21.1

344 344 345 346

The Edwards-Wilkinson equation 21.1.1 Derivation 21.1.2 Results 21.1.3 Numerical simulations

Contents 21.2

22

23

24

ix

The Kardar–Parisi–Zhang equation 21.2.1 Derivation 21.2.2 Scaling behaviour in one dimension 21.2.3 Conservative growth 21.2.4 Experiments 21.3 Directed paths in random media 21.3.1 The Cole–Hopf transformation 21.3.2 Directed polymers 21.3.3 The replica approach 21.3.4 Many directed polymers 21.4 Perspectives 21.4.1 Sequence alignment 21.4.2 Textural growth References

347 347 349 350 351 352 352 354 355 357 359 359 360 361

Reprint: Theory of powders S. F. Edwards and R. B. S. Oakeshott

363

Building a thermodynamics on sand J. Kurchan

375

23.1 Introduction 23.2 Compact granular matter is an athermal glass. 'Granularizing' simple glass models 23.3 The assumption 23.4 Caveat I. Flat distributions are not generic out of equilibrium 23.5 Caveat II. Convection currents, shear bands, inhomogeneities, insufficient relaxation 23.6 Encouraging news from the analytic front 23.6.1 Closure approximations 23.6.2 The athermal situation 23.6.3 Intrinsic limitations of the approach 23.7 Towards realistic models and experiment 23.7.1 Effective temperatures 23.7.2 Tests of the flat measure hypothesis 23.8 Inherent structures 23.9 Counter-examples 23.10 Conclusions References

375

379 380 380 383 384 384 385 385 386 387 388 388

Reprint: The transmission of stress in an aggregate S. F. Edwards and R. B. S. Oakeshott

391

377 378 378

Contents

x

25

Granular media: Three seminal ideas of Sir Sam

397

J.-P. Bouchaud and M. E. Gates

25.1 25.2

Introduction Statistical mechanics ol granular matter 25.2.1 Force chains and arching granular assemblies 25.2.2 Marginal coordination of granular packings 25.2.3 Thermodynamics without temperature: the Edwards ensemble 25.3 Some related developments 25.3.1 The discrete scalar model 25.3.2 Continuum closure schemes for granular stresses 25.3.3 Slow compaction and dynamics 25.4 Concluding remarks References

397 399 399 400 401 403 404 406 410 411 412

Chapters on the Edwardsian approach to research

417

26 The case for Edwardsian research in solid mechanics: A sermon

419

J. S. Langer

References

426

27 A scientist for all seasons G. Allen

428

Editors' acknowledgements

437

Index

439

PAPERS OF SIR SAM EDWARDS REPRINTED, WITH PERMISSION, IN THIS VOLUME A new method for the evaluation of electric conductivity in metals

Edwards, S. F. Philosphical Magazine, 3, 1020-1031 (1958). Reproduced by permission of Taylor and Francis Group. The journal's web site can be found via: http://www.tandf.co.uk Appears in this volume on pages 10-22. The statistical dynamics of homogeneous turbulence

Edwards, S. F. Journal of Fluid Mechanics, 18; 239-273 (1964). Reproduced with the permission of Cambridge University Press. Appears in this volume on pages 30-65. The statistical mechanics of polymers with excluded volume

Edwards, S. F. Proceedings of the Physical Society, 85; 613-624 (1965). Reproduced by permission of The Institute of Physics.1 Appears in this volume on pages 86-98. The theory of polymer solutions at intermediate concentration

Edwards, S. F. Proceedings of the Physical Society, 88; 265-280 (1966). Reproduced by permission of The Institute of Physics.1 Appears in this volume on pages 108-124. Statistical mechanics with topological constraints: II

Edwards, S. F. Journal of Physics A: General Physics, 1, 15-28 (1968). Reproduced by permission of The Institute of Physics.1 Appears in this volume on pages 144-158. Theory of spin glasses

Edwards, S. F. and Anderson, P. W. Journal of Physics F: Metal Physics, 5, 965-974 (1975). Reproduced by permission of The Institute of Physics.1 Appears in this volume on pages 164-174. 1 An archive of Institute of Physics journals can be found via the Institute's web site: http://www.iop.org

xii

Papers of Sir Sam Edwards

The theory of rubber elasticity Deam, R. T. and Edwards, S. F. Philosophical Transactions of The Royal Society of London. Series A, Mathematical and Physical Sciences, 280, 317-353 (1976). Reproduced by permission of The Royal Society. Appears in this volume on pages 237-274. Dynamics of concentrated polymer systems Part 2.—Molecular motion under flow Doi, M. and Edwards, S. F. Journal of the Chemical Society, Faraday Transactions II, 74, 1802-1817 (1978). Reproduced by permission of The Royal Society of Chemistry. Appears in this volume on pages 300-317. The surface statistics of a granular aggregate Edwards, S. F. and Wilkinson, D. R. Proceedings of The Royal Society of London. Series A, Mathematical and Physical Sciences, 381, 17-31 (1982). Reproduced by permission of The Royal Society. Appears in this volume on pages 328-343. Theory of powders Edwards, S. F. and Oakeshott, R. B. S. Physica A: Statistical and Theoretical Physics, 157, 1080-1090 (1989). Reprinted with permission from Elsevier. Appears in this volume on pages 363-374. The transmission of stress in an aggregate Edwards, S. F. and Oakeshott, R. B. S. Physica D: Nonlinear Phenomena, 38, 88-92 (1989). Reprinted with permission from Elsevier. Appears in this volume on pages 391-396.

CONTRIBUTING AUTHORS Geoffrey Allen University of East Anglia, Norwich NR4 7TJ, U.K. Philip W. Anderson Department of Physics, Princeton University, Princeton, New Jersey 08544, U.S.A. Robin C. Ball Department of Physics, University of Warwick, Coventry CV4 7AL, U.K. Jean-Philippe Bouchaud Service de Physique de 1'Etat Condense, CEA, Orme des Merisiers, 91191 Gif-sur-Yvette CEDEX, France Michael Gates School of Physics, University of Edinburgh, JCMB King's Buildings, Mayfield Road, Edinburgh EH9 3JZ, Great Britain Pierre-Gilles de Gennes Institut Curie Recherche, 11 rue Pierre et Marie Curie, 75231 Paris CEDEX 05, France Gregory L. Eyink Department of Applied Mathematics and Statistics, The Johns Hopkins University, 3400 North Charles Street, Baltimore, Maryland 21218-2682, U.S.A. Paul M. Goldbart Department of Physics, University of Illinois at Urbana-Champaign, 1110 West Green Street, Urbana, Illinois 61801-3080, U.S.A. Nigel Goldenfeld Department of Physics, University of Illinois at Urbana-Champaign, 1110 West Green Street, Urbana, Illinois 61801-3080, U.S.A. Wlliani W. Graessley Princeton University; 7496 Old Channel Trail, Montague, Michigan 49437, U.S.A. David Khmelnitskii Cavendish Laboratory, University of Cambridge, Cambridge CB3 OHE, U.K.

xiv

Contributing Authors

Mehran Kardar

Physics Department, Massachusetts Institute of Technology, Cambridge, Massachusetts 02139, U.S.A. Jorge Kurchan

Laboratoire de Physique et Mechanique des Milieux Heterogenes, Ecole Superieure de Physique et Chimie Industrielles, 10 rue Vauquelin, 75231 Paris CEDEX 05, France James S. Langer

Department of Physics, University of California, Santa Barbara, California 93106, U.S.A. Thomas C. B. McLeish

IRC in Polymer Science and Technology, Polymers and Complex Fluids, Department of Physics and Astronomy, University of Leeds, Leeds LS2 9JT, U.K. Marc Mezard

CNRS, Laboratoire de Physique Theorique et Modeles Statistiques, Universite Paris Sud, Bat. 100, 91405 Orsay CEDEX, France Takao Ohta

Yukawa Institute for Theoretical Physics, Kyoto University, Kyoto 606-8502, Japan Yoshitsugu Oono

Department of Physics, University of Illinois at Urbana-Champaign, 1110 West Green Street, Urbana, Illinois 61801-3080, U.S.A. Giorgio Parisi

Dipartimento di Fisica, INFM, SMC and INFN, Universita di Roma La Sapienza, P. A. Moro 2, 00185 Rome, Italy David Sherrington

Rudolf Peierls Centre for Theoretical Physics, University of Oxford, 1 Keble Road, Oxford OX1 3NP, U.K. Katepalli R. Sreenivasan

Abdus Salam International Centre for Theoretical Physics, Strada Costiera 11, 34100 Trieste, Italy Edward Witten

School of Natural Sciences, Institute for Advanced Study, Einstein Drive, Princeton, New Jersey 08540, U.S.A.

FOREWORD

From silex to plastics our species has struggled with many forms of condensed matter. But this study has been through a special time, in the 20th century, after the explosive growths of quantum mechanics and statistical physics. Among the pioneers, I usually quote four names to my students: John Bardeen, Phil Anderson, Leo Kadanoff, and Sam Edwards. Sam is a fruitful example. He has been driven (I think) by two distinct passions: a passion for architecture, and a passion for unicorns. Architecture. The aim is to inject reality inside a formal construction. Sam started with electrons in nearly pure metals (with a few impurities providing collisions). By accident I heard of this problem in pre-Edwardsian times (~1953) from talks by Rudolf Peierls. Peierls carefully pointed out the many unclear assumptions which were required to build up a Boltzmann equation: the whole approach was then not much more than a guess. But, soon after, Sam produced a diagrammatic study, selecting the diagrams which were important at low impurity concentration. This gave an elegant (and rigorous) derivation of the transport equation. There was more to come. A few years later, Abrikosov and Gorkov, using the same tool, were able to give a precise description of 'dirty superconductors.' Adding non-magnetic impurities to a metal decreased the coherence length and increased the penetration depth. Magnetic impurities had a more wicked effect— killing the superfluidity for s-wave pairs. Sam had been (I presume) fond of the Feynman diagrammatic formalism. But choosing electrons in metals, he had given a certain flesh to the statue, and initiated a new form of sculpture. Something similar happened later with path integrals. Through his friends (Allen, Gee,...) Sam heard about linear, flexible polymer chains. And he rapidly realised that their statistical properties could be expressed in terms of path integrals. He then came up with an analogy between a chain conformation and the trajectory of a (non-relativistic) quantum mechanical particle. The analogue of time, here, is an index n labelling the successive monomers along the chain. The analogue of a quantum mechanical propagator is a statistical weight. And in both systems the probability of presence is proportional to the square of an amplitude. When the polymer chain is faced with an external potential U(r), its statistical weight is ruled by a Schrodinger equation, where the potential comes in exactly as it does for the quantum mechanical analogue! This had a considerable impact on polymer science. All of a sudden, 50 years of theoretical experience in quantum mechanics could be transposed to polymer chains... And again the fruits were multiple. Some were obvious from the start

2

Foreword

(e.g. the concept of self-consistent fields) and some were more delicate. Here I personally think of three items, all due to Edwards a) the idea of a screening length in polymer solutions, b) the statistics of ideal chains faced with a topological constraint (e.g. a cyclic polymer making a prescribed number of turns around a rigid rod) c) the effects of internal cross-links in a chain, and their similarity with poor solvent conditions. Unicorns. Feynman diagrams and path integrals are typical examples of 'architectures' which were transformed into living cities by Edwards. But he also has another passion, which I call 'The search for unicorns.' To chase unicorns is a delicate enterprise. Medieval Britons practised it with great enthusiasm (and this still holds up to now: read Harry Potter). Sir Samuel is not far from the gallant knights of the twelfth century. Discovering a strange animal, approaching it without fear, then not necessarily harnessing the creature, but rapidly drawing a plausible sketch of its main features. One beautiful unicorn prancing in the magic garden of Physics has been named 'Spin glass.' It is rare: not many pure breeds of Spin glasses have been found in Nature. But we have all watched the unpredictable jumps of this beast. And we have loved its story—initiated by Edwards and Anderson. Let me hop to another image. A number of difficult objects from condensed matter physics have undergone what we might call 'Edwards ripening' (not to be confused with Ostwald ripening): spin glasses and granular matter being two major examples, where the pre-Edwards questions are drastically different from the post-Edwards questions. What are the ingredients of this success? I do not know. But what I find unique in the Edwards case is a certain combination of abstract thought and Welsh common sense. My students learn a lot from reading Edwards. P.-G. de Gennes January 2003

PREFACE

The collection of trails blazed by Sam Edwards during half a century of fundamental research in theoretical physics is truly astonishing. Sam has led theoretical physics into uncharted territories—from his roots in quantum field theory— beginning with his seminal work on the transport properties of disordered metals, and continuing to the present day with his ground-breaking efforts to create a statistical mechanics of granular materials. Along the way, he and his collaborators developed the first modern theory of polymers in solution and in the rubbery state; created and explored the tube concept, which has had momentous implications for understanding the viscoelasticity of polymer melts; formulated the spin-glass problem and provided its first solutions using the method of replicas— work that has had profound implications in areas as diverse as combinatorial optimization, neural networks, as well as glassy systems; made important contributions to the still-unsolved problem of Navier-Stokes turbulence; and initiated the recent explosion of activity in the dynamics of growing interfaces. The span of his intellectual landscape is so vast that researchers in distinct subfields of condensed matter physics may be forgiven for not realizing that their Edwardses are one and the same. Sam's impact has been so large for many reasons. Not only are there the diversity and significance of the issues tackled, the originality of the approaches taken, the conceptual advances furnished, and the technical power brought to bear. But also there are Sam's instincts for identifying fertile ground and artfully selecting—or, when necessary, inventing—precisely the right tools. Sam doesn't simply open up doors into new vistas for theoretical physics—he kicks them down! As an example of his influence, consider the historical development of the field that came to be known as soft condensed matter physics. While much of the theoretical physics of the 1960s was focused on high-energy and solid state physics, Sam's work opened up new directions in which physicists would focus on strongly fluctuating matter made up of constituents with non-trivial internal degrees of freedom. For polymers, this means the random character of chain-like objects, but Sam quickly turned his attention to networks, rodlike molecules, polyelectrolytes etc.: indeed, the field of soft condensed matter physics, or complex fluids, finds its precursors in Sam's works from the 1960s and 1970s. Sam was one of the first—along with Pierre-Gilles de Gennes—to realize the richness of systems that possess a coupling between internal modes (such as polymer chain conformations) and conventional ordering in space. Soft condensed matter physics has grown into an essential part of modern-day physics, and its scope has grown from polymers and liquid crystals to encompass such diverse

4

Preface

topics as granular materials and biological media. By his leadership in building theoretically-sophisticated models of more and more complex materials, Sam has grown his chosen field from its birth as 'solid state physics' into the much deeper and richer field of condensed matter physics that it is today. Rene Magritte's paradoxical surrealist painting La Trahison des Images features a realistically drawn pipe above the handwritten words 'Ceci n'est pas une pipe' (This is not a pipe). In the spirit of Magritte's paradox, consider the volume before you: it is intended as a tribute to Sam, but it is not a Festschrift. We thought that it would be uncharacteristic of Sam to look in the rear view mirror, as it were, and celebrate the achievements of the past. Instead, we wanted to keep the focus on the future, and in particular the innovations in condensed matter that continue to the present day as his scientific legacy. As we have already touched upon, Sam tackles problems that previously had not even been recognized as being within the purview of theoretical physics. Consequently, his work has been enormously influential, as subsequent workers have struggled with—and in some cases solved—the problems that Sam has introduced and formulated for the community. Accordingly, every contributor to this volume was chosen as a representative of one of the fields that Sam has initiated or profoundly influenced. Each was requested to look forward from their field of activity and speculate about the future, as well as to document, as they saw it, the ways in which the field is flowering from the seeds that Sam planted. In addition, we present a selection of Sam's most seminal papers, each of which can justly lay claim to having paved the groundwork for what is, to this day, a thriving sub-field of condensed matter physics. Each paper is reprinted here, together with the original chapters that we commissioned. The immense scope of Sam's ceuvre means that this book provides, in one place, an unusually broad survey of the present state of some aspects of condensed matter physics. Moreover, the approach taken by the contributors means that the reader gains unparalleled insight into the historical and intellectual development of their subject. In some cases there has been an orderly and logical development of ideas; in others, to quote one of our contributors ' . . . it is impossible not to notice that all simple steps were taken with a delay, all possible mistakes were made, and the whole logic of research was often missing.' Let us now turn to the subject-matter of the book, and briefly introduce the reprinted papers together with an outline of the chapters that accompany them. A new method for the evaluation of electric conductivity in metals

Edwards, S. F. (1958). Phil. Mag., 3, 1020-1031. Although Sam's work is primarily in what is now known as soft-condensed matter theory, this influential paper calculated the Drude conductivity of an electron elastically scattered by a random potential using a diagrammatic analysis. This technique, independently and simultaneously developed by Abrikosov and Gorkov in the Soviet Union, has formed the foundation for modern theories of superconducting alloys, weak localization and mesoscopics. David Khmelnitskii

Preface

5

of Cambridge University explores the impact and consequences of this work, starting on p. 23. The statistical dynamics of homogeneous turbulence

Edwards, S. F. (1964). J. Fluid Mech., 18, 239-273. Turbulence is one of the great, unsolved problems in classical physics. Is there a limiting, universal state at sufficiently high Reynolds numbers, characterized by its correlation functions and spectral properties? How can one compute macroscopic properties of the flow, such as drag phenomena, and what is the nature of the interaction between small and large scales? In recent years, sophisticated approaches to some of these issues have been proposed, and there has been considerable progress on the variety of data that can inform theoretical research in this area. But in 1964, all this was in the future. Sam had learned from the failures of meson theory in the 1950s that perturbative field-theoretic approaches to statistical problems are worthless in a strong-coupling regime, and recognized that the statistical characterization of turbulence was another example of such a problem. The solution that he espoused—and still, arguably, the method of choice to this day—is self-consistent, or variational field theory. Much of the machinery that is now part of the standard theorist's toolkit originated in ideas contained in Sam's landmark 1964 article on turbulence. On p. 66, Katepelli Sreenivasan of the International Centre for Theoretical Physics and Gregory Eyink of the Johns Hopkins University take us on a tour of this article, and recapitulate how it has influenced recent progress in turbulence theory. The statistical mechanics of polymers with excluded volume

Edwards, S. F. (1965). Proc. Phys. Soc., 85, 613-624. In 1965, Sam Edwards formulated the problem of the single self-avoiding walk as path integration involving a Hamiltonian that now bears his name. He used pseudo-potential theory to model the excluded volume interaction with deltafunctions, and solved the theory self-consistently to obtain the dilute solution polymer size exponents. This technique has been widely and profitably used in every polymer system where thermal fluctuations allow the polymer chains to explore phase space. Although there are corrections to the results of Edwards' method due to anomalous dimensions, these are small, and subsequent renormalization group approaches to this problem have not detracted from it being the standard technique to use in the statistical physics of complex polymeric fluids. Robin Ball of Warwick University presents a thought-provoking discussion of this work, starting on p. 99. The theory of polymer solutions at intermediate concentration

Edwards, S. F. (1966). Proc. Phys. Soc., 88, 265-280. In 1966, Sam considered the behaviour of an ensemble of interacting polymers in solution, and used collective coordinates to extract the equation of state and exhibit the phenomenon of screening. The predictions of this work, that chain dimensions shrink as the concentration increases (conjectured earlier by Flory)

6

Preface

and the universal equation of state itself are in impressive agreement with experiments carried out twenty years later. This work led to the scaling theory of polymer solutions, as developed extensively by de Gennes and others. Yoshitsugu Oono of the University of Illinois at Urbana-Champaign and Takao Ohta of Kyoto University analyze the foundations and implications of this work, starting on p. 125. Statistical mechanics with topological constraints: II

Edwards, S. F. (1968). J. Phys. Al, 15-28. Sam's approach to polymer systems was to begin with dilute solutions and work his way up in concentration. Problems of topological entanglement arise in crosslinked materials, and Sam made several conceptually different attempts to find a way to overcome this profound difficulty. The earliest of these was to formulate topological entanglement as a sequence of constraints, utilizing topological invariants. The resulting field theories, it must be said, did not ultimately prove to have lasting value in polymer science for a variety of reasons, not least of which was their complexity. However, these ideas were to prove prescient, as many years later, physicists and mathematicians encountered similar mathematical challenges in quantum field theory. Edward Witten of the Institute of Advanced Study, Princeton has traced the logical continuity of these ideas from Sam's work to the present day. His article can be found on p. 159. Theory of spin glasses

Edwards, S. F. and Anderson, P. W. (1975). J. Phys. F5, 965-974. This is the first in a series of papers that formulated the spin-glass problem, first applied the replica method to a disordered magnetic system, and obtained the first mean field theory of the spin-glass state. With thousands of citations it is hard to exaggerate the importance of this paper, with its notions of ergodicitybreaking, the use of replicas, and the introduction of the Edwards-Anderson order parameter. The discovery by Giorgio Parisi of replica symmetry breaking in the infinite-range spin glass is only one of the most well-known of the numerous subsequent works that were inspired by the Edwards-Anderson paper. Spin glass concepts and techniques—even the Hamiltonian itself—have been of seminal importance not only for glassy materials but also for such diverse topics as neural networks, combinatorial optimization, and even protein folding. Several articles with different emphases accompany our reprint. Philip Anderson of Princeton University describes the genesis of the Edwards-Anderson paper, starting on p. 175. David Sherrington of the University of Oxford presents historical background to the work, emphasizes the central role of the Edwards-Anderson model as a representation of certain complex systems, and summarises the development of mean field theories of other spin glass models, starting on p. 179. Giorgio Parisi of the Universita di Roma describes the impact of the overlap order parameter and its subsequent developments, starting on p. 192, and Marc Mezard of the Centre National de Recherche Scientifique and Universite Paris Sud describes

Preface

7

the general approach to random solid states—in a sense the application of the Edwards-Anderson ideas to real glasses—starting on p. 212. The theory of rubber elasticity

Beam, R. T. and Edwards, S. F. (1976). Phil. Trans. R. Soc. Lori. A 280, 317-353. Starting in the late 1960's and continuing into the late 1970's, Sam tackled the problem of an ensemble of interacting polymers with quenched disorder in the form of permanent cross-links, i.e. a minimal model for a gel or rubber. Edwards was the first person to recognize that statistical mechanical systems with quenched disorder required special treatment, and he introduced the replica method—in the context of rubber elasticity—as a general tool for disordered systems. The landmark paper by Beam and Edwards brings to bear on this problem a host of theoretical innovations, including the replica method and early attempts to recognize and account for the effects of topological entanglements created by the cross-link constraints. This and subsequent works endeavoured to explain the elastic response of networks and the formation of the solid rubbery state itself. Paul Goldbart and Nigel Goldenfeld of the University of Illinois at Urbana-Champaign describe the Beam and Edwards paper and the current status of network theory, starting on p. 275. Dynamics of concentrated polymer systems

Boi, M. and Edwards, S. F. (1978). J. C. S. Faraday Trans. II 74, Part 1: 1789-1801, Part 2: 1802-1817, Part 3: 1818-1832; ibid. 75, Part 4: 38-54 (1979). Reprinted here is Part 2.—Molecular motion under flow.

This landmark series of papers elaborated on the notion that in a concentrated solution or melt, the effect of the surrounding polymers on a given test chain can be thought of as a confining tube, which must itself be self-consistently found from the motion of confined polymer chains. The Boi-Edwards papers calculated in complete detail the consequences for the transport coefficients describing the polymer self-diffusion and the viscosity, as well as stress-strain characteristics, damping function, etc. This tour de force and its subsequent elaborations remains the premier example of a transport constitutive equation being derived from a first-principles molecular model, and its predictions have been amply verified in considerable detail. It is no exaggeration to state that the Boi-Edwards theory is the cornerstone of modern polymer rheology, and it has been applied not only to the originally-considered case of linear polymers but comb, branched and 'H-polymers' with impressive quantitative agreement with experiment. Here, we reprint the second paper in the series, which represents for many the pivotal step in the development of the theory. William Graessley of Princeton University and Tom McLeish of the University of Leeds review the Boi-Edwards papers and report on the many subsequent developments, starting on p. 318.

8

Preface

The surface statistics of a granular aggregate

Edwards, S. F. and Wilkinson, D. R. (1982). Proc. R. Soc. Lon. A 381, 17-31. How can the random deposition of particles on to a surface give rise to the long-range correlations and self-affinity that are found in Nature and computer idealizations of surface dynamics? Edwards and Wilkinson answered this question in a much-cited and influential paper, which had as one of its broader impacts the emphasis on using symmetries and conservation laws alone to formulate the stochastic equations governing the coarse-grained dynamics. Building on this framework, Kardar, Parisi and Zhang (KPZ) showed that in two or fewer dimensions, the Edwards-Wilkinson fixed point can be unstable towards an alternative fixed point due to a nonlinear gradient term. Their work helped established a close connection between the interface problems originally considered by Edwards and Wilkinson and other problems in disordered systems: directed polymers in random media, Burgers' turbulence and even the statistics of DNA sequence alignment. In recent years, Edwards has also adapted self-consistent mode-coupling techniques from his 1964 work on Navier-Stokes turbulence to provide tractable approximations to the KPZ equations in the strong-coupling regime. Mehran Kardar of the Massachusetts Institute of Technology was one of the first to appreciate the potential of the Edwards-Wilkinson work, and provides an overview of its ramifications, starting on p. 344. Theory of powders

Edwards, S. F. and Oakeshott, R. B. S. (1989). Physica A 157, 1080-1090. By the late 1980s, Sam's interest in granular materials had moved on from deposition and formation processes to issues of the mechanics of powders. In the first of the two articles with Oakeshott reprinted here, and elsewhere, Sam turned his attention not to the mechanics but to the statistical mechanics of granular matter. Sam recognized that a dense static granular medium is in some sense a glass, leading him to make the remarkable suggestion that the principle of equal a priori probabilities governs configurations of this highly nonequilbrium system. Subsequent computer simulation work has supported this conjecture, although the precise reasons for its success are still not understood, and allowed a substantial, predictive theoretical development to occur. The way in which Sam's ideas apply to granular materials, together with a critical analysis of the thermodynamic foundations of granular material, is explained in the article by Jorge Kurchan of the Ecole Superieure de Physique et Chimie Industrielles, Paris, starting on p. 375. The transmission of stress in an aggregate

Edwards, S. F. and Oakeshott, R. B. S. (1989). Physica D 38, 88-92. Perhaps the most paradoxical obervation in powders is that the vertical normal stress at a point under a sandpile does not track the height of the sandpile above the point. On the contrary, below the highest point, the stress exhibits a minimum. In this, the second of the two articles with Oakeshott reprinted here, Sam was one of the first to put forward the notion of force chains, which link points

Preface

9

of strong contact, and to explore the consequences for the mechanics of powders. This has become an experimentally rich subject with numerous interesting phenomena, which is reviewed by Jean-Phillipe Bouchaud of the Commissariat a 1'Energie Atomique, Prance, and Michael Gates of the University of Edinburgh, Scotland, starting on p. 397. As is clear from this litany of Sam's contributions, his style is rather unique, perhaps 'making a silk purse out of a sow's ear! In other words, Sam mixes powerful theoretical ideas with a penchant for identifying physics in problems that seem to lesser mortals to be 'not worthy,' or more charitably, 'not ripe' for advanced theoretical analysis. This style has infected much of the condensed matter physics community, and none more so than his students and associates over the years. James Langer, of the University of California at Santa Barbara, himself a pioneer in the extension of physics into non-traditional areas of science, was a graduate student at Birmingham while Sam was a lecturer there. Jim has provided an object lesson in how to pursue the 'Edwardsian' style of research, using for illustration his own work on fracture and plasticity, starting on p. 419. The book concludes with a perspective on Sam's career by Sir Geoffrey Allen, Chancellor of the University of East Anglia. Sir Geoffrey was one of the earliest scientific influences on Sam, and fittingly, he has provided us with a fascinating and personal account of polymer science in the U.K. in the 1960s, as well as Sam's impact on industrial scientific research and his leadership role in the management of academic scientific research in the U.K. during the 1970s. This volume has a somewhat unusual title. So perhaps it will be no surprise for the reader to learn that it comes from Sam himself. One cold and damp winter's afternoon in Cambridge, Sam and one of us (NG) were discussing somewhat self-consciously the way in which physicists should chose research problems. For Sam there was no question about the right strategy: 'The first person into the bank vault steals the gold' was his memorable advice. He should know: Sam has been the first so many times, as this book attests. For us as editors, the task of compiling and organizing this volume has been a unique and wonderful privilege. It has given us the opportunity to revisit the many gems that Sam Edwards has as his lasting legacy to the scientific community; indeed, we were fortunate enough to suffer the embarrassment of riches, and there were a number of papers that we were tempted to reprint but resisted due to length constraints. We have enjoyed interacting with a marvelous group of authors whose scholarship, scientific vision and passion for their subject shines through in their articles. Pierre-Gilles de Gennes was one of the early supporters of this project, and we are very pleased that he has provided such a warm and interesting foreword. We take this opportunity to thank all the contributors for their hard work and valuable articles. And lastly, we thank Sam for making condensed matter physics such a fascinating discipline to practice. Paul M. Goldbart and Nigel Goldenfeld Urbana

David Sherrington Oxford

1 REPRINT A NEW METHOD FOR THE EVALUATION OF ELECTRIC CONDUCTIVITY IN METALS by S. F. Edwards Philosophical Magazine, 3, 1020-1031 (1958).

11 [

1020 ]

A New Method for the Evaluation of Electric Conductivity in Metalsj By 8. F. EDWARDS Department of Mathematical Physics, University of Birmingham [Received May 23, 1958] ABSTRACT

A method is developed which allows the evaluation of the closed formal expressions for electrical conductivity which have recently been developed by several authors. The case of a random set of scatterers is treated in detail and the formal solution made to yield directly the solution to the Boltzmann equation. A brief mention of the application of this method to liquids and alloys is made.

§ 1. INTBODUCTION RECENTLY it has been realized by several workers (Nakano 1956, Kubo 1956, Kohn and Luttinger 1957, Greenwood 1958), that the electric conductivity in, say, a metal can be written down in a closed formal expression, without going through the intermediate form of deriving a transport equation, and moreover these closed forms are exact. The usual derivation of a transport equation (cf. Peierls 1955) is rather limited in its applicability and cannot in any simple way be extended to the cases of alloys and liquids etc., and moreover, even where it is usually used, it is not at all clear (see e.g. Peierls 1955, p. 123) that there are not temperature dependent corrections which would entirely invalidate the usual solution. Now the formal exact solutions avoid all this, but carry the difficulty that they are still in a rather abstract form, and it is not clear how they are to be evaluated. This paper is concerned with the evaluation of these formulae, and will show that they can readily give the same result as the usual transport equation where the latter has been assumed to be correct, and thus dispose of the possibility of temperature dependent corrections. The use of the exact formulation in new problems will only be very briefly mentioned in this paper, and since the present object is only to illustrate the method, the simplest problem, that of the conductivity of electrons scattered by a random set of scattering centres, will be discussed. § 2. FORMULATION OF THE PEOBLEM The starting point will be the formula of Greenwood and Peierls, which states that the conductivity tensor is given by

Communicated by Professor E. E. Peierls, C.B.E., F.R.S.

12

On a New Method for the Evaluation of Electric Conductivity in Metals 1021 where vmi/ is the matrix element of velocity, / the electron distribution function, and the S function is to be understood in the sense that the limit En-+Em is taken after the system is considered so large that always there are many levels between En and Em. This form as it stands is not suitable for computation, so it is rearranged by first writing it out in full:

Introduce units so that (2m/fia)~^= 1, then

Now

is a Green function, the solution of the

homogeneous Schrfjdinger equation. Jj is

If the Schrodinger equation for

and the Green functions G+, G_ are given by

where e is an infinitesimal quantity used to define the contour defining the G+, G_, then

From these the sum and the difference can be made

P standing for principal part. In. the absence of potentials these functions are just

assuming that one already is dealing with an infinite system, i.e., a continuum of energy so that the sums over n become integrals. So if G is used for the difference of G+ and G_, (8) the sine like function, and also

13

1022

S. P. Edwards on a New Method

we now specialize to the case of a diagonal a, we have

where Gn(x, x') is O(En; x, x'). The problem now is to find G and / in the presence of the scattering potential, and finally to average a over all configurations of this potential^. It is convenient to express this in the following way: the Schrodinger equation is now where _£a being the positions of the scattering centres, and u the potential they exert on the electron. It is convenient to use the Fourier transform of this potential, defining

Now if the Xa are random, it can be shown by standard probability theory that the distribution function for the p's is

where N is the total number of scatterers, and £ the normalization to give total probability unity. When N is large, this can be used with fc running over the whole continuum of k space. So we reach the final formula

This form has the great advantage that it is essentially the same form as that of electrons interacting with the quantized electromagnetic field, and so techniques for evaluating it are already in existence. Moreover, there are none of the divergence problems of electrodynamics here and the various approximate techniques of electrodynamics can be applied with confidence. f The meaning of the averaging is discussed in detail by Kohn and Luttinger (1957).

14

for the Evaluation of Electric Conductivity in Metals

1023

§ 3. E VALUATION The essential difference between (1) and (17) is that the averaging over the scatterers can be carried out before the integrations over coordinates and allows manipulations which are meaningless when applied to (1). Although one can, on the basis of (17), derive integral equations for the average of G(x, x'}, G(y,y1}, it is simplest to consider the perturbation expansion of the G?'s, from which the structure of the integral will become clear. However, one should emphasize that there is no need to approach the evaluation by a perturbation approach and the results to be obtained below can be got directly. Consider firstly a simpler problem, that of obtaining the average of just one G alone. This is the difference of the averages of G+ and G_, which are more convenient to consider. Now in perturbation theory one can write, using (?(0> for the p-independent functions (10),

Upon averaging, using brackets for average value

and so on, neglecting terms relatively of order N~1. This can be obtained directly of course, without using the expansion (16). This gives

This is conveniently expressed in diagrams, which are slightly different from those of electrodynamics. Consider G(x,x') before averaging, draw a full line for every