Statistical Mechanics of Driven Diffusive Systems: Phase Transitions and Critical Phenomena (Vol 17)

Statistical Mechanics of Driven Diffusive Systems Statistical Mechanics of Driven Diffusive Systems Phase Transition...

Author: B. Schmittmann | R.K.P Zia | C. Domb | J.L. Lebowitz

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Statistical Mechanics of Driven Diffusive Systems


Phase Transitions and Critical P h e n o m e n a Ed ited by C. Domb

Department of Physics, Bar-Ilan University, Ramat-Gan, Israel and

J. L. Lebowitz

Department of Mathematics and Physics, Rutgers University, New Brunswick, New Jersey, USA

Volume 17 Statistical M e c h a n i c s of Driven Diffusive Systems B. Schmittmann and R. K. P. Zia

Center for Stochastic Processes in Science and Engineering and Physics Department Virginia Polytechnic Institute and State University Blacksburg, VA 24061-0435, USA

ACADEMIC PRESS Harcourt Brace & Company, Publishers

London San Diego New York Boston Sydney Tokyo Toronto

ACADEMIC PRESS LIMITED 24-28 Oval Road London NWl 7DX

U.S. Edition published by A C A D E M I C PRESS INC. San Diego, CA 92101

This book is printed on acid free paper

Copyright 9 1995 ACADEMIC PRESS LIMITED

All Rights Reserved No part of this book may be reproduced or transmitted in any form or by any means, electronic or mechanical including photocopying, recording, or any information storage and retrieval system without permission in writing from the publisher

A catalogue record for this book is available from the British Library

ISBN 0-12-220317-8

Typeset by Macreth Media Services, Hemel Hempstead, Herts Printed in Great Britain by WBC Book Manufacturers Ltd, Bridgend, Mid Glamorgan

Contents G e n e r a l P r e f a c e ................................................................................................................

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P r e f a c e to V o l u m e 17 ......................................................................................................

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C o n t e n t s of V o l u m e s 1-16 ..............................................................................................

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Statistical Mechanics of Driven Diffusive Systems B. SCHMITTMANN a n d R.K.P. ZIA 1 I n t r o d u c t i o n ...................................................................................................................

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2 T h r e e p e r s p e c t i v e s o f the s t a n d a r d m o d e l ................................................................

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3 L o n g r a n g e c o r r e l a t i o n s a b o v e criticality ..................................................................

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4 Critical p h e n o m e n a ......................................................................................................

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5 Physics b e l o w criticality ...............................................................................................

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6 V a r i a t i o n s of the s t a n d a r d m o d e l ...............................................................................

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7 R e l a t e d n o n - e q u i l i b r i u m s t e a d y state systems .........................................................

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8 S u m m a r y a n d o u t l o o k .................................................................................................

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9 A c k n o w l e d g e m e n t s ......................................................................................................

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R e f e r e n c e s .....................................................................................................................

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I n d e x ...............................................................................................................................

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General Preface This series of publications was first planned by Domb and Green in 1970. During the previous decade the research literature on phase transitions and critical phenomena had grown rapidly and, because of the interdisciplinary nature of the field, it was scattered among physical, chemical, mathematical and other journals. Much of this literature was of ephemeral value, and was rapidly rendered obsolete. However, a body of established results had accumulated and the aim was to produce articles that would present a coherent account of all that was definitely known about phase transitions and critical phenomena, and that could serve as a standard reference, particularly for graduate students. During the early 1970s the renormalization group burst dramatically into the field, accompanied by an unprecedented growth in the research literature. Volume 6 of the series, published in 1976, attempted to deal with this new literature, maintaining the same principles as had guided the publication of previous volumes. The number of research publications has continued to grow steadily, and because of the great progress in explaining the properties of simple models, it has been possible to tackle more sophisticated models which would previously have been considered intractable. The ideas and techniques of critical phenomena have found new areas of application. After a break of a few years following the death of Mel Green, the series continued under the editorship of Domb and Lebowitz, Volumes 7 and 8 appearing in 1983, Volume 9 in 1984, Volume 10 in 1986, Volume 11 in 1987, Volume 12 in 1988, Volume 13 in 1989, Volume 14 in 1991, Volume 15 in 1992 and Volume 16 in 1994. The new volumes differed from the old in two new features. The average number of articles per volume was smaller, and articles were published as they were received without worrying too much about the uniformity of content of a particular volume. Both of these steps were designed to reduce the time lag between the receipt of the author's manuscript and its appearance in print. The field of phase transitions and critical phenomena continues to be active in research, producing a steady stream of interesting and fruitful results. It is no longer an area of specialist interest, but has moved into a central place in condensed matter studies. The editors feel that there is ample

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General Preface

scope for the series to continue, but the major aim will remain to provide review articles that can serve as standard references for research workers in the field, and for graduate students and others wishing to obtain reliable information on important recent developments. CYRIL DOMB JOEL LEBOWlTZ

Preface to Volume 17 The field of non-equilibrium statistical mechanics continues to grow in scope and depth. Driven diffusive systems introduced almost a dozen years ago to model fast ionic conductors, have become a paradigm of stationary nonequilibrium states of particles conserving dynamics, which do not satisfy detailed balance. Their study led to the discovery of the existence of longrange spatial correlations in such systems. This genetic scale invariance, reminiscent of self-organized criticality, is one of the reasons why these systems are at the forefront of present work in this field. The current state of affairs in this area, including some previously unpublished material, is excellently described by Schmittmann and Zia in this volume. The major difficulty encountered in studying non-equilibrium, as compared to equilibrium phenomena, on a microscopic level is that while the latter are always described (in principle) by suitable averages over welldefined Gibbs ensembles, the far richer repertoire of non-equilibrium behaviour has, at least for the present, no such universal description. This is true even in the case of stationary situations which ought to be more similar to equilibrium systems and therefore easier to analyze. Thus there is currently no established systematic analytical procedure for deriving properties of these systems from averages over known ensembles. We have to rely instead on numerical evidence and heuristic physical arguments. This requires that great care be taken to sort out the wheat from chaff and not confuse the spurious with the essential- a task carried out admirably by Schmittmann and Zia who are themselves among the principal contributors to the field. The lack of systematic formalism for these systems remains true even when the dynamics is only slightly perturbed from one which yields stationary states not having any correlations. In particular we do not know how to obtain a convergent perturbation expansion in some appropriate small parameter. This is in contrast to the high temperature and low fugacity expansions available for equilibrium systems. Despite these difficulties, there continues to be made some real, albeit slow, progress in the development of a microscopic statistical mechanical theory of non-equilibrium process. The present volume will help in this task and the

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P r e f a c e t o V o l u m e 17

authors are to be thanked for taking the time to review the whole field and present a clear systematic exposition of the current state of the art. To make progress one is obliged to consider models of microscopic dynamics which are just caricatures of the real thing. There is evidence, however, which the authors carefully describe, that as long as the model dynamics preserve the essential features of the real dynamics, they provide qualitative and even quantitative information about the behaviour of real macroscopic systems. In this way driven diffusive systems serve as a model for elucidating cooperative phenomena in steady-state non-equilibrium situations analogous to the Ising model for equilibrium systems. CYRIL DOMB JOEL L. LEBOWlTZ

Contents of Volumes 1-16 Contents of Volume 1 (Exact Results)* Introductory Note on Phase Transitions and Critical Phenomena. C. N. YANG. Rigorous Results and Theorems. R. B. GRIFFITHS. Dilute Quantum Systems. J. GINIBRE. C* Algebra Approach to Phase Transitions. G. EMCH. One Dimensional ModelsmShort Range Forces. C. J. THOMPSON. Two Dimensional Ising Models. H. N. V. TEMPERLEY. Transformation of Ising Models. I. SYOZI. Two Dimensional Ferroelectric Models. E. H. LIES and F. Y. Wu.

Contents of Volume 2* Thermodynamics. M. J. BUCKINGHAM. Equilibrium Scaling in Fluids and Magnets. M. VICENTINI-MISSONI. Surface Tension of Fluids. B. WIDOM. Surface and Size Effects in Lattice Models. P. G. WATSON. Exact Calculations on a Random Ising System. B. McCoY. Percolation and Cluster Size. J. W. ESSAM. Melting and Statistical Geometry of Simple Liquids. R. COLLINS. Lattice Gas Theories of Melting. L. K. RUNNELS. Closed Form Approximation for Lattice Systems. D. M. BURLEY. Critical Properties of the Spherical Model. G. S. JOYCE. Kinetics of Ising Models. K. KAWASAKI.

Contents of Volume 3 (Series Expansions for Lattice Models)* Graph Theory and Embeddings. C. DOMB. Computer Enumerations. J. L. MARTIN. Linked Cluster Expansions. M. WORTIS. Asymptotic Analysis of Coefficients. D. S. GAUNT and A. J. GUIq'MAN. Ising Model. C. DOMB. Heisenberg Model. G. A. BAKER, G. S. RUSHBROOKE and P. W. WOOD. Classical Vector Models. H. E. STANLEY. Ferroelectric Models. J. F. NAGLE. X - Y Model. D. D. BEarS. *Out of print.

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Contents of Volumes 1-16

Contents of Volume 4*

Theory of Correlations in the Critical Region. M. E. FISHER and D. JASNOW. Contents of Volume 5a*

Scaling, Universality and Operator Algebras. LEO P. KADANOFF. Generalized Landau Theories. MARSHALLLUBAN. Neutron Scattering and Spatial Correlation near the Critical Point. JENSALS-NIELSEN. Mode Coupling and Critical Dynamics. KvozI KAWASAKI. Contents of Volume 5b*

Monte Carlo Investigations of Phase Transitions and Critical Phenomena. K. BINDER. Systems with Weak Long-Range Potentials. P. C. HEMMERand J. L. LEBOWITZ. Correlation Functions and Their Generating Functionals: General Relations with Applications to the Theory of Fluids. G. STELE. Heisenberg Ferromagnet in the Green's Function Approximation. R. A. TAHIR-KHELI. Thermal Measurements and Critical Phenomena in Liquids. A. V. VORONEL. Contents of Volume 6 (The Renormalization Group and its Applications)*

Introduction. K. G. WILSON. The Critical State, General Aspects. F. J. WEGNER. Field Theoretical Approach. E. BREZlN, J. C. LE GUILLOU and J. ZINN-JOSTIN The 1/n Expansion. S. MA. The e-Expansion and Equation of State in Isotropic Systems. D. J. WALLACE. Universal Critical Behaviour. A. AHARONY. Renormalization: Ising-like Spin Systems. TH. NIEMEIJERand J. M. J. VAN LEEUWEN. Renormalization Group Approach. C. DI CASTROand G. JONA-LASINIO. Contents of Volume 7 *

Defect-Mediated Phase Transitions. D. R. NELSON. Conformational Phase Transitions in a Macromolecule: Exactly Solvable Models. F. W. WIEGEL. Dilute Magnetism. R. B. STINCHCOMBE. Contents of Volume 8

Critical Behaviour at Surfaces. K. BINDER. Finite-Size Scaling. M. N. BARBER. The Dynamics of First Order Phase Transitions. J. D. GUNTON, M. SAN MIGUEL and P. S. SAHNI. Contents of Volume 9 *

Theory of Tricritical Points. I. D. LAWRIE and S. SARBACH. Multicritical Points in Fluid Mixtures: Experimental Studies. C. M. KNOBLER and R. L. SCOTt. Critical Point Statistical Mechanics and Quantum Field Theory. G. A. BAKER, JR.

Contents of Volumes 1-16

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XlII

Contents of Volume 10

Surface Structures and Phase TransitionsmExact Results. D. B. ABRAHAM. Field-Theoretic Approach to Critical Behaviour at Surfaces. H. W. DIEHL. Renormalization Group Theory of Interfaces. D. JASNOW. Contents of Volume 11

Coulomb Gas Formulation of Two-Dimensional Phase Transitions. B. NIENHUIS. Conformal Invariance. J. L. CARDY. Low-Temperature Properties of Classical Lattice Systems: Phase Transitions and Phase Diagrams. J. SLAWNY. Contents of Volume 12 *

Wetting Phenomena. S. DIETRICH. The Domain Wall Theory of Two-Dimensional Commensurate-Incommensurate Phase Transitions. M. DEN NIJS. The Growth of Fractal Aggregates and their Fractal Measures. P. MEAKIN. Contents of Volume 13

Asymptotic Analysis of Power-Series Expansions. A. J. GUTrMANN. Dimer Models on Anisotropic Lattices. J. F. NAGLE, C. S. O. YOKOI and S. M. BHATTACHARJEE. Contents of Volume 14

Universal Critical-Point Amplitude Relations. V. PRIVMAN, P. C. HOHENBERG and A. AHARONY. The Behaviour of Interfaces in Ordered and Disordered Systems. G. FORGACS, R. LIPOWSKYand TH. M. NIEUWENHUIZEN. Contents of Volume 15

Spatially Modulated Structures in Systems with Competing Interactions. W. SELKE. The Large-n Limit in Statistical Mechanics and the Spectral Theory of Disordered Systems. A. M. KHORUNZHY, B. A. KHORUZHENKO, L. A. PASTUR and M. V. SHCHERBINA. Contents of Volume 16

Self-Assembling Amphiphilic Systems. G. GOMPPER and M. SCHICK.

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S t a t i s t i c a l M e c h a n i c s of Driven Diffusive Systems B. Schmittmann and R. K. P. Zia Center for Stochastic Processes in Science and Engineering, and Physics Department, Virginia Polytechnic Institute and State University, Blacksburg, VA 24061-0435, USA

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Introduction

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Three perspectives of the standard model 2.1 T h e m i c r o s c o p i c d r i v e n lattice g a s 2.2 Dynamic mean-field theories 2.3 A mesoscopic Langevin equation .

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Long range correlations above criticality . . . . . . 3.1 G e n e r i c s i n g u l a r i t i e s in t w o - p o i n t f u n c t i o n s . . . . 3.1.1 Power law and exponential decays . . . . . 3.1.2 A fixed line o f G a u s s i a n d y n a m i c m o d e l s . . 3.2 Three-point correlation functions . . . . . Critical phenomena . . . . . . . 4.1 Simulation studies . 4.1.1 Signals of a phase transition . . . . 4.1.2 Two-point correlations . . . . . 4.1.3 Higher correlations . . . . . 4.1.4 F i n i t e - s i z e effects in the s t a n d a r d m o d e l . . 4.2 Theoretical investigations . . . . . . 4.2.1 General scaling laws with strong anisotropy . 4.2.2 Field theoretic renormalization group analysis . Physics below criticality . 5.1 The co-existence curve

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Copyright 9 1995 Academic Press Limited All rights of reproduction in any form reserved

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Dynamics of phase separation . . . . I nte r f a c ia l p r o p e r t i e s in steady states . . . 5.3.1 S u p p r e s s i o n o f interfacial r o u g h n e s s 5.3.2 Shifted b o u n d a r y c o n d i t i o n s a n d " t i l t e d " 5.3.3 N u m e r i c a l a n d theoretical studies . .

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V a r i a t i o n s o f the s t a n d a r d m o d e l . . . . . 9 6.1 R a n d o m drive a n d m u l t i p l e t e m p e r a t u r e m o d e l s . 9 . . 6.1.1 Collective b e h a v i o u r a b o v e Te . . . . 6.1.2 P h a s e t r a n s i t i o n s a n d critical p r o p e r t i e s o f the b u l k . 6.1.3 A n o m a l o u s c o r r e l a t i o n s o f interfacial f l u c t u a t i o n s . . 6.1.4 C o m b i n a t i o n s o f direct a n d r a n d o m drives . . . 6.2 C h e m i c a l p o t e n t i a l g r a d i e n t s a n d drive defects . . . 6.2.1 Systems with a c h e m i c a l p o t e n t i a l g r a d i e n t only . . 6.2.2 F i n g e r f o r m a t i o n in c o m b i n e d drives . . . . 6.2.3 Effects o f line defects in external fields 6.2.4 Stability o f a n interface in a t r a n s v e r s e ch em i ca l p o t e n t i a l gradient . . . . . . . . . . 6.3 T h e t w o - l a y e r system . . . . . . . . . 6.4 Multi-species m o d e l s . . . . . . . . . 6.4.1 T h e b l o c k i n g t r a n s i t i o n . . . . . . . 6.4.2 T h e p o l a r i z e d lattice gas . . . . . . . Repulsive i n t e r a c t i o n s . . . . . . . . . 6.5 Q u e n c h e d r a n d o m impurities . . . . . . . 6.6 Special limits . . . . . . . . . . 6.7 6.7.1 E x t r e m e a n i s o t r o p i c rate . . . . . . . 6.7.2 M o d e l s in one d i m e n s i o n . . . . . . . R e l a t e d n o n - e q u i l i b r i u m s t e a d y state systems . . . . . 7.1 M o d e l s with c o m p e t i n g c o n s e r v e d a n d n o n - c o n s e r v e d d y n a m i c s 7.2 M u l t i - t e m p e r a t u r e m o d e l s with G l a u b e r d y n a m i c s . . 7.3 M o d e l s for driven interfaces . . 7.4 Gel electrophoresis a n d p o l y m e r s in s e d i m e n t a t i o n . . 7.5 Self-organized criticality a n d o t h e r m o d e l s o f generic scale invariance . . . . . . . . . . . 7.6 Liquids in n o n - e q u i l i b r i u m steady states . . . . 7.6.1 L i n e a r i z e d h y d r o d y n a m i c s far f r o m criticality . . 7.6.2 P h a s e t r a n s i t i o n s u n d e r shear . . . . . .

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References Index

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1

Introduction

Most phenomena in nature involve so many degrees of freedom that it is impossible to account for all of them. Fortunately, it is often adequate to treat only a few of them in detail and regard the rest as "noise", with a certain postulated distribution. Thus, methods of statistical mechanics become essential for any quantitative description of complex behaviour. Typical questions concern the identification of appropriate variables, as well as the study of their stationary and time-dependent properties, such as means, fluctuations, correlations and responses to external forces. For systems in thermal equilibrium, this approach is well established, by virtue of the framework provided by Gibbs (1902): Once the microscopic Hamiltonian, 9~', of the system is specified, the stationary distribution over configuration space is known, in terms of the familiar Boltzmann factor, e -~e. Thus, averages of time-independent observables can be computed, at least in principle. The remaining difficulties are "merely" technical, in the sense that configurational sums may not be exactly obtainable. Yet, given our basic understanding of the task at hand, powerful approximation schemes, such as series expansions (see Guttmann, 1989, for a recent review, and further references) and renormalization group methods (e.g. see Amit (1984), or Zinn-Justin (1989)), have been developed. Thus, in the literature, articles on equilibrium systems form the large majority. In nature, by contrast, equilibrium systems are the exception rather than the rule: non-equilibrium phenomena are overwhelmingly more abundant. The reasons for this disparity between physical reality and the focus of current research are immediately obvious: given the wide range and the enormous complexity of non-equilibrium behaviour, a reasonable classification scheme has yet to be found. A similar gap exists in one-particle mechanics: while motion in non-conservative force fields, especially if dissipation is present, is much more common in nature, the topic of motion in conservative fields commands far more attention in the literature. In equilibrium, the distribution over configuration space, e - ~ e , is stationary. In contrast, for non-equilibrium systems, the distribution will generically be time-dependent, obeying, say, a master equation. Not surprisingly, little is known in general. One possible specific inroad is to study systems which have "settled down" into non-equilibrium steady states, so that the distributions,

B. S c h m i t t m a n n and R. K. P. Zia

while non-Hamiltonian, have become time-independent. Of course, this class is still enormous, and with many examples in biology, chemistry and physics (Nicolis and Prigogine, 1977; Haken, 1978, 1983), non-equilibrium steady states abound not only in the realm of theorists' imagination. Here, we will be concerned with systems coupled to two reservoirs of energy in such a way that there is a steady energy flow through the system. A trivial example is a resistor in steady state, gaining energy from a battery and losing it to the atmosphere. Even for this restricted class, however, there is no equivalent of Gibbs' framework and, typically, distributions cannot be expressed solely in terms of the internal energies of the system. Thus, in addition to the "technical difficulties" associated with computing averages in a many-body system, we must first solve the "more fundamental" problem of finding the stationary distribution. For systems which are only weakly perturbed so that they remain "close to equilibrium", much is known at the level of linear response (e.g. see de Groot and Mazur, 1984). Our focus here rests on steady states "far from equilibrium" where such schemes break down. Against this backdrop of a vast theoretical terra incognita, a reasonable approach consists in investigating systems which, while retaining the essence of the difficulties of "far from equilibrium" states, are as simple as possible. In this very spirit, Lenz suggested the Ising (1925) model in an attempt to understand the nature of ferromagnetic phase transitions. Here, this philosophy provides one of the main motivations behind the introduction of a simple non-equilibrium system (Katz et al., 1983, 1984), which we refer to as the "standard model". It consists of an ordinary Ising lattice gas (Yang and Lee, 1952) in contact with a thermal bath, but with one "minor" modification. Instead of having particles hop to nearest-neighbour unoccupied sites with a rate specified only in terms of the energetics of the inter-particle interactions, an external uniform driving field is introduced, through a bias in the rates for hopping along the field. Imposing periodic boundary conditions, the microscopic master equation satisfies translational invariance, so that this model is far simpler than its cousins with open boundaries. On a finite lattice, the system evolves towards a non-equilibrium steady state with a non-zero particle current. Furthermore, unlike its equilibrium counterpart, this lattice gas exchanges energy with two reservoirs, associated with the driving field and the heat bath, so that there is a steady energy current through the system. The other motivation for studying this model comes from the physics of superionic conductors or solid electrolytes (Geller, 1977; Salamon, 1979; Bates et al., 1982; Perram, 1983). In this class of materials, one or more species of ions enjoy high mobility, due to a large number of available interstitial or vacant lattice sites. The (zero-field) conductivities of some superionic conductors, like solid AgI, are comparable to those of melts of

Statistical mechanics of driven diffuse systems

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"normal" ionic crystals such as NaC1. Further, the strong temperature dependence of the ionic conductivity (Salamon, 1979; Bates et al., 1982) is characteristic of phase transitions. For example, in AgI, it increases abruptly over several orders of magnitude, believed to be a signal of an order-disorder transition in the site occupancies (Sato, 1977; Dieterich et al., 1980). Net interactions between the conducting ions, which can be repulsive or attractive, supposedly play a crucial role. If they are short ranged, then the phase transitions may be well modeled by the Ising lattice gas (Salamon, 1979; Dieterich et al., 1980). On the other hand, a gauge symmetry in the Ising model maps systems with repulsive interactions into attractive ones, casting doubt on the validity of the simple lattice gas description: the presence of an external field breaks the gauge symmetry, so that the transitions in repulsive and attractive systems may well be quite different. Further, a driven lattice gas will fall into the realm of non-equilibrium systems, and thus entirely new phenomena, requiring entirely new approaches, may arise. We caution, however, that our interest here is not directed towards the understanding of real materials. Indeed, since most of the existing experimental data are taken in small external fields, linear response theories are quite adequate for that purpose. Rather, we are interested in "far-from-equilibrium" physics, requiring large driving fields which, at least for superionic conductors, lie outside the experimentally accessible regime. Moreover, most of the novel behaviour occurs only if inter-particle interactions are attractive, which is not the case for a major fraction of the solid electrolytes. Much progress has been made since the previous reviews (Spohn, 1987; Marro, 1987; Schmittmann, 1990), especially for systems with attractive inter-particle forces. However, with each step forward, several new phenomena and many new questions emerged, so that the subject is far from "closed". Thus, we write this book with two missions in mind: to attempt a pedagogical summary of what is known; and to make a partial listing of exciting new frontiers. If the latter succeeds in raising sufficient interest in readers to devote some of their research efforts to this subject, this review will have served its purpose. The bulk of this book is devoted to a review of the standard model, starting with the specifications at the "microscopic" level in terms of a master equation (Section 2.1). In principle, we only need to solve this linear equation and find the steady state distribution and, from that, thermodynamic quantities and phase transitions. Since this task is insurmountable in practice, we must turn to approximations, the first of which is a dynamic version of mean-field theory (Section 2.2). Since we are mainly interested in collective behaviour in the long wavelength, low-frequency limit, a good approximation is the mesoscopic, continuum description in terms of a Langevin equation. This approach forms the basis of the extremely


successful field theoretic analyses of critical phenomena, and we provide some details in Section 2.3. In the rest of this book, we review simulational and theoretical results for a few selected quantities only. Many others have been measured and deserve more attention; yet, due to space limitations, we will only list them here: total current (Katz et al., 1984; Dickman, 1988) and its power spectrum (Leung, 1991b); site number distribution and fluctuation (Andersen et al., 1991; Cheng et al., 1991a); and finally, transitions across the co-existence curve and metastability (Marro and Vallrs, 1987). For all temperatures above criticality, where the equilibrium Ising model is analytic, typical correlations in the driven system are singular in momentum space (Zia et al., 1993 a, b), leading to power law decays (Zhang et al., 1988; Garrido et al., 1990) in configuration space. We confine our discussion, in Section 3, to only the two lowest correlations, i.e. two- and three-particles ones. In Section 3.1.2 we review the formulation of the Langevin approach in terms of a Boltzmann-like weight, e - j , where J is the dynamic functional (Janssen, 1976; de Dominicis, 1976) whose role is formally analogous to that of ~ in equilibrium systems. In this sense, it may be regarded as a Gibbs' distribution, albeit for a d + 1 dimensional system. However, the steady state distribution for the d dimensional system is no more explicit in this formalism than in the statement that it exists as a particular solution to the master equation. Section 4 is devoted to critical behaviour, with the first part (Section 4.1) focused on simulation results. Here we encounter a major distinction between the driven and the ordinary Ising model, i.e. the presence of strong anisotropy, implying that correlations along the field scale differently from correlations along transverse directions. In Section 4.2, we provide a general framework for a scaling analysis appropriate for systems with strong anisotropy, showing, for example, that there are four different ~7-1ikecritical exponents. The final part of this section (4.2.2) is devoted to the present status of theoretical understanding of critical singularities. Since singularities arise from infinite fluctuations, computing them analytically is formidable indeed, a goal which took nearly a century to attain, even in equilibrium systems. Thus, it is remarkable that, thanks to field theoretic techniques in renormalization group analysis, most of the critical exponents were obtained analytically (Janssen and Schmittmann, 1986b; Leung and Cardy, 1986) before they were measured in simulations. In these studies, the WilsonFisher (1972) fixed point, which controls the critical properties of the undriven system, is found to be unstable. Instead, a new fixed point, which is non-Hamiltonian, is found. Surprisingly, even though this approach is based on an expansion in powers of ( 5 - d), critical exponents can be computed exactly to all orders, as a result of a continuous symmetry associated with this fixed point. On the other hand, early Monte Carlo simulations


followed standard routes of finite-size scaling in isotropic systems and found a sizable discrepancy between the data and the predicted value for the exponent ~/. Only a proper accounting of the effects of strong anisotropy could resolve the discrepancy (Leung, 1991 a, 1992). Since, at present, there are no known disagreements between theory and simulations of critical properties, we venture that this aspect of the standard model is well understood. The physics far below criticality, on the other hand, is rather poorly understood. Thus, Section 5 is devoted mainly to simulation results. A major source of the difficulty is that steady states with half-filled lattices are inhomogeneous, i.e. the system phase segregates. Though some work has been done to explore the co-existence curve (Section 5.1), there are still many open questions. The next section (5.2) is a brief detour into the dynamics of phase separation. Even though these studies are not strictly associated with steady states, we include them here, since self-similarity in time and dynamic scaling are closely related subjects. Finally, once the system has settled into an inhomogeneous steady state, most of the interesting behaviour is associated with interfaces. In the last section (5.3), we review a number of unfamiliar features, such as suppression of roughness, dependence of bulk properties on interface orientation and novel transitions into multi-strip domains. Parallel to investigations of the standard model, numerous studies of its variations and generalizations were performed. Some of these are briefly reviewed in Section 6. Since the original model, with a uniform electric field wrapped around a torus, is hardly physical, it is of some interest to explore more realistic models. If we relax to cylindrical boundary conditions, it is possible to impose an electric field by applying a changing magnetic field through the cylinder. However, a DC drive would be difficult to maintain for long periods of time. It is much easier, in this example, to impose an AC or a random drive. Since AC fields are not scale invariant in time, and typically lead to much more complex behaviour, Schmittmann and Zia (1991) studied a model driven by fields which are random in time, with zero mean. The global current vanishes, but particle jumps in the drive direction suffer an "extra randomness". In this way, this model is similar to a two-temperature lattice gas, in which particle hops along one particular lattice axis are coupled to a heat bath at a higher temperature (Garrido et al., 1990; Cheng et al., 1991 a). Since the underlying symmetry differs from the standard model, the Langevin equation must be modified (Section 6.1), leading to vanishing oddpoint functions above To, a new fixed point for critical behaviour, as well as more tractable interfacial properties in the inhomogeneous phase. Returning to uniformly driven systems, the first parts of Section 6.2 are devoted to another approach toward more achievable realizations of the standard model, i.e. open boundaries. When particle sources and sinks are placed at opposite edges of the lattice, a density gradient is set up, leading to a steady


current. Various types of behaviour are encountered, depending on the presence or absence of a bulk drive. Systems with sources and sinks at the boundaries are similar to systems with periodic boundary conditions and a single row of "defective" drive. Such generalizations are described in Section 6.2.3. In the next section (6.3), we investigate a pair of standard d = 2 models stacked to form a multi-layer system. There are several motivations for studying such variations, none of which prepared us for the surprising result that, even in the absence of any inter-layer interactions, there is an additional low temperature phase, characterized by homogeneous densities in both layers. Partly motivated by superionic conductors with more than one type of ions, another generalization is the study of driven systems with several particle-species. Clearly, this horizon is immense, even for equilibrium systems which include the Ashkin-Teller (1943) model, all the Potts (1952) models, the Blume-Emery-Griffiths (1971) model, and various vertex models (Lieb and Wu, 1972). Not much of this new terrain has been explored. We are only aware of investigations of driven systems with twospecies of particles, identical except for being oppositely "charged" (Section 6.4). An intriguing extension of this model is to reduce the mobility of one species to zero, so that, starting from a random configuration, these particles behave as quenched obstacles for the mobile ones. Clearly, the behaviour of models with quenched impurities is of great interest, since almost no physical system is pure while almost no theoretical result remains unaffected by the inclusion of impurities. In Section 6.6, we review two attempts at addressing this issue for the driven case. All the models studied in the preceding sections involve attractive interparticle forces, yet repulsive interactions dominate between ions in a large number of superionic conductors. For an Ising lattice gas in equilibrium, repulsive interactions can be mapped into attractive ones by a gauge transformation. However, an "electric" field spoils particle-hole symmetry and, if driven, a repulsive system behaves very differently. Specifically, the order parameter here is the "staggered" density, which obeys no conservation law, so that, for example, the leading critical singularities in this non-equilibrium system are identical to those in the equilibrium Ising model, as long as the second order phase transition is present. Generally, much less is known about these systems. Section 6.5 is devoted to a brief review. In the final part of Section 6, we review two special limits of the standard model for which exact solutions exist. In an early attempt to predict phase transitions, van Beijeren and Schulman (1984) proposed a "fast rate limit", in which jumps along the field direction occur infinitely more frequently than others. It is argued that the system will "equilibrate" between consecutive transverse jumps, so that an L x M lattice gas reduces to a chain of L sites


9

with occupation variables in the range of [0, M]. All details of the drive disappear from this effectively one-dimensional system, which is solved exactly for a special case. Remarkably, order-disorder transitions still exist, even though all longitudinal correlations vanish (Section 6.7.1). Diametrically opposed to this limit is a "standard model in d = 1", where all transverse directions are absent. Clearly, for periodic boundary conditions, no phase transition can exist, so that the focus shifts to time-dependent phenomena. Alternatively, phase transitions can be induced through modifications, such as open boundaries or local obstructions. Both aspects of these one-dimensional systems yield interesting results, which are reviewed in Section 6.7.2. In Section 7, we attempt to provide some contact between driven diffusive models and the vast world of non-equilibrium steady state phenomena, by reviewing a handful of systems. By no means a comprehensive catalogue, this section should be regarded as a collection of special topics, selected mainly because these systems are most closely related to ours, in one aspect or another. They range from mathematical models with no known physical realization, to readily accessible systems with a wealth of experimental data. The relation of our models to three of these subjects is especially intimate, in that the macroscopic equations of motion are very similar. These topics are: (i) driven interfaces, modelled by the KardarParisi-Zhang (1986) equation; (ii) drifting polymers and gel electrophoresis; and (iii) some models of self-organized criticality with conserved quantities. We hope that, as a result of the similarities, there will emerge a rich cross-fertilization of ideas and techniques. In a final section (8), we summarize the present status of driven diffusive systems by highlighting some of the progress in the past decade. We conclude by listing a number of open questions and speculating on possible applications to a wide variety of non-equilibrium steady state phenomena, such as staging in intercalated compounds, flux flow in type II superconductors, hot electron transport, dynamics of fracture and granular material flow.

2

Three perspectives of the standard model

Since the introduction of the prototype (Katz et al., 1983, 1984), a large number of driven diffusive models have appeared. For clarity, we describe in some detail the simplest of these, which we call the "standard model". In Section 2.1, the rules governing the "microscopic" dynamics and some general features of its simulation are presented. Section 2.2 is devoted to the mean-field perspective. In practice, this approach can only account for

10


correlations of a few neighbours. To study the collective behaviour in the long wavelength, low-frequency limit, a coarse-grained Langevin equation of motion is much more suitable. Being quite successful, especially in predicting critical properties, this approach is reviewed in the last section.

2.1

The microscopic driven lattice gas

Motivated by both theoretical simplicity and the physics of fast ionic conductors, we start with the usual Ising lattice gas, defined on a d-dimensional hypercubic lattice with N sites. Thus, each lattice site can exist in two states, occupied or empty, labelled by an occupation variable ni = 1 or 0, for site i. This choice already implies an excluded volume interaction, since there can never be more than a single particle on each site. A complete set {ni}, i = 1,..., N, specifies a particular configuration, (s Clearly, we may just as easily employ, instead of ni, Ising spins: s~ = 4-1, with si = 2n~ - 1, so that particle and spin language will be used interchangeably. Since our particles are meant to model ions, we will only consider configurations with fixed total particle number ~-~ n i. In spin language, this translates into fixed magnetization. This conservation law will turn out to be crucial for the existence of a number of interesting phenomena. The internal energy of a particular configuration resides in additional interactions between the particles. To keep the model simple, they are assumed to affect nearest neighbours only, leading to the Ising Hamiltonian

,~[IE] = - J ~ ninj, (~,./>

(2.1)

where (i, j) denotes the sum over nearest-neighbour sites. A positive J models an attractive (ferromagnetic, in spin language) interaction, while negative Js imply repulsive (antiferromagnetic) interactions between the particles. The interactions of our system with its environment are represented by a coupling to a heat bath at temperature T. Thus, all equilibrium properties can be computed by weighting with the canonical distribution: Peq (I~) -- e-#~e/Z, where Z is the partition function and/~ = 1/kB T. Note that the constraint on particle number, or magnetization, while implying a conservation law for the dynamics, has no beating on static equilibrium properties, due to the equivalence of canonical and grand-canonical ensembles in the thermodynamic limit. At half density, for d > 1, the Ising lattice gas exhibits the usual second order phase transition at a critical temperature Te, which is positive. In the celebrated d = 2 case, on the square lattice, Te takes the value (Onsager, 1944) 0.5673 J/kB. All critical properties are universal, belonging to the "Ising universality class" (Kadanoff, 1971).


11

Since our interest is in non-equilibrium behaviour, we will need to specify how a given configuration ~ evolves into a new one, ~'. This canbe done by specifying a set of transition rates W[~ ~ ]. As a consequence, we must now deal with a time-dependent probability distribution P(~, t), which obeys a master equation in continuous time: ~t

P(fi;, t) - Z

{ W[ff;' ~ ~]P(ff;' t) - W[~ ~ ~']P(ff;, t)}

(2.2)

or a similar balance equation in discrete time. The specific choice of W determines the model dynamics and, jointly with the boundary conditions, the steady state distribution: P*(fi;)= P(ff,,t---, oe). For equilibrium systems, our choice of rates must be somewhat restricted, to ensure that P* ((g) - Peq(E). This constraint is most conveniently imposed in the form of the detailed balance condition,

W[(~' ---+ if;] eeq ((~) = ~ W[(~ ~ (~'] Peq((~') "

(2.3)

The important point here is that the ratio Peq ((~)/Peq (if t) is explicitly known. It is just exp ( f l A g ) , with A~rF -- ~r - ~ff((~). One may thus choose rates of the form W[(s ~ (~'] = w ( 3 A g ) , with an appropriate w satisfying

w(-x)/w(x)

= e x.

(2.4)

For a diffusive system, driven or not, particle number is conserved. Therefore, we choose a Kawasaki (1966, 1972) dynamics to describe the hopping of particles to empty nearest-neighbour sites. Jumps with longer finite ranges, while adding to the complexity of the model, are not expected to lead to significantly different collective behaviour. Next, we introduce the driving field, which we label by E and refer to as the "electric" field while imagining that the particles are "charged". In the standard model, it is uniform in both space and time, and points along a specific lattice axis. It biases the jump rates, favouring (suppressing)jumps along (against) its direction while leaving jumps in the transverse directions unaffected. Locally, this bias affects the particles much like a gravitational field, so that, for certain boundary conditions (Katz et al., 1984), it can be incorporated into the Hamiltonian as "potential" energy. In those cases, the steady state of the system is indeed an equilibrium state described by a Boltzmann distribution. However, our interest here is to pursue precisely the kind of boundary conditions which do not permit a global, Hamiltonian representation, and thus induce a non-equilibrium steady state. Whether a global potential energy is defined or not, it still appears reasonable to choose rates which include the work done locally by the field. Thus, we

12


simply add a term gE to Ajvf in the rate functions, where g = (-1,0, +1)

for jumps (along, transverse to, against) f

(2.5)

while E represents the product of the magnitude of E, the "charge" of the particles and a lattice constant. Thus, we choose W[($ ~ ($'] - w[/3(A~ + gE)],

(2.6)

with w still satisfying (2.4), a "local detailed balance" condition. The specific choice of w determines the rate's completely. For most simulations, Metropolis (Metropolis et al., 1953) rates, i.e. w(x)= min {1,e-X}, were used. Examples of other rates include w(x) - 1/(1 + e -x) and, more significantly, inherently anisotropic ws such that jumps transverse to E are strongly suppressed (van Beijeren and Schulman, 1984; Krug et al., 1986; and Vall~s and Marro, 1986; see also section 6.1.1). Finally, we must specify the boundary conditions. Since our interest is in non-equilibrium steady states, the simplest choice here are periodic boundary conditions (PBC). Clearly, these will induce a non-trivial global current while respecting translational invariance. With these conditions, the lattice topology becomes a torus, with a uniform field E "looping" around it. Note that this field cannot be written as the gradient of an electrostatic potential; rather, it can be generated by a magnetic flux increasing linearly with time. For the same reason, it is clearly impossible to imagine gravity as the source of this drive, with the possible exception of the world of M. C. Escher who portrayed our model beautifully in a lithograph Ascending and Descending (reproduced on the cover) 23 years before Katz, Lebowitz and Spohn! Returning to our world, the combination of uniform E and PBC is a key feature of the standard model. It excludes (Katz et al., 1984) the existence of a simple effective Hamiltonian which incorporates both the configurational energy of the particles and their potential energy in the external field. Of course, since P*, the steady state distribution, still exists, we may label -kBT(lnP*) as the effective "Hamiltonian". But that is just as much an unknown as P*, and certainly not simply related to W and E. Instead, it is expected to be both T-dependent and long ranged. Thus, there is little to gain by writing P* in Boltzmann form. Let us point out here that it is precisely these key features which make the standard model difficult to realize experimentally. Either we give up the PBC in favour of the uniform field (since linearly increasing magnetic flux can only be maintained for relatively short times), or we need to allow for time-dependent fields. Both of these scenarios will be discussed later. With the model completely specified, it is possible, in principle, to find P* by solving the linear equations: y ~ , { W[~' ~ (~]P* ( ~ ' ) W[ff---,ff']P*(ff)}-0. In practice, this task is unsurmountable, with two


13

mm

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(a)

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Fig. 2.1 "Ground states" for a half-filled 2 x 4 periodic lattice, driven with an infinite E (a) and in equilibrium (b). exceptions. The first is systems with J = 0, i.e. biased diffusion of particles with no interactions other than excluded volume. Then, P* cx 1 (Spitzer, 1970), though there are some non-trivial t-dependent effects which will be discussed in Sections 3.1.2 and 6.7.2. Without some modifications of the boundary conditions or the microscopic rates (Section 6.7.2), there are no phase transitions in such systems. Turning our focus to J > 0 models, the other solvable case involves very small systems, e.g. 2 • 3 and 2 • 4 (Zhang, 1987). While it is difficult to glean reliable information on collective behaviour and singularities in thermodynamic functions, exact solutions to such systems do provide insight into some of the dramatic differences between equilibrium and non-equilibrium steady states. As an example, for a half-filled 2 x 4 lattice with T ~ 0, only one configuration (modulo translations) survives

Tc(E)

E Fig. 2.2 Schematic phase diagram. Tc(0)-~ configurations at points a, b and c.

0.5673J/kB. See Fig. 2.3 for typical

B. S c h m i t t m a n n

14

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Fig. 2.3 T y p i c a l c o n f i g u r a t i o n s f o r a 30 x 30 s y s t e m , at T "-~ 2.2, E = c~ (a); T "~ 1.1, E = 0o (b); a n d T "~ 1.1, E = 0 (c). T is q u o t e d in u n i t s o f t h e O n s a g e r v a l u e . F r o m K a t z et al. (1984).


15

in either case; but these "ground states" are distinct for large E (Fig. 2.1)! Another example is the difference between ( ~vf2) - ( g )2, i.e. the fluctuations in the "internal energy", and - ~ ( ~ ) , the "specific heat". In a 2 x 3 lattice with E - or there is a familiar peak at finite T for the latter. However, the former is proportional to x(1 + 6x)/(1 + 9x) 2, where x - e -8~J (Zia and Blum, 1995). In other words, the energy fluctuations are monotonic in T! Careful analysis shows that the difference between the two quantities is also monotonic in T, while in the equilibrium case it is, of course, zero. To build a better understanding of collective behaviour such as phase transitions, we turn to Monte Carlo simulation methods. As most studies are carried out in d - 2, T will be given in units of the Onsager value. Typically, a half-filled lattice ( E n i - N/2) is chosen so that, at the least, the equilibrium critical point is accessible. The most prominent feature, namely a transition from a disordered to an ordered phase, survives for all fields (Katz et al., 1983, 1984). Schematically, the phase diagram is shown in Fig. 2.2. Note that To(E) appears to be monotonic in E, saturating at approximately 1.4 times the Onsager value (Leung, 1991 a). In practice, these large fields are realized by E / J - 50 or 100, which essentially suppresses all jumps against the drive, while those along E are always allowed. Most of the subsequent simulation studies use such fields (referred to as "saturation" or "infinite" fields), in the belief that they maximize the effects of non-equilibrium dynamics. Two typical configurations (Katz et al., 1984) associated with the disordered and ordered phases, for a 30 • 30 lattice driven at infinite E, are shown in Figs. 2.3 a, b. Note that, as in the equilibrium system, particle number conservation forces the ordered state to consist of co-existing phases of particle-rich and hole-rich regions (referred to as "strips"). In the driven case, the strips are always parallel to E; stable strips orthogonal to E have never been observed. In contrast, in the undriven model on a square lattice, strips may be oriented in either direction. For comparison, Fig. 2.3 c shows a typical disordered configuration in an equilibrium Ising model. A casual glance may lead us to the conclusion that the driven and undriven systems display similar features. However, on closer examination, novel behaviour can be found at all temperatures, as we will show in the following three sections. Before proceeding to discuss mean-field theories and the mesoscopic model, let us point out the symmetries of this system, which will play a decisive role in determining the universal properties. In particular, both the microscopic lattice version and its mesoscopic continuum counterpart must exhibit the same symmetries if they are to belong to the same universality class. For the lattice model, all rates chosen according to (2.6) are invariant under the following operations:

16

(i) (ii)

(iii)


translations, modulo the periodicity of the lattice; field reversal and "charge" conjugation (c), replacing all particles ("positive charges") by holes ("negative charges"), i.e. E r and ni r 1 - ni; and field reversal and reflection in y (R), the coordinate of the drive direction, i.e. E r - E and y r -y.

Combining the last two yields invariance under CR transformations, i.e. particle r hole and y ~ - y .

2.2

Dynamic mean-field theories

Starting from a microscopic model, to define a suitable order parameter and to predict the existence of phase transitions are difficult tasks, in general. For a qualitative, first impression of the effects of collective behaviour, mean-field approaches have proved quite successful. In the case of an Ising model in equilibrium, Bragg and Williams (1934, 1935) implemented the molecular field notion of Weiss (1907), and found a phase transition for the d = 2 model. This initial effort was refined by Bethe (1935) to include the effects of correlations with nearest neighbours. Later, the cluster variational method was introduced (Kikuchi, 1951; Morita, 1957) to account for further correlations, among spins in a cluster of k sites. Though some improvements were found, these refinements were naturally unable to predict critical singularities, which are effects of an infinite correlation length. Nevertheless, mean-field theories are practical tools for finding a rough phase diagram. We should mention that the most recent developments of this approach, which incorporate the ideas of finite-size scaling (Suzuki et al., 1987), are so advanced that critical exponents can be estimated to within a few per cent (Katori and Suzuki, 1987, 1988)! In this respect, the mean-field approach should be thought of as a sequence of approximations which lead to quantitatively reliable results. In the same spirit, Kikuchi (1966) introduced the path probability method for studying non-equilibrium systems. Applying it to solid electrolytes, Kikuchi and Sato (1971) computed diffusion coefficients and ionic conductivities for /3-alumina. However, these studies focused on the linear regime, considering only the effects of small fields away from criticality. More recently, a similar generalized mean-field theory was designed specifically for investigating the standard model (Dickman, 1988; Pesheva, 1989; Pesheva et al., 1993) and non-equilibrium phase transitions. In this section, we briefly review the methods and some results. Adopting the ideas of conventional equilibrium mean field theories, this


17

approach considers a finite cluster of particles which is treated exactly, assuming its environment to be homogeneous. A dynamic version of the cluster variational method, this formulation involves finding a hierarchy of evolution equations for the probability distributions of configurations within a cluster of k sites: Pk[~]. Symbolically, we write

dek dt

(2.7)

=

An "/-level approximation" consists of neglecting all correlations in Pk, if k > 1, so that P 1 , . . . , Pt form the degrees of freedom in such a treatment. To derive the functions Fk, a finite system just large enough to include at least one /-cluster and its environment is considered, while all configurations and their subsequent transitions, according to the master equation, are analysed. In this manner, Fk depends upon J, E and T through the rates w. Lastly, setting Fk = 0 provides a set of equations for determining the steady state values: P*k(J, E, T). Note that, due to the conservation law, dP1/dt - O, since P1 is related simply to the fixed, overall density. Clearly, one of the underlying assumptions of this approach is spatial homogeneity. As a result, it is unsuitable for studying the phase separated states occurring at temperatures below Tc. In practice, this program is quite labour-intensive, even for small clusters. Since the dynamics conserves particle number, the lowest level approximation allowed is the pair (1 = 2). By contrast, for Glauber spin-flip dynamics, the smallest cluster is just a single spin. Furthermore, since the driven system is anisotropic, we must consider two distinct pair probabilities, for the transverse and longitudinal directions. In d - - 2 , these clusters and their environments are shown in Fig. 2.4. Beyond the scope of this review, the details for deriving the F2's may be found in Pesheva et al. (1993). Apart from this lowest level, the square cluster with four sites has also been considered (Dickman, 1988). The crucial problem of locating phase boundaries is addressed by two

I I o

C

",

Fig. 2.4 Clusters used for the pair approximation in a dynamic mean-field theory. (From Pesheva et al., 1993.)

18


different methods. Dickman (1988) relies on the local stability properties of the homogeneous phase, and focuses on the current generated in response to a small density gradient. Above criticality, the current opposes the density gradient and restores homogeneity. When such a current vanishes, the system is "critical". One disadvantage of this method is evident, i.e. inability to distinguish a second order transition from a spinodal point. In the standard model with a half-filled lattice, there is a known second order transition so that, by setting P1 = 1/2, a phase diagram in the T - E plane can be obtained. At the pair level, the agreement with d = 2 simulations is quite reasonable. Tr is found to be monotonically increasing, overestimating the exact value at E = 0 by 27%, yet coming to within the errors of simulations at infinite E. The results from the square approximation are even more remarkable, improving to 4% at E = 0 without spoiling the agreement at E = e~. Unfortunately, the results for d = 3 (1 = 2) are disappointing: Tr is found to decrease with E, in conflict with the data. So far, higher level approximations have not been explored to see if there are improvements. Also, the reason behind the qualitative differences is not well understood. Apart from the phase diagram in d = 2, the agreement between theoretical pair probabilities above Tr and the data is also quite good. For T < Tr and P1 ~: 1/2, the system is still in a homogeneous phase provided the densities lie outside the co-existence curve. Thus, this approach can still be used; however, it yields the "spinodal" line, rather than the co-existence curve itself. To address the issue of the co-existence curve, Pesheva et al. (1993) introduced an extra hypothesis to arrive at an equation which determines P1- Defining a generalized entropy, -kB ~ Pk In Pk, these authors postulated a "free energy" function ~({Pk}) in analogy with equilibrium cases. Extending the maximum entropy principle of Jaynes (1957), they proposed that, for non-equilibrium steady state systems, 9 be minimized subject to the constraints Fk = 0. Solving these equations, all Pk>l can be eliminated in favour of P1, so that 9 is now a function of P1 alone. The mean-field approximation appears only as a means to compute Fk, leading to an approximate ~MF(P1). At the pair-level, this function displays the familiar transition from a single well to a double well structure, and provides us with the co-existence curve. In this sense, this approach is more powerful than Dickman's. Not surprisingly, agreement with the d - 2 data is also qualitatively good. Systems with d > 2 or l > 2 have not yet been analysed to see if the qualitatively poor predictions of Dickman's method persist. We end this section with some remarks. In addition to the obvious advantages in dealing with the standard model, this approach can be easily adopted for investigating a host of other questions. For example, since the phase transition has a dynamic origin, the phase diagram should depend upon the details of the dynamics chosen. In particular, it was found that


19

dramatic differences occur (Pesheva, 1989; Pesheva et al., 1993) if van Beijeren and Schulman (1984) rates were used instead of Metropolis ones. Another example involves a standard model with repulsive inter-particle interactions (Dickman, 1990), more details of which will be given in Section 6. On the other hand, there are disadvantages in this approach. Already mentioned is the failure of Dickman's (1988) method to deal with a system below criticality. For the alternative method, it is unclear whether generalizations to other models will continue to yield good agreements with simulations. Also unclear at present is whether the maximum entropy principle will find a firm foundation, similar to its counterpart in equilibrium statistical mechanics. A more serious difficulty is whether generic long range correlations, which will be discussed in detail in the next section, can be properly described using only small clusters. Finally, to find critical properties by either method, larger and larger clusters must be used, which can be quite laborious without being transparent. At a deeper level, even after critical properties are computed, this approach certainly does not address the issue of universality.

2.3 A mesoscopic Langevin equation The microscopic model gives us both a well defined starting point and the necessary rules for performing Monte Carlo simulations. While mean-field theories do provide us with an avenue to "predict" a phase diagram and some correlations at the microscopic level, they are very limited for describing the physics of collective behaviour at the macroscopic level. To comprehend the latter fully, it is far better to start with a "mesoscopic" picture. Like hydrodynamics, this approach has been quite successful, at a phenomenological level, for understanding the slowly-varying long wavelength properties of a many-body system. Beyond that, it has seen spectacular results within the context of renormalization group analyses of critical behaviour. For the equilibrium Ising model, the mesoscopic picture is given by the LandauGinzburg-Wilson Hamiltonian (Amit 1984). In this spirit, we introduce the mesoscopic dynamics, via a Langevin equation of motion for an appropriate mesoscopic field, which is, in this case, particle density p(s t). In principle, starting from the master equation for the microscopic occupation variables ni, a coarse-grained version for the mesoscopic p(~?, t) can be derived by some suitable averaging procedure (Eyink et al., 1995). In practice, this route proves to be difficult for most systems (Forster, 1975; Spohn, 1991) so that, instead, phenomenological equations are postulated. Typically, symmetries and physical considerations, such as the appropriate choices of order parameter and other slow variables, dictate their form.


20

Indeed, the Landau-Ginzburg Hamiltonian was originally proposed in this manner, while the more rigorous route came later. The principal symmetry in a diffusive system is particle conservation. So, all equations of motion for the density field take the form of the continuity equation: OtP = - V . J . Different models are distinguished in two ways" firstly, they may require additional fields, as in the case of repulsive interactions between particles where the order parameter field (the staggered particle density) plays as important a role as p. In this section, we focus on attractive interactions, where the density in fact is the order parameter, and this single field suffices. Secondly, models may differ in the form of the current, J[p], which is generally composed of a systematic, deterministic part and a noisy, random part. In the absence of E, the standard form for the systematic current is -A ~7#, where A is a transport coefficient and # is the chemical potential, both of which are functions of p in general. For studying critical dynamics (Hohenberg and Halperin, 1977) near equilibrium, it is sufficient to let A be a constant and assume # - rS~[p]/6p, where ~ [ p ] is the mesoscopic Hamiltonian. A nontrivial p-dependence of A, while generically present, has no bearing on universal properties. Conversely, we may want to focus on system properties away from criticality. Then, the densitydependence of A can no longer be neglected. In particular, A vanishes as p approaches zero or one, the latter being a consequence of the excluded volume constraint. The "noisy" part of J is referred to as a Langevin force JL- It models the relevant effects of the fast microscopic degrees of freedom after the coarse graining. In most applications, it is assumed to be Gaussian distributed, with zero mean, at each space-time point (~, t). Statistical averages of p and other quantities then follow from these distributions, since the Langevin equation relates p to JL. To make more quantitative statements, we specialize to densities near 1/2, pertaining to most simulations. Near equilibrium, the coarse-grained dynamics of the Ising lattice gas with conserved density is invariant under translations and rotations, as well as particle-hole exchange. It is known as Model B in the Hohenberg-Halperin (1977) terminology. Expressing p(s t) in terms of the local magnetization: ----+

~b(Y, t) - 2p(Y, t) - 1

(2.8)

and taking Yg[4~] to be the Landau-Ginzburg-Wilson Hamiltonian:

U (/)4+ "''}, o,~[q)]- J ddx{l (~ ~)2 -+-lT~2-~--~..

(2.9)


21

the equation of motion becomes

e,e

- vi

avi-

+

9

(2.10)

In (2.9), ~- (x (T - Tc) measures the deviation from the critical point, while u (> 0) is necessary for stability below Tc and for describing the spontaneous magnetization. The subscript i (= 1,..., d) in this section will stand for the vector index. To respect the symmetries of the underlying Ising model, ~[qS] must be even in ~band isotropic in s (the only second rank tensor respecting four-fold symmetry of the square lattice being 6ij). The Langevin current is denoted by ~i here. Being Gaussian with zero mean, its distribution can be specified by the following:

((i(s (~i(X, t)~j(.~', t')) - Nij6(.~- s

(2.11 a)

t'),

(2.11 b)

where Nij is a noise correlation matrix. Since the above dynamics is constructed to describe critical fluctuations around thermal equilibrium, Nij must be chosen such that the fluctuation-dissipation theorem (FDT) is satisfied (Kubo, 1966), i.e.

Nij = 2AkBT6ij.

(2.12)

This ensures that the stationary distribution is given by the usual Boltzmann weight e -~g. Equations (2.9-2.12) form a complete set of specifications, from which both static and dynamic critical properties may be calculated. One of the standard routes employs renormalized perturbation theory for dynamics (Janssen, 1976, 1979; Bausch et al., 1976; de Dominicis, 1976; de Dominicis and Peliti, 1977, 1978). Via a renormalization group treatment, by expansion around the upper critical dimension d c - 4, this route leads to the usual Wilson-Fisher (1972) fixed point. Due to the conservation law, the dynamic exponent z has a mean-field value of 4. For d below dc, it is exactly 4 - ~7, where ~ is the exponent controlling the power law decay of the pair correlation function. Next, we consider the influence of the driving field E. At the least, it will generate an additional contribution, Je, to the current J. In principle, j~ may again be explicitly calculated by coarse-graining the microscopic dynamics. In practice, we will postulate the simplest form of re(p, E) consistent with the symmetries. Being a vector, it must be proportional to E. The remaining factor must be isotropic and invariant under {particle r hole and/~ ~ - E ~, as discussed in the last section. Thus, this factor must be even in both ~band E. Further, it must vanish if no particles (p - 0) or no holes (1 - p - 0) are present locally. The simplest choice satisfying these constraints is p(1 - p)g. Here, g is a new, coarse-grained (odd) function of the microscopic E, whose

22


precise form need not be known, as is the case for the parameters 7- and u in 9~'[~b]. Thus, absorbing any other constants in the definition of ~, we write je = 4p(1 - p)~ = (1 - ~b2)~.

(2.13)

For the region of our interest (T > Tc), we may neglect higher powers of ~b. Indeed, for the study of universal behaviour, we can show that all the neglected powers are irrelevant in a technical sense. A case involving odd powers of ~b- which is unphysical in the context of the standard model - has been considered by Honkonen and Kupiainen (1987). As a first guess at the equation of motion, it is tempting simply to supplement the right-hand side of (2.10) with an additional term

- f .jE = r

2,

(2.14)

where o~ is the magnitude of r and ~ denotes the gradient operator in the field direction (henceforth called the longitudinal direction). However, there is every reason to believe that full isotropy, exemplified in (2.10) and (2.12), would not survive the process of coarse-graining a strongly anisotropic microscopic dynamics such as ours. In other words, terms like r will generate anisotropies in both (2.10) and (2.12). Thus, simply adding (2.14) might be adequate only for considering the effects of E at the lowest order. For general cases, especially for saturation fields, we should widen our parameter space and consider couplings associated with longitudinal gradients being different from those controlling transverse gradients. Similarly, the noise matrix is expected to deviate from unity. Due to CRinvariance, even (odd) powers of ~ must be accompanied by even (odd) powers of 8. Finally, all properties associated with the ( d - 1)-dimensional transverse subspace are isotropic. The resulting Langevin equation of motion can now be written down. Denoting the transverse gradients by V, we have t) - ),

- v2)v2

+ ("is -

) 2eP - 2 ,, 2v2eP

U (V2~3 q-/'i;~2q~3)nt- e ~ 2 } -- (V" --* ~q- ~r q- 3.1

(2.15)

Note that we have absorbed the coefficient of V44~ into )~ and that of V2~b3 into u. Since we expect the noise correlations to be anisotropic, we denote the transverse and longitudinal Langevin currents by { and if, respectively. N U is still diagonal, but isotropic only in the transverse subspace. Denoting the two distinct values by n• and n I, we have, instead of (2.11 b) and (2.12), ...+

(V .{~Y, t ) V ' . ~'(E", t')) = n•

-- .~')(5(t - t')

(2.16a)


23

and ,')/-

,ll

-

- t').

(2.165)

Not surprisingly, equations (2.15, 2.16a, b) are special cases of the very general class of "driven diffusion equations" which may be derived, in principle (Eyink et al., 1995), from the microscopic model. Thus, we should regard all parameters appearing in these equations as functions of J, E and T. While we do not need to know these dependences in detail, a few basic properties must be established. Firstly, equations (2.15-2.16) may appear to describe nothing more than an equilibrium system with anisotropic couplings in (2.1) and/or anisotropic transport coefficients. For such a system, however, FDT will continue to hold. Thus, at the very least, the noise matrix must be proportional to the diffusion matrix, Dij. (Note that, defined through ate - ViDijX~j r + . . . , D is parametrized by the AT's here.) In other words, for anisotropic systems near equilibrium, we must have T• -- nj_/nll. By contrast, driving with E takes the system out of equilibrium. Though the system will come to a steady state, its non-equilibrium character typically breaks the FDT. Thus we have every reason to expect, generically, 7• # n__z~' TII nil

(2.17)

In Section 3, we will explore the consequences of equations (2.15-2.17) for some of the simplest correlation functions. In particular, we will see that the FDT is broken at all T above criticality. Secondly, we wish to point out another major difference between this system and an anisotropic system near equilibrium. It lies in how the parameters carry information about the second order phase transition. For the latter, thanks to the FDT, both 7-11and 7• are inversely proportional to the static susceptibility. Thus, both must vanish at Tr with the same critical exponent 7, though their amplitudes may differ. Below Tc, both may be chosen negative, so that the Langevin equation admits inhomogeneous stationary solutions which are consistent with the phenomenology of the separation of phases by interfaces oriented in any direction. In contrast, the FDT does not apply in non-equilibrium systems, and, theoretically, three possibilities exist (Leung and Cardy, 1986; Janssen and Schmittmann, 1986 b; Gawedzki and Kupiainen, 1986) as T is lowered to Tr (a) (b) (c)

T• ~ 0 but TII > 0; zi > 0 but Zll ~ 0; and both vanish.

In practice, these parameters must be chosen so as to reproduce, at least qualitatively, what is observed. In simulations, only phase separation with

24


interfaces parallel to E is seen, leading us to adopt the choice: TII > 0 with T•

(2.18)

for describing criticality in the standard model. On the other hand, both nil and n• being variances of noise currents, must remain positive. Thus, FDT violation is, in the sense of (2.17), "maximal" at T~. Beyond criticality, many unusual properties of the system below Tr can be described by the choice Tll > 0 with T• < 0, as we will show in Section 5. While we take the position that (2.18) is "phenomenologically given", we recognize that there is some theoretical basis for this choice. In Section 4 we will give a perturbative argument, due to Leung and Cardy (1986), for TH being larger than T• by a positive contribution. Here, we argue on a more intuitive level as follows. Being the coefficients of the second order derivatives, Tii and 7• may be identified with the bare diffusion coefficients in the longitudinal and transverse directions. Clearly, they will be renormalized by both nearest-neighbour interactions and the drive. The former typically lower the diffusion coefficient, until it vanishes at T~. In the longitudinal subspace, however, the effect of interactions is reduced by the field which tends to break, or establish, nearest-neighbour pairs at random. Thus, we argue that, as T is lowered, the transverse diffusion coefficient should vanish first. This Langevin equation (2.15), together with the noise correlations (2.16) and FDT violation (2.17), forms the basis for most analytic studies of the standard model. In the critical region, T• is small, so that both the noise and the non-linearities must be considered. Since standard perturbation expansions fail in this region, one must take recourse to renormalized field theory, which will be discussed in Section 4. Well above criticality, the fluctuations of the local order parameter are small, so that the effects of the non-linearities are also negligible. The theory then becomes essentially Gaussian and is easily solved. In particular, we will see in Section 3 that the violation of the FDT leads to rather dramatic effects, such as long range correlations in the high temperature regime. Below criticality, we obtain a mean-field type description of the ordered phase if we neglect the noise terms. The deterministic part of the Langevin equation gives us a nonlinear partial differential equation whose solutions characterize the mean-field profile of interface configurations. This aspect of the Langevin equation, along with its consequences and further developments, will be discussed in Section 5. For later reference, this equation is invariant under the following operations" (i) (ii) (iii) (iv)

translations, in both space and time; "charge" exchange, i.e. E r - E and r ~ - r reflections of both the longitudinal coordinate y and the field E; and rotations in the ( d - 1) transverse directions.


25

The first three symmetries are inherited from the lattice gas model. In addition, the effects of lattice anisotropies should be irrelevant in the long wavelength limit, so that (iv) is present, as usual. As in the lattice model, combining (ii) and (iii) yields invariance under CR transformations, i.e. 4~r -4~ and y r - y . Finally, we end this section with a caution on terminology. We have referred to the inequality (2.17) as "FDT-violation", which, strictly speaking, is not correct. The FDT may be derived from detailed balance of the stationary distribution P*: W[(s - (s163

- W[(s --0 (s163

(2.19)

for allpairs of configurations. In a coarse-grained picture, we necessarily deal with a different set of configurations. Thus, both W and P* will be different also. Since some information is surely lost in this complex procedure, it is not surprising that detailed balance may hold at the mesoscopic level, even if it is violated at the microscopic level. Indeed, there are many examples of driven Ising models in which renormalization group methods allow us to compute, quantitatively, how FDT-violating contributions vanish as we focus on larger and larger length scales. In such cases, the reader may be confused by apparently contradictory statements in the literature concerning detailed balance in the "same system". For example, it is trivial to see that (2.19) is violated in the J - 0 standard model, given that P* c~ 1 and W[fi;' --0 ts r W[~ --0 1$'] in general. Yet, for d < 2, detailed balance is shown (Janssen and Schmittmann, 1986 a) to prevail in the coarse-grained system. Two further complications are: (i) using the Fokker-Planck equation, Graham (1980) proved that all non-equilibrium steady states satisfy detailed balance, provided the time reversal symmetry operation is appropriately generalized; and (ii) for any non-equilibrium system with P* c~ 1, through transforming into a rotating frame in {~} space by eAt, where A is the antisymmetric part of W[($ ~ (~' ], (2.19) can be satisfied with the transformed rates (Rficz and Zia, 1993). In view of the complex landscape surrounding this issue (Deker and Haake, 1975; Zia and Schmittmann, 1995; Eyink et al., 1995), we will choose a simplistic approach for this book. We will use the term "FDT-violation" to mean systems satisfying (2.17), even though some examples to be discussed (Sections 3.1.2 and 6.1.2) are indistinguishable from certain equilibrium Ising systems.

3

Long range correlations above criticality

It is well known that, except at the critical point, correlations in an equilibrium Ising model are short ranged, controlled by a finite correlation length ~.

26

B. Schmittmann and R. K. P. Zia

Specifically, they decay exponentially when r, a typical inter-particle separation, becomes large compared to ~. Translated into momentum space, this property takes the form of "analyticity at the origin". For the standard model, however, simulation data revealed that correlations behave quite differently. In this section, we consider these novel properties in the twoparticle correlation function (Section 3.1). Further, we call attention to the existence of correlations between odd numbers of particles, since the drive breaks Ising symmetry. Section 3.2 is devoted to the simplest example: the three-point function.

3.1

Generic singularities in two-point functions

Concentrating on the two-particle correlations, we first review the remarkable phenomenon of power law decays for all T > Tc. Meanwhile, their Fourier transforms, the structure factors, display a discontinuity singularity at the origin. However, we show that these singularities are quite distinct from the critical ones, that a finite correlation length is present, and that a hidden exponential decay, similar to equilibrium behaviour, may be extracted. Tracing the generically scale invariant behaviour to the violation of FDT, we discuss (Section 3.1.2) the connection with the existence of a fixed line, in the sense of the renormalization group, of Gaussian dynamic models. We should mention that generic power law decays in two-body correlations are observed in other non-equilibrium systems (e.g. Schmitz, 1988). In Section 7.6, we discuss in more detail the distinguishing features of the standard model within the wider context of these other systems.

3.1.1

Power law and exponential decays

For more than a decade, a number of authors (Onuki and Kawasaki, 1979; Onuki et al., 1981; Procaccia et al., 1979; Ronis et al., 1979; Kirkpatrick and Cohen, 1980; Kawasaki et al., 1980; Beysens and Gbadamassi, 1980; Tremblay et al., 1981; Desai and Grant, 1981; Duffy and Lutsko, 1986; Schmitz, 1988; Law and Sengers, 1989), interested in the behaviour of liquids under non-equilibrium conditions, began to discover that, in general, singular structure factors and power law decays in two-point correlations should be expected. Instead of a complex system such as a real liquid, Spohn (1983) considered a stochastic lattice gas, driven by density gradients only at the boundaries, and found long range correlations. Thus, such behaviour is, in a sense, expected in the standard model. Though Katz, Lebowitz and Spohn did not emphasize this point in their original papers, the singular property of the structure factor was already observed, at a temperature 60% above


27

criticality (Katz et al., 1984). Subsequently, power law decays in configuration space were explicitly measured (Zhang et al., 1988). Theoretically, they are found to lowest order in expansions around the high temperature limit (Zhang et al., 1988) and the E = 0 limit (Garrido et al., 1990). Here, we approach the subject from the mesoscopic point of view, starting with the Langevin equation (2.15). For temperatures far above Tc, a number of properties can be understood from a linearized equation of motion. This approach follows the spirit of fluctuating hydrodynamics (e.g. Cohen and Schmitz, 1985), in which linearized equations provide adequate descriptions for systems near equilibrium and far above the critical point. Clearly, the success will depend upon our choosing certain ranges of values for the parameters. Of course, this approach amounts to a "phenomenological" one, in which we learn qualitative features from careful scrutiny of the "data". Our starting point is:

~2 )v2r

i32)~2~b_ 20exi32V2r

-(V .{+ ~)

(3.1)

with the noise satisfying ({) = ( ff) = 0 and (2.16 a, b). Note that, for aesthetic reasons, we do not absorb c~• into A here. It is easy to find all correlations from a linear Langevin equation (Lax, 1966). Here, we provide a few details for convenience. Define qS(Y,t) ei(k'~+~t)05(/~,~o), so that the solution to (3.1) is simply dp(k.w) = {iw + a(/~)}-] (-i){/~• . ( + kllff}

(3.2)

A(/~) - A { r • 2 + rlik ~ + ( a • 4 + 2a xkilk• 2 2 + C~llk~)}.

(3.3)

where Since the noise has zero mean, we immediately obtain the expected (4~) = 0. Beyond that, we use (2.16 a, b) to obtain the full dynamic structure factor: S(k, a;) =_ (c~(k, w)c~(-k, - w ) ) =

k.N.

k

~2 q_ A2(]~) ~

(3.4)

where N denotes the (diagonal) noise matrix, i.e. lc. N . lc - n• 2 + nllk ~.

(3.5)

From (3.4), we arrive at the steady state structure factor S(k), the analogue of the static structure factor for equilibrium systems, by integration over ~v: S ( k ) - k . N . k / 2 A ( k ).

(3.6)

28


Writing 2A(/~) = / ~ . D . / ~ + O(k4), where D is the diffusion matrix, i.e.

kT. D. kT= 2A(r• 2 + rllk~) ,

(3.7)

we find the small k behaviour of the structure factor:

S(k) ~

k.~.k _. k.D.k

(3.8)

_..

For a non-equilibrium steady state, we should expect "FDT-violation" (2.17): 9f D, (3.9) so that (3.8) depends upon 0, the angle between k and E. In other words, S has

a discontinuity singularity at the origin. Unlike critical singularities, this discontinuity is finite and can be characterized by the ratio R =

S(k• ~ 0 k l l - 0) _.

'

.

(3.10)

S(k• = 0, klf ~ 0) Clearly, ]R I is the upper bound on the ratio of any other two limits of S(k). Before discussing simulation data, which show that R > 1 always, we digress for several remarks.

................

~-,-..~..

. / -~" .

.

.

.

.

.

.

....

",.,,

i ,/ .

9.,.,,

.... ...J

-.

t (a)

j..-'

>k•

(b)

Fig. 3.1 Schematic contour plots of the structure factor, S(k), for T > Tc, in systems with ordinary anisotropy (a) and anisotropy due to FDT violation (b).


29

Within the linear approximation (3.1), FDT is not strictly violated even though t~ ~ ~. Keeping in mind the comments at the end of Section 2 and in anticipation of dealing with non-linear terms (cf. 2.15), we will refer to (3.9) as an "FDT-violating" condition. If, on the other hand, we were to describe a system near equilibrium, then FDT would require ~ cx [D, so that R would equal unity and S ( k - ~ 0) would be a 0-independent constant, the static susceptibility. In particular, for an isotropic system, we would have T• = ~-II--T and N cx 0, the unit matrix (cf. 2.12). Thus, the familiar OrnsteinZernike form of the static structure factor, 1//3(~-+0(k2)), would be recovered. To emphasize the difference between our non-equilibrium steady state and an equilibrium system with anisotropic interactions or diffusion constants, schematic plots of contours of S(k) are shown in Fig. 3.1a, b. Note, in particular, that S in Fig. 3.1b is not monotonic in k• for fixed nonzero kll, in each quadrant. In contrast, it is monotonic in kll, for fixed k j_. This behaviour is one of the clearest signals of a discontinuity like (3.8) and (3.9). Since the structure factor is proportional to the intensity of radiation scattered from the system, it would be extremely interesting to observe such a pattern in an experimental realization of the standard model. We should point out (Lindg~rd, 1990; Bruce, 1992) that scattering patterns like Fig. 3.1 b were observed in, for example, uniaxial ferroelectrics with dipolar interactions (Skalyo et al., 1970). Similar behaviour is also found in structural phase transitions (Cowley, 1980; Bruce, 1980). In those cases, however, they arise from microscopic Hamiltonians with long range forces. There is a model (Kasteleyn, 1963; Sutherland, 1968) with short range interactions which displays long range correlations in equilibrium, but it relies on the presence of a global constraint which plays the role of long range forces. In this sense, a "singularity" already exists, for all these cases, at the microscopic level. In contrast, all the microscopics of the standard model are short ranged and analytic. Further, the similarity between these systems ends here. Underpinned by FDT violating dynamics, the driven system displays singularities in other quantities (see, for example, Section 3.2). Returning to our model, the singular structure is definitely observable in Monte Carlo simulations. In Table IIc of Katz et al. (1984), where S(k; T - 2.2) is presented, we see that R ~ 4. At T = 1.6 (15% above Tc), this ratio grows to ~10 (Hwang, 1993). In the thermodynamic limit, R is expected to diverge as T is lowered to Tc. In this sense, we may regard Tc as a point of maximal FDT-violation. In Fig. 3.2, we show the data for a 60 x 60 system at T - 1.6. Note that, by symmetry, only one quadrant needs to be displayed. To compare with the schematic plot Fig. 3.1 b, it is best to focus on ---4

30


/ /

/

s(k)

// 1,/,

/ /

12-////

/

10"////

8-// / / // / "'///m 'i// oLs

0

1

,

/ 3 2

3

4

5

6

7

8

9

10 11 12 13 14

4

roll

mj. Fig. 3.2 Simulation data of S(k) in a 60 x 60 system at T = 1.6. Here, m• - 60k• and mll - 60kll/27r. the fourth quadrant. In particular, the non-monotonic dependence on k• is manifest in this data. Next, let us show that this discontinuity singularity translates into power law decays in configuration space. Due to translational invariance, the twopoint correlation depends only on (s t), the spatial and temporal separation between the points, and is given by G(~, t ) Thus, the

(~b(~, t)~b(0, 0)) c(

equal-time correlator

Iei(fc'g+~t)S(lc, w).

(3.11)

is

G(.~, O)oc I eis

(3.12)

the large r (r = I~l) behaviour of which is controlled by (3.8), i.e.

G(r ~ c~) c( "J eir k. N. k

(3.13)

Since D is positive and diagonal, we can rescale the momenta and the corresponding co-ordinate by defining fit = I]])1/2. k

and

g' = [I])-1/2. .~'.

(3.14)

Thus, the denominator is just k '2 while the numerator becomes kT'. M./~', with M = D-1/21~D -1/2. (3.15)

Statistical

mechanics

of driven

diffuse

systems

31

Extracting this factor from the integral, we have 9--.! ...I

G(r' --, oo) c~ V '

i etk .x .~

. f

'

~ ,"~

k, 2

c< V

" V '(r')

2-d

(3.16)

For an equilibrium state, with or without anisotropies, the F D T ensures cx Q, so that G(r ~ c ~ ) = 0. (The reader should not be alarmed at this result; the neglected O(k 4) terms will produce exponential decay.) This proportionality is expected to fail for our non-equilibrium steady state, so that we can extract a non-trivial traceless part of ~ which we write as ~ . Denoting a unit vector in the radial direction by f' and carrying out the differentiation, we arrive at ^/ r - ~ , r

G(r'~

oo) cx

^/

(r') d

,

(3.17)

which is the celebrated power law decay. Note that rescaling does not affect powers, though amplitudes do change. In Fig. 3.3, we reproduce the data, for a system with T ranging from near criticality up to 6.0, from Zhang et al. (1988). We see clearly the power - 2 , displayed by the two-particle correlations for, in this case, longitudinal separations. Though much more difficult, correlations in the transverse direction for this system were also measured, with the main conclusion that they are negative for large separations. This fact is consistent with the traceless nature of M. The same authors also studied a 16 x 16 x 256 system at many temperatures above To, and found excellent

. . . .

-2

--

I . . . .

,

I

. . . .

i

. . . .

9 : = ~

log G(r) 9

_

.

o,

--

"

",",,,

'..

-4 9

ldll~ll~ i "

9

.

, , -...

9

~:~

9.

,

0

,

i

1

2

3

i I~I

4

log r

Fig. 3.3 Simulation data of G(r) in a 14 x 300 system at various T > Tc. The solid lines are of slope -2; the dashed one, -2/3. From Zhang et al. (1988).

32


agreement with the power - 3 . Though no simulations of systems in higher dimensions are available, there is every reason to expect the power law to remain valid. While we see how long range correlations appear in this case, it is natural to ask if there is a necessary set of conditions for their occurrence. Though no rigorous proofs exist, strong arguments have been advanced (Zhang et al., 1988; Garrido et al., 1990; Grinstein, 199 l) in favour of the following: (i) (ii) (iii)

a conserved dynamics; a non-equilibrium steady state; and spatial anisotropy associated with the dynamics.

The first ingredient already produces power law decays even for equilibrium systems, in G(~, t) for fixed ~ and large times, i.e. G(t ~ ~ ) o( t -d/2. Since t scales like r 2 in diffusive processes, we may naively expect G(r ~ oc) oc r -d from scaling, instead of the usual e -r/~. This expectation would be correct, if it were not for the validity of FDT, which forces the amplitudes of such terms to vanish. In this sense, an equilibrium state is "special", "atypical", or "constrained". The role of the second ingredient is to lift this "constraint", so that power laws in r are again generic. The third ingredient is more subtle, necessary only for producing power laws in the two-body correlations (Grinstein et al., 1990). To end this section, we exploit the tracelessness of ~ and extract a hidden exponential decay G. Now, power law decays in two-point correlations are reminiscent of the behaviour in the equilibrium Ising model at criticality. However, there is a significant difference. In the latter, the power law decay leads to an infinite integral f Gdd x, which is the (static) susceptibility. Naively, the same may be expected for the driven case. However, the amplitude of the r -d is far from positive definite, as can be seen in (3.17). Indeed, the traceless nature of ~ leads to a null amplitude for r -d once an average over the angles of f' is performed. Thus, f Gddx(oc ~ Gddx ') is convergent, though we should not identify this quantity with the susceptibility. Recognizing this property, we may indeed recover the conventional wisdom that "correlations away from criticality decay exponentially." To do so, we define an averaged correlation function by (~(r') - J" df~' G()?' 0) -~d ' '

(3.18)

where Sd = 27rd/E/F(d/2) is the surface of a unit sphere in d dimensions. Note that, in terms of the original coordinates, (3.18) corresponds to an average over an ellipsoid. Clearly, this average destroys the power law (3.17). To see the exponential in the remainder, we onlyneed to study the Fourier transform of (3.18), i.e. a similar average of S(k), (3.6). Thus, we


33

consider _

r

S(k') =- J Sd 1 + k'2(a~ sin 4 0 + 2a" sin 2 0cos 2 0 + all cos 4 0)'

(3.19)

where a~ - a• a" =_ a•177 and a l l - all/2~rl~. The numerator is simply n'~ sin 2 0 + nll cos 2 0, where n~ - n•177 and nll= nll/2~rll. Being interested only in behaviour to lowest order in k', we may expand the denominator, carry out the angular average and recast the result as -

1 -Jr-~ 2 k t 2 '

(3.20)

where )~ = [ ( d - 1)n'L + nil]/d and ~2 is [(a + 3)(d 2 - 1)a'~ + (d 2 - 1) (2a" + ail)]n~_ + [3(a- 1) (a~_ + 2 a ' ) + 15ail]nll (d + 2)(d + 4)[(d - 1)n~_ + nll] Note that O(k 4) terms in the denominator are neglected. Again, we emphasize that the quantities in (3.20) should not be thought of as nonequilibrium versions of equilibrium ones. Since the FDT is violated, there is no simple relationship between response and correlation functions (Zimmer, 1993). However, for a check on (3.20), we may set all anisotropies to zero (i.e. Tj _ - TII- T, etc.) and recover the equilibrium results with 2 being the static susceptibility and ~ the correlation length (divided by 2AT). Having found S to be of the Ornstein-Zernike form, it is a standard step to arrive at the exponential decay in configuration space: -

e-r'/~ [d-l]/2"

l

G(r --, oo) oc

(3.21)

(r')

3.1.2

A fixed line of Gaussian dynamic models

In the last section, we found that the singularities in S are finite, unlike the case for equilibrium systems at criticality. Nevertheless, the large distance behaviour of G is dominated by power laws. We have learned, since the introduction of renormalization group ideas in statistical physics (Patashinskii and Pokrovskii, 1964; Kadanoff, 1966; Fisher, 1967; Wilson, 1975; Wilson and Kogut, 1974), to associate such scale invariant behaviour with fixed points. Here, we demonstrate that this is possible. Further, the high temperature properties of the standard model are found to be controlled by a line of fixed points, in contrast with a single, T = c~, fixed point for equilibrium Ising models (Zia and Schmittmann, 1995). To investigate the effects of the drive, van Beijeren et al. (1985) considered a

34


simplified version of (3.1) with the additional t e r m ~0q~2" ~t ~b(s t) = A{T(V 2 + 02)~b + g~b 2} - ( V . ( + ~(),

(3.22)

with isotropic noise. They argued that these dynamical equations are appropriate for a driven lattice gas with no inter-particle interactions, a system which may be regarded also as the high temperature strong field limit (T,E-+ c~, with fixed E/T) of the standard model. Using modecoupling techniques to lowest order, anomalous diffusive behaviour is found for d < 2, with logarithmic corrections appearing in d = 2. Recalling that the steady state distribution for a J = 0 driven gas is a constant (Section 2.1), it is remarkable that, nevertheless, there are singularities in this system. Of course, these anomalies are displayed only in the t-dependence of some quantities and, in the form of anomalous dimensions, for only one physically accessible dimension. Deferring the discussion of these and related phenomena in d = 1 systems to Section 6.7.2, we note that there are no "transverse" dimensions here. As a result, none of the interesting features associated with anisotropy can exist. Motivated by the discovery of anomalous diffusion, and as a prelude to studying critical properties in the standard model, Janssen and Schmittmann (1986a) re-analysed this problem within the framework of dynamic renormalization group. To check the validity of using isotropic noise and diffusion matrices and to study the renormalization due to 8, these authors considered the fully anisotropic case, i.e. (3.22) with two n's and 7-'s. Verifying dc, the upper critical dimension, to be 2, a systematic expansion in powers of dc - d is set up and a "non-trivial" fixed point is found, for d < dc. Under the renormalization group, both diffusion and noise matrices develop anisotrophies. At the fixed point, however, the two matrices are proportional, i.e., couplings associated with the breaking of the FDT are shown to be irrelevant. Thus, even though the isotropic form of (3.22) is not validated, the FDT does hold, at the fixed point. In other words, for d < 2, the lowfrequency and long wavelength behaviour of the "free" gas and the standard model above Tc are identical, both given by an anisotropic, 8 # 0, theory. On the other hand, for d > 2, the coupling g is renormalized to zero. In the language of dynamicfunctionals, to be described in some detail below, such a linear Langevin equation is represented by a Gaussian distribution. Not to be confused with static Gaussian distributions, we will refer to our cases as dynamic Gaussian models. In particular, the static Gaussian model (Berlin and Kac, 1952) at criticality is controlled by a single fixed point (Fisher, 1974a; Wilson and Kogut, 1974). In contrast, we have here a fixed line of Gaussian theories, each being associated with non-equilibrium, FDTviolating dynamics.


35

To study Gaussian theories, it seems excessive to use a dynamic renormalization group approach. On the other hand, this approach is invaluable for investigating both anomalous diffusion of the "free" gas in d < 2 and critical behaviour of the standard model. Thus, we present some details here. Our interest will be averages of various functions of ~b, over the Gaussian distributed noise (2.16). A convenient representation of these averages is to introduce a Martin-Siggia-Rose (1973) response field, ~b(~, t), which takes account of the noise, and to write a weighted configuration sum over ~ and ~b: ("/-

[" ~ ~

b" e-j[6'r

(3.23)

where J [ r r is the dynamic functional. In this representation, dynamical averages are computed as in equilibrium statistical mechanics, with J and exp ( - J ) playing the role of the Hamiltonian and the Boltzmann factor, respectively. To cover all the cases of interest here, it is necessary to study a functional (Janssen and Schmittmann, 1986a) which corresponds to the anisotropic version of (3.22),

~f[~, ~] -- J dtddx{(b((b - A[T• V2q~ +

+ l$(n•

Til~2q~+ e~b2])

2 + nil ~2)q~},

(3.24)

where ~ - ~4)/~t. Beyond the g term, all operators with higher powers of gradients and fields, irrelevant in the renormalization group sense, can be ignored. Note that J f [ g - 0 ] is quadratic in the fields, so that e x p { - J / [ g - 0]} is just a Gaussian distribution. Indeed, we used the subscript f with the notion of "free gases" in mind. Analysing the scaling properties of i f , a number of parameters can be absorbed, so that only three essential couplings remain:

p = "rII//T_L., V O( g2/p3/2

w - "r•

and

(3.25)

The first measures anisotropic scaling. Its non-trivial renormalization for d < 2 accounts for anomalous diffusion (only in the longitudinal direction). The effects of the drive, in a perturbative treatment, enter solely through the second parameter. All theories in the v - 0 subspace are naturally called Gaussian. The last coupling is a measure of "FDT-violation", via ( w - 1). However, for a theory with w - 1 and 8 r 0, it is unclear whether FDT would be valid. Certainly, detailed balance is violated at the microscopic level, since P* cx 1 and W is not symmetric. Thus, it is significant that, at the mesoscopic level, such a theory can be cast in a form which manifestly satisfies detailed balance. To be specific, we write

Jz[w -1] - Jdtddx{gp[

-

-

]+

(3.26)

36


where )~ - n•177 -

+

~2) +

e(

(3.27)

e +

and

f W/[qS]- J dax{ lqfl}.

(3.28)

We believe that the driven "free" gas (J - 0) is described by this functional, while (3.28) is entirely consistent with the microscopic result P* o( 1. Returning to the general case (3.24), we study, in a loop expansion, the Wilson functions/3p,/3w and 3v, which control the renormalization group flow of the couplings. For d < 2, the Gaussian fixed line (v* - 0) is found to be unstable, with flow into a "non-trivial" v * - 0 ( 2 d). From ~p(v*), the anomalous diffusive behaviour can be computed; in particular, the density-density correlation function is found to scale as S(kll,k• ) f(k~tl+ZX,k2t), with an anisotropy exponent A - ( 2 - d ) / 3 , which is known exactly to all orders in ~ - 2 - d. The results, evaluated in the limit of d - 1, are identical to those using other approaches in d - 1 (Section 6.7.2). Such an agreement is surprising, since the methods here rely on a subtle interplay between longitudinal and transverse couplings, whereas in d - 1 there are simply no transverse dimensions. If this equality is not merely coincidental, then there must be a deeper "universality" associated with d - 1 systems. Turning to the FDT-violating parameter, the effect of v* -r is to drive w to 1, provided w < 3 initially. This remarkable FDTrestoring feature vindicates the use of a formulation like (3.22) to find the leading asymptotic behaviour of systems with N 9( D. However, we caution that FDT-violating operators, though irrelevant in this sense, are dangerously so (Fisher, 1974 b), just like ~4 in equilibrium Ising models in d > dr - 4. In particular, two-point correlations will still decay with power laws, only as r -d-~/6 rather than as r -a (Zia and Schmittmann, 1995). The results for systems in d > 2 are diametrically opposite. Here, v* - 0 is stable, so that the infrared behaviour of a driven diffusive system is described asymptotically by a Gaussian theory! Before jumping to the premature conclusion that the infrared properties of a driven system and a free gas in equilibrium are the same, we must consider the flow of the other two parameters. Specifically, both/3p and/3w are proportional to v. For d > 2, v flows to zero sufficiently rapidly so that both p and w will "freeze" at some nontrivial p* and w* which depend upon initial values. These final values will be "generic", so that in general, we would have p*-140,

w*-l#0

and

v*-0.

(3.29)

In other words, we have a fixed line of Gaussian models (Zia and Schmittmann, 1995). The anisotropy proportional to p * - 1 is associated


37

with a marginal operator (Bruce, 1974), and may be eliminated by a trivial re-scaling of, say, kll. However, the operator associated with w* - 1, is nontrivial. Mathematically, w* cannot be removed by rescaling. Physically, it is a signal of FDT-violation, distinguishing these models from the ordinary T - o c fixed point in a typical equilibrium system. Finally, we must again be careful in interpreting v* - 0 and refrain from concluding that all properties in these systems are identical to those in ~ - 0 ones. In particular, ( ~ 2 is a dangerous irrelevant operator (Fisher, 1974 b). The d - 2 case, being at a critical dimension, is more subtle. Here, v flows to 0 so slowly that w - 1 also vanishes! As in the d =/=2 cases, though the fixed point itself is identical to an equilibrium Gaussian theory, the operators associated with both v and w - 1 are dangerously irrelevant. In particular, both flows are controlled by logarithms (Zia and Schmittmann, 1995) typical of d - d e systems. Therefore, the effects of FDT violation should vanish rather slowly. At a naive level, we could expect M ~ 1/In r, for example. Thus, we should not be surprised to find pure 1 / r 2 behaviour in the relatively small d - 2 systems studied to date. It would be interesting to perform simulations on much larger lattices and compare with this prediction of cross-over. To summarize, this renormalization group analysis shows that the lowfrequency and long wavelength properties of the standard model, for all T > Tc and d > 2, are given by dynamic Gaussian models, parametrized by a T-dependent w*. These are the fixed points responsible for scale invariant behaviour such as r -a. Being Gaussians, these theories predict simple exponents. However, their FDT-violating character is crucial for the amplitudes of the spatial power laws to be non-vanishing. For d - 2 systems, though the fixed point itself is precisely an equilibrium Gaussian model, its approach is so slow that the present data is well fitted by r -2 alone.

3.2

Three-point correlation functions

In general, a multi-particle dynamic correlation function is composed of a multi-point vertex function (one-particle irreducible) and various two-point correlations and responses. Thus, we should expect the singularities in S(k) to induce some others in the multi-body correlations, even if the vertices are analytic. In this section, we investigate the simplest case: the three-point function $3. In particular, we show that it is divergent (Hwang et al., 1991, 1993) in the limit of small, generic momenta. Yet, it vanishes when the origin is approached in special directions! Not surprisingly, the behaviour also

38


implies long range correlations in configuration space (Garrido et al., 1990). Apart from such drastic discontinuities, the very existence of $3 is of interest. In the absence of the drive, the Ising symmetry in our model would ensure that all odd-point functions vanish, especially for T > T~. In Section 2, we see that, when the system is driven, this symmetry is replaced by CR invariance. Therefore, odd-point functions can exist, provided they are also odd in y or kll. Restricting our discussions to correlations at equal times, we consider G3({~}) = (r162 0)r In k space, this is denoted by

S3(kl,k2,k3) ~ J. (q~(kl,oJ1)q~(k2,od2)q~(k3,oJ3)).

(3.30)

Starting from (3.23 and 3.24), we seek $3 as a power series in 8, a procedure that is completely reliable for finding the long wavelength behaviour at T > T~ and d > d~. Note that this is not an expansion for small driving fields E. We emphasize again that the effects of E have been coarse-grained into the parameter set {r•177 8}, all of which may be considered to be O(1). However, renormalization group analyses assure us that 8 generates no infrared singularities above dr so that higher orders in a perturbative expansion merely "renormalize" the amplitudes, while the large distance behaviour is completely captured at the lowest non-trivial order. In this case, we need only O(8), and the result is

S3({k})

-2i 6(k, +

+

A(k~) + A(k2) + A(k3)

{k,

+ cyclic perms ,"

(3.31)

Note that, as required by CR, this quantity is odd under {kll } r {-kll }, while reality of G 3 forces it to be purely imaginary. Being interested in the long wavelength behaviour of $3, we need S(k) only to O(1), i.e. (3.8). Had FDT been valid, this would be a simple k-independent constant. The three terms in the last factor would add to {kill + k211+ k311}, which is zero due to the 6 function. To obtain a nontrivial result (Garrido et al., 1990), O(k 2) terms in S(/~) would then be needed, so that $3 vanishes as O(k) in the limit of small momenta. In contrast, let us consider the effects of FDT violation, which is characterized here by S(k) being O(1) and dependent on the direction of k. In general, the cyclically permutated terms sum to some non-zero value. Thus, $3 diverges as O(1/k), for generic, small momenta. On the other hand, there are many special directions (e.g. all momenta being transverse) for which $3 is identically zero. In this sense, $3 exhibits an "infinitely strong" discontinuity as {k } approaches the origin.


39

These predictions were tested (Hwang et al., 1991, 1993) in simulations on several L x L periodic lattices, using saturation E fields. After reaching steady state, G 3 is measured. For simplicity, and to exploit the difference between the momenta, we_,choose k 1 to be purely longitudinal and k 2 to be purely transverse. Since k 3 - - k l k2 by translational invariance, it will have both components. The_.three terms in (3.31) reduce to two, while $3 is proportional to S(kl) - S(k3), which is in turn proportional to ( R - 1), the factor that succinctly displays the importance of FDT-violation. Since $3 is even in {k• }, we may simplify further by integrating over k2• via setting x 2 • x3• To maximize the effect of the divergence at small momenta, we use the smallest allowed (positive) value of kl, i.e. 27r/L. The resultant is denoted by S. As predicted by (3.31), its real part is consistent with zero while the imaginary part is non-trivial. We show only the latter, plotted against T, in Fig. 3.4. Note that it is negative definite, having a pronounced dip near Tc. Being proportional to ( R - 1), we believe that this dip reflects the "maximal" violation of the FDT at criticality. Further, note that ISI increases with L, a behaviour consistent with $3 ~ O(1/k), since S is obtained from $3 by evaluating at the smallest momentum, 27r/L. In contrast, it would be difficult to explain this behaviour with $3 ~ O(k). We have seen that there are dramatic singularities in the three-point correlation functions, even in the disordered phase far from criticality. They depend sensitively upon a subtle interplay between the breaking of spatial isotropy, the Ising symmetry and the FDT. If isotropy were imposed (Grinstein, 1991), whether its origin lies in an isotropic microscopic dynamics or in an approximation scheme (Garrido et al., 1990), less drastic singularities, such as finite discontinuities in the derivatives, would result. Beyond the 1 .......... ~ - - g . - | - - - ~ - - - : : - - - : : - - a - - - a - - - ~ - - - - , , - - - ; ~ - - - a - - - a - - - - i i o

,~(T)

........ = ......... m----

o qm 0

[] "

:

cl

o

o "

.

,I, q,

-10 0.5

1 [5

9_5 T

Fig. 3o4 Simulation data of the imaginary part of S(T), for square lattices with L - 10 (m), 30([]) and 60 (§ From Hwang et al. (1993).

40


three-point function, we expect similar singularities in higher, odd-point correlations, though none have been investigated yet. Finally, there is the most intriguing question: How does one observe a three-point correlation in physical systems directly? All existing "measurements" of this correlation (Ravech6 and Mountain, 1972; Egelstaff, 1983), using light or neutron scattering, are performed on equilibrium systems in which FDT holds: thus, it is obtained as the derivative of a two-point correlation with respect to density. Indeed, in contrast to the two-point function, which is directly accessible in scattering experiments, the three-point function is rarely discussed in the literature (Frisch and McKenna, 1965; Jacobaeus et al., 1980; Lohmann and Wirnitzer, 1984; Itzykson and Drouffe, 1989). Recently, Dietrich and Fenzl (1989) proposed a method to extract a three-point correlation from the nonlinear lattice deformations produced by the particles. Though ingeneous, this method is not generally applicable, e.g. not in liquids. It would be most desirable to develop experimental techniques to measure these correlations directly.

4

Critical phenomena

In a celebrated paper, Onsager (1944) demonstrated that, in the equilibrium Ising model, a second order phase transition occurs at Tc-~ 0.5673J/kB and Pc = 1/2. We saw in Section 2 that a transition from a disordered to a phase separated state survives the addition of a driving field. In this section, we study the nature of this transition in more detail, using both simulations (Section 4.1) and field theory techniques (Section 4.2). The former show that the transition remains second order, so that the powerful methods of field theoretic renormalization group can be exploited. The novel features provided by the drive here include, for instance, the longitudinal and transverse momenta scaling with different powers, resulting in the upper critical dimension being 5 instead of 4. All can be traced to the "maximal" violation of the FDT, or equivalently, to a breaking of the supersymmetry (Gawedzki and Kupiainen, 1986) which underlies equilibrium dynamics.

4.1

Simulation studies

The typical configurations shown in Figs. 2.3, as well as similar ones in d - 3 cubic lattices, give a hint that the phase transition in undriven systems persists


41

at all E ~ 0. In this section, we review simulation studies of a number of key correlations which show the quantitative aspects of this transition. We concentrate on d = 2 results only, since very few simulations have been carried out in d = 3 systems (Marro et al., 1987; Zhang et al., 1988). All numerical references to T are given in units of the Onsager Tc.

4.1.1

Signals of a phase transition

For the equilibrium Ising model with non-conserved dynamics, the signal of the phase transition is the spontaneous development of a non-zero uniform magnetization: M, the average of ~ - ~_,i si/N. Since the symmetry is broken spontaneously, both 93/> 0 and 9J/< 0 states are equally probable, so that, for a finite system, the signal of the transition is not clearly carried by M. Instead, (93~2) is used for detecting the onset of order. Above Tc, the 0(1 IN) term is proportional to the susceptibility, which is divergent as T is lowered to Tc. Thus, monitoring the finite-size dependence of ( ~j~2 ) will reveal considerable information about the transition. In our case, with conserved dynamics, 93/is just a constant. To find the critical point, 93/must be set at zero, or, in the lattice gas language, the overall density must be set at 1/2. The signal of a transition is phase separation in which ~b(s (or the local density) becomes inhomogeneous. Thus, instead of ~ , we must use ~b(k) with a non-zero k. Now, all simulations show, especially for large E, that the system separates only in the transverse direction. Further, as the typical configurations on the square lattice with PBC show, the ordered state consists of two strips, one with average "magnetization" being positive and the other negative (or density being greater/less than 1/2). To distinguish this state from a homogeneous one, consider 4~10= 4~(k• = 27r/L, kll--0), which would be non-zero for the former state and zero for the latter. As in the non-conserved case, monitoring (4~10) would not lead to a clear signal of the transition. In this case, the phase of 4~0 is connected to the position of the high density strip, "washing out" (q~10). Instead, (I ~10[2) plays the same role as (9312), and can serve as an order parameter (Katz et al., 1983, 1984) in identifying the transition in the standard model.

4.1.2

Two-point correlations

Of course, just as (gJ/2 ), (] 4~10]2 ) is a special two-point correlation function. In a simulation, all two-point correlations are just as easily studied. This section is devoted to the critical behaviour of a selected number of these. In the first three subsections, we concentrate on large distance properties, while in the last, we study correlations of the nearest neighbours. Since we make use

42


of (2.1) in specifying our microscopic model, we may define an average "internal energy", U, in terms of nearest neighbour correlations.

Anisotropic structure factors, the order parameter and the exponents A,/3 Following Katz et al. (1983, 1984), the steady state structure factor in simulation studies of an L x L system is defined, in spin language, by S(]~; L) = L -2

y ~ s(.~) e ikT"~

,

(4.1)

X

where, instead of i, the site label is denoted by Y, a pair of integers ranging from 1 to L. Similarly, k = (k• is also discrete:

k• = 27rm•

and

kll = 27rmll/L ,

(4.2)

with the m's being integers from 0 to L - 1. For convenience, we will use the notation S(mx,mll ) interchangeably. No confusion should arise, since these arguments are integer valued. The conservation law forces S(O) to be zero, in case the overall density is chosen as 1/2. It is easy to check that ([q~10[2} is just S(1,0)/L 2. In the thermodynamic limit, we expect (I ~10[ 2 ) to be zero above Tc and positive below Te. In Fig. 4.1, we reproduce the data of Katz et al. (1984) for S(1,0) with L -- 30 and E/kBT - 0, 0.75 and "oo". The dramatic rise near 0.3 was the first indication that the standard model also displays a second order transition. Of course, singling out S(1,0) for study is guided by the observation of phase separation only in the transverse direction. For an equilibrium model with large anisotropies, this phenomenon will also occur. However, in that case the "longitudinal" structure factor, S(0, 1), will display the same type of singularity (though with a different amplitude) as T is lowered toward Tc. Thus, it is important to explore S(0, 1) as well. So far, only one study focused on this quantity. A plot of both S(1,0) and S(0, 1) is given in Fig. 4.2 (Hwang, 1993). From the data, we may conclude that, unlike its counterpart in the equilibrium model, S(0, 1) is either non-singular or weakly singular. Though a more refined study is needed, this drastic difference leads us to believe that R--, o~ (in the thermodynamic limit) and that (2.18) is an appropriate description of our model near criticality. To re-emphasize the difference between our system and equilibrium Ising ones with anisotropies, we will introduce the term strong anisotropy in reference to cases where some ratios of the susceptibilities (1/r) diverge as T ~ To. In contrast, all ratios remain finite in the latter systems, for which we will use the term "weak anisotropy". Closely related is the existence of different exponents associated with transverse and longitudinal momenta, so that an independent anisotropy


43

300

s(~,o) E=o 200

100

E=0

0.0

0.2

0.4

0.6

J / 4 k BT

Fig. 4.1 The structure factor S(1,0), which could serve as an order parameter, plotted against the inverse temperature. The exact value for the equilibrium model is J/4kBTc ~ 0.4407. From Katz et al. (1984).

16"

10-

Ill

S(0,1) 9 0

4.

4,

9

9

,:4

T Fig. 4.2 The structure factors S(1,0) (m) and S(0, 1) (§ as a function of T for a 60 x 60 system driven at infinite E.

44


exponent, A can be defined:

kll ~ k l+~ .

(4.3)

This notion will be discussed in more detail in Section 4.2.1. Here, we return to the simulation data of an order parameter. Though S(1,0) could serve as an order parameter, Katz et al. (1984) introduced

M* - V/(gJ/~} - (9J/2},

(4.4)

to make closer contact with the magnetization deep within one of the strips. Here, 9J/~ and 99l2 measure, respectively, the square of the magnetization in the longitudinal and transverse directions. For example, 9J/~- L - ' E •

[

L-IZ

s(Y)

,

(4.5)

II

where the sums are over x• and Xll. Plotting M* (in the L - 30, E - c~ case) as a function of T, they concluded that/3, the magnetization exponent, is greater than its equilibrium value of 1/8. Subsequently, Vall6s and Marro (1987) carried out an extensive study. Using an isotropic finite-size scaling analysis, they concluded that, at saturation fields, / 3 - 0.230 + 0.003 and Tc(~) - 1.355 + 0.003. This value of/3 lies well outside 0.5, the prediction of field theoretic renormalization group analyses (Janssen and Schmittmann, 1986 b; Leung and Cardy, 1986). A more recent effort (Leung, 1991 a, 1992) takes into account the implications of strong anisotropy which calls for, naturally, rectangular lattices with LII c~ L~_+A. If A _7/=0, the choice (4.4) for the order parameter suffers in several ways (Wang et al., 1989; Leung, 1992). Instead, the average of 9~1 - sin (Tr/Li)2Lll E~ s(.~)ei/~'~] ", / ~ - (27r/L• 0)

(4.6)

i

is used. Note that 9J/ is proportional to the magnitude of r so that it is directly related to the structure factor (4.1) by S(1,0) o((9J12). Deferring details of this type of anisotropic finite-size scaling analysis to a later section (4.1.4), we quote the findings. The data are consistent with /3-1/2

and

A-2,

(4.7)

while arriving at a new value of the critical temperature: Tc(oC) - 1.41 + 0.01.

(4.8)

Furthermore, Leung (1991 a, 1992) showed that it is possible to extract other


45

values of an effective ~ with other choices of lattice geometries, which "explains" the discrepancy between his result and earlier ones using square lattices. (In some sense, A is effectively 0 for square lattices.) Thus, a stronger statement cannot be made at present (Wang, 1995). It must wait for a better understanding of finite-size effects in a non-equilibrium system such as ours. Finally, we remark on a susceptibility-like exponent -~ which is associated with the divergence of S(1,0) as T ---, Tc. A measure of the fluctuations, would be the usual 7 for equilibrium systems, thanks to the FDT. Here, -~ and may be distinct and we will focus on the former. Though no one has reported a value for "~, we believe that Leung's data for T > Tc provide some evidence for ~ = 1. The uncertainty lies in the possible difference between ( ~ ) 2 , which is the square of the measurements and (O~l2), which is proportional to the structure factor. A reliable study of S(1,0) would obviously be very desirable. Critical power law decays and the exponents rl, u The previous subsection is devoted to the critical properties of the two "lowest" components of the structure factor: (1,0) and (0, 1). Beyond these, critical fluctuations affect all small k components, building up anomalous power law behaviour, for d Tc. At To, however, critical fluctuations should modify the power - d . In particular, for T > Tc, it is natural to expect that, up to some correlation length, ~, G(V) will decay as r raised to some power other than d. In analogy with equilibrium cases, we label this power by 2 - d - rl'. Due to strong anisotropy, r/' is not identical to r/, a symbol we reserve for the anomalous dimension of the field q5 (cf. Section 4.2.1). For r > ~, a crossover to r -d is expected. Extending the initial effort of Vall6s and Marro (1987), Zhang et al. (1988) demonstrated that this expectation is correct for the "longitudinal correlation" Gl(r) - GIs177= 0, Xll - r). Referring to Fig. 3.3, we see that Gz is given essentially by r - for T >> Tc. Meanwhile, by comparing the dashed line with the data for the lowest T (1.37 is slightly above Tc), it is entirely reasonable to conclude that there is a distinct break from r -2/3 to r -2. Though the authors (Zhang et al., 1988) made no claims along these lines, we believe that the data are consistent with

r/iI - 2/3,

(4.9)

where we have inserted a subscript to indicate its origin in the longitudinal correlation. Certainly, the authors drew a similar conclusion for their data in d - 3, i.e. a crossover from r -~ to r -3. To extract a correlation length, Gz was fitted phenomenologically to a

46


Lorentzian in r (Vall6s and Marro, 1986, 1987): 1/[1 + (r//~ll)2]. Making a log-log plot of~l I against r = (T/Tc) - 1, Zhang et al. (1988) concluded that Ull = 0.7.

(4.10)

Since the Lorentzian does not take into account the crossover to a critical power law, one may question the reliability of (4.10). Unfortunately, no other forms have been used to fit Gt, so that no comparisons can be made. Furthermore, the added complications due to strong anisotropy are not fully understood, so that the relationship between this definition of Ull and the one associated with longitudinal momenta (kll ~ 7-vii) remains murky. An anisotropic scaling study of Gt would be very desirable. The picture for "transverse correlations" is even less clear. In all the simulations to date, these correlations appear to be barely above the noise. One source of the difficulties lies in the negative amplitude associated with the r -2 correlations. As T ~ Te, positive correlations at small r build up and compete. A possible definition of the correlation length is the point where G vanishes, though none has been suggested. To quote Zhang et al. (1988), "A determination of the transverse correlation length is very difficult." A direct determination of v• will require considerably more effort. Nevertheless, bypassing the measure of correlation lengths, indirect evidence of both v's exist. It comes from investigations of finite-size effects with strong anisotropy (Leung, 1991 a, 1992; Wang, 1995), in which the variables rLi/Vll _and rL 1/~1 enter. All data collected for both the order parameter and (gJ~) are consistent with u11=3/2

and

u•

(4.11)

Unlike in (4.10), the exponent here can be identified with the one longitudinal momenta, since the smallest non-zero kll is precisely 27rLii.

Response functions and susceptibilities Measuring the time dependence of (s (~, t)s (0, 0) ), it is possible to extract the response function, which in turn will yield susceptibilities x(k, ~o). To find the critical properties, we naturally concentrate_, on the response of the system to a static "magnetic" field, i.e. x(k) - x(k, w - 0). But the dynamics conserve total magnetization, so that our system will not respond to a uniform "magnetic" field and X ( 0 ) = 0. On the other hand, like the structure factor, we should expect X(1,0) and X(0, 1) to be dominant. Finding out how these diverge as T ~ Tc will provide a natural definition of the exponents "),• and 3'11,respectively. Since the FDT is violated, these functions could provide information not carried by the steady state (t = 0) structure factors. Such investigations remain to be pursued.


47

Internal energy and specific heat Another well-known signal of a second order transition is the divergence of C, the specific heat. By definition, it is OU C _- ~ T '

(4.12)

where U is an internal energy per site. Since we employ (2.1) in the specifications of our microscopic model, we will defined U by

U-

(5) -

/

Y ~ s ( Y ) s ( Y + d) x~a

)

,

(4.13)

where {d} is a set of lattice spacing vectors. Thus, U is proportional to the correlation of the nearest neighbour spins. Note that .~ is just J*f~/N + J/2. Measuring U(T) for an L = 30, E = c~ system, Katz et al. (1984) found that C peaks around roughly the same Tc. A more extensive study (Vall6s and Marro, 1987), also conducted with saturation fields, gave convincing evidence that C diverges (Fig. 4.3). However, no critical exponent c~ has been reported. A novel feature in the driven system is anisotropy, so that correlations of nearest neighbours in the longitudinal and transverse directions are not the same. Several studies (Katz et al., 1984; Vall6s and Marro, 1987; Zhang et al.,

'

'ii il II

2-

1

0

0.~

[

1.0

|

1.3

,

,,

T/Tc(O) Fig. 4.3 Specificheat, found by measuring U(T), for systems with various L, ranging from 15 to 100, and E = co. Extrapolations to an infinite system are denoted by II. From Vall6s and Marro (1987).

48


1988) collected data on these separate "energies'" UII and U• When plotted vs. T, peaks in the slopes have been noted (Zhang et al., 1988). In principle, two c~'s can be defined. Large scale simulations are needed to clarify the nature of such novel behaviour. Another interesting phenomenon connected with the anisotropic dynamics is the change in "energy" during a Monte Carlo step (MCS). By definition of steady state, the change associated with all jumps is zero. However, if we separate the change associated with longitudinal jumps from those associated with transverse jumps, we find that (Hwang, 1993) (~/~t)l I - -I~/~t)• > 0, (4.14) where the notation is self-explanatory. We believe that (4.14) is one of the unmistakable hallmarks of a non-equilibrium steady state. In equilibrium, both are zero as a result of detailed balance. In our model, the inequalities in (4.14) are entirely consistent with the notion that the drive feeds energy into the system via jumps along E while energy is lost to the thermal bath via transverse jumps. In a steady state, the two rates balance, of course. Presumably, these quantities also develop critical singularities. A detailed study of (~fO/Ot)l I - ( ~ / ~ t ) • which is, like the total current, identically zero in equilibrium systems, would be exceedingly fruitful for providing insight into the critical properties of this non-equilibrium steady state system.

4.1.3

Higher correlations

Apart from two-point functions, correlations of higher numbers of particles are expected to display singularities near Tc also. For equilibrium systems, thanks to the FDT, derivatives of two-point functions with respect to T or other control variables will generate a subset of these higher correlations. A well known example is the specific heat in the Ising model, which, being related to the energy fluctuations, is (the integral) of a specific four-spin correlation. In a non-equilibrium case, the lack of FDT implies that we can expect new information when higher correlations are measured.

Three-point function As we have discussed in Section 3.2, the three-point correlation is also a direct measure of the FDT-violating non-equilibrium aspect of the standard model. Simulation data on S, a particular three-point function, show that it displays generic singularities at all T > Tc (Hwang et al., 1991). Beyond that, the prominent feature near Tc (Fig. 3.4) gives a hint that S is quite sensitive to critical fluctuations. At present, a detailed study of this peak, especially including anisotropic scaling, is lacking. Having little data to report, we will briefly discuss an example of the wealth of information which can be extracted from the three-point correlations.


49

Some of the singularities of $3, as we mentioned, can be traced to those in the two-point functions. Thus, it is important to "factor out" such contributions from the data. The remainder is then assigned to various three-point vertex functions: •3-n,n. Understandably, the most important one for our model is F1,2, associated with the operator ~ 4 ~ 2. In the renormalization group sense, this operator is precisely the most relevant and plays the same role as the four-point coupling u4~4 for the equilibrium case. Thus, for d < 5 (the critical dimension here), it will take on a non-zero fixed point value. Correspondingly, a universal value exists for an appropriate combination of two- and three-point correlations at Tc. Measurements of this kind of quantity, in the spirit of Binder' s (1981 ) study of cumulants in the equilibrium model, should yield an accurate fix on Tc as well as certain universal properties of the system. Fluctuations & the structure factor and the &ternal energy Strictly speaking, the steady state structure factor has no fluctuations, since it is by definition (4.1), the average of two-particle operators. When the squares of these operators are averaged, a certain four-point correlation is measured. From that, we can study AS 2, the "fluctuations in the structure factor". Away from criticality, these appear to be just as large as S (Hwang, 1993). This seemingly surprising result turns out to be a simple consequence (van Beijeren, 1990) of a Gaussian theory (Section 3.1.2). Based on this fact, Leung (1991 a, 1992) collected data on the moments (~f~4) and (~f~2), in order to study the cumulant

2 - (9~4)/(9~2),

(4.15)

within the context of finite-size scaling with strong anisotropy. A generalization of Binder's (1981) cumulants, this quantity is found to vanish as T ~ c~, rising to 1 as T ~ 0. Plotted against T, curves associated with different sizes cross at Tc, allowing a precise determination of the latter. Moreover, in the critical region, though there are weak finite-size dependences (Wang, 1995) the data for various sizes collapse (Fig. 4.4a), showing consistency with renormalization group results (4.11). Similar t o AS 2, fluctuations in the internal energy, A U 2 - - ( ~ 2 ) _ (~)2, are related to a special four-point correlation. For systems in equilibrium, there is a trivial relationship between the specific heat, C, and A U 2. Indeed, in simulations, a standard way to check on whether the system has "equilibrated" or not is to measure both C and C ~ - A U 2 / k B T 2 and to check if C = C'. In non-equilibrium systems, there is no a priori relationship, so that A U 2 should give us new information on the system.


50

0.7 9

I

I

0.6

gL

(a)

mA

+

o Am

0.5

i

I

O.3 0.2

o.,

o~-

o

§ +

0.4'

+

a-%

(~)

Q

a

9

tLII 213 0.7 zlo~,,1/2 / (b)

log( roll]

,., m+,4.+M SI" -0.1 .0.3 +.5 .0" '

"'

9

-1/2

(b) -i.s

-1

4).5

log( t L II

0

2/3

0.5

1

)

Fig. 4.4 Scaling plots of the renormalized coupling (a) and the order parameter (b). Here, gL is 8/3 times the cumulant (4.15), m is (gJl), and t = (Tc - T) / Tc is the reduced temperature. From Leung (1992).

We are aware of only one published result (Vall6s and Marro, 1987) on C', found in systems with L - 10, 15, 30, 50 and 60. It shows a dramatic difference from C, i.e. C' is consistent with being a monotonic function of T in the range of 0.7 to 1.5, showing no trace of the large fluctuations which usually accompany second order transitions! However, runs with over 2 x 106 MCS on a system with L = 10 (Hwang, 1993) appear to show a moderate peak near Tc. Detailed comparisons between these data and C have not yet been made, but they definitely show a less dramatic form of FDT violation than that found by Vall6s and Marro (1987). Obviously, a careful study of C' is desirable, both for clarifying the differences between C and C' and for extracting singular behaviour of this particular four-point function.


4.1.4

51

Finite-size effects in the standard model

Mathematical singularities do not occur in thermodynamic quantities of a finite system; they emerge only in the thermodynamic limit. To make meaningful comparisons between Monte Carlo simulations and theoretical predictions, it is crucial to have a firm grasp on finite-size effects. For equilibrium systems near criticality, there is a well established framework in which these effects are exploited to give information about singularities (Fisher, 1971; Barber, 1983; Privman, 1990). For non-equilibrium systems in general, much less is understood. For the standard model in particular, only phenomenological approaches have been explored, while a foundation based on the renormalization group, in the spirit of Privman and Fisher (1983) and Br6zin and Zinn-Justin (1985), remains to be established. We briefly summarize the results to date. For more details, the reader may consult a recent review on finite-size effects in this model (Leung, 1992). Initial efforts (Katz et al., 1983, 1984; Marro et al., 1985; Vall6s and Marro, 1987) followed the standard routes used in analysing equilibrium systems. Most of the simulations were performed on square lattices of various sizes, using isotropic finite-size scaling. These analyses led to, for example, a numerical estimate of /3 which is quite distinct from the field theoretic prediction. Recognizing the importance of strong anisotropy in the standard model and of similar features at Lifshitz points (Hornreich et al., 1975; Mukamel, 1977) and K models (Bhattacharjee and Nagle, 1985) in equilibrium, Binder and Wang (1989) proposed a generalized free energy and made some speculations for a system at Tc. More recently, Leung (1991 a) carried out a detailed study of anisotropic finite-size scaling, using rectangular lattices in a phenomenological approach with some field theoretic input. Making contact with the notion that longitudinal and transverse momenta scale with different powers at Tc, Leung employed a series of rectangular lattices respecting the relation (4.3), namely LII cx L l+a.

(4.16)

In this manner, any singular dependence on the "shape factor", LII/L[ , is then eliminated from various thermodynamic quantities. Specifically, Leung chose 20 x 20, 26 x 44, 32 x 82, 40 x 160, relying initially on the prediction A = 2 in d = 2 (Janssen and Schmittmann, 1986b; Leung and Cardy, 1986). Subsequently (Leung, 1991 b), A = 2 was determined independently from a study of the power spectrum of the average current. Measuring the first, second and fourth moments of 9~ in these lattices, scaling plots of renormalized coupling and order parameter (Fig. 4.4 a, b) provide strong support that the critical properties of the microscopic and the mesoscopic

52


models belong to the same universality class. In contrast, using square lattices of various L's (Achahbar and Marro, 1995) means that the shape factor vanishes as L ~ ~ , so that extra singularities are induced. A simple isotropic scaling analysis of the data can yield effective exponents which are combinations of two distinct exponents. Recently, a much more extensive study was carried out by Wang (1995), using runs up to 108 MCS and systems up to 64 x 1024. The conclusions are essentially the same as Leung's: the data are consistent with field theoretic predictions. To make stronger statements, it would be necessary to go beyond this phenomenological approach. Though renormalization group analyses have been applied to dynamic finite-size scaling (Goldschmidt, 1987; Niel and Zinn-Justin, 1987; Diehl, 1987), all are concerned with pure relaxation near equilibrium (model A). A thorough investigation of model B critical dynamics may be pursued as a prelude to establishing a theoretical foundation for finite-size effects in the standard model in which the dynamics is both FDT-violating and strongly anisotropic.

4.2

Theoretical investigations

The central hallmarks of critical behaviour in an equilibrium system are scaling, universality and non-analyticity. Initially formulated as a unifying principle for a wealth of experimental data, the scaling hypothesis (Widom, 1965a, b; Domb and Hunter, 1965; Kadanoff, 1966; Patashinskii and Pokrovskii, 1966; Fisher, 1967; Griffiths, 1967; Ferrell et al., 1967; Halperin and Hohenberg, 1967) states that the leading singular parts of all thermodynamic quantities are generalized homogeneous functions of their arguments. Subsequently, renormalization group analyses (Kadanoff, 1966; Wilson and Fisher, 1972; Domb and Green, 1976) succeeded in providing a framework in which scaling properties can indeed be derived, universality is understood and singularities can be computed. Here, we will first (Section 4.2.1) follow this development and discuss scaling properties of systems with strong anisotropy, such as the standard model and a range of other nonequilibrium steady state systems. The novelty here is A, which emerges as a new, independent exponent, in addition to the familiar set. We should mention that such a concept is also present in some equilibrium systems, e.g. uniaxial ferroelectrics/ferromagnets with dipolar interactions (Larkin and Khmel'nitskii, 1969; Aharony, 1973; Br6zin and Zinn-Justin, 1976), structural phase transitions (Cowley, 1980; Bruce, 1980), and uniaxial Lifshitz points (Hornreich et al., 1975). However, its consequences and significance were not extensively explored until recently (Binder and Wang, 1989; Binder, 1990). Finally, we will review the field theoretic renormalization group study of the standard model in Section 4.2.2.


4.2.1

53

General scaling laws with strong anisotropy

The objectives of this section are threefold: to postulate appropriate anisotropic scaling forms; to define a set of associated critical exponents; and to study the scaling laws that link them. To provide contrast for the notion of strong anisotropy, we will begin with a brief review of the concept of weak anisotropy, which characterizes the dynamic scaling properties of an equilibrium Ising model with anisotropic exchange interactions. Another difference here is the absence of a free energy, which is a customary starting point in discussions of scaling. We also avoid beginning with the magnetization, since ~J~ - 0 in the standard model. More suitable candidates are the two-point correlations and the structure factor. A further advantage is that, in these functions, strong and weak anisotropic scaling are clearly distinguishable. To re-emphasize, the behaviour described in this section applies only to the leading singular parts, while subleading singularities are known as "corrections" to scaling. As T approaches Tc, the correlation length, ~, in the equilibrium Ising model diverges, so that thermodynamic quantities are dominated by this length rather than microscopic details, and exhibit scaling behaviour. A convenient way to characterize such homogeneous functions is to introduce, say, an external momentum scale #. We begin with two examples: the dynamic structure factor and its spatial Fourier transform, the two-point correlation function. These obey

S(k, t, 7-)

#-2+~S(k/#, t# z, ~_/#1/~),

(4.17 b)

G(Y, t, ~-) - #d-2+OG(.~#, t# z, T/#I/"),

(4.17 b)

-

where k is the wave-vector, Y is the distance between the two-points, t is time, ~- is the critical parameter ~ ( T - T~), while ~/, u and z are the familiar critical exponents (see, e.g., Stanley (1971)). Note that S is written in the time domain here, whereas in (3.4) it is expressed in ~o-space. Several comments are in order. First, note that all components of k scale linearly with #, i.e. this structure holds for a//momenta, independent of their orientation. A similar structure holds for all lengths. Second, we emphasize that the presence of anisotropies in the exchange interactions leads to no new scaling forms or universal behaviour, apart from a simple re-scaling of the different components ofk or Y (Landau and Swendsen, 1988). In particular, a single exponent u suffices to characterize the correlation length, and only one z is needed for the relaxation times, so that the only remnant of the anisotropies lies in the amplitudes of the power laws. Finally, note that the same exponent ~ is used to describe the behaviour of the t = ~- = 0 correlations for both small momenta and large distances. Reminiscent of the correlations in


54

the standard model at high temperatures (Section 3.1.1), this type of anisotropy will be referred to as "weak". The main feature is that the exponents and the scaling functions, apart from a trivial rescaling, are independent of the anisotropy. In the language of the renormalization group, such anisotropies are associated with marginal operators (Bruce, 1974). In contrast, in systems with "strong" anisotropies, different components of k or )7 scale with different exponents. For simplicity, we first investigate the case of having only one component ("longitudinal") of/~scale as #I+A, with A > 0, while the remaining d - 1 "transverse" ones still scale linearly. The new index A characterizes the anomalous dimension of the longitudinal momenta, and will be called the anisotropy exponent. The results are applicable to both the standard model and the equilibrium systems mentioned above. At the end of this section, we will discuss a generalization to systems in which n "longitudinal" components scale non-trivially. We will see that S and G will be associated with different ~7-1ikeexponents, so that here, we first focus on the structure factor. Instead of (4.17 a) we now have:

S(k~,kll, t, 7 )

=

#-2+nS(k•

t#z, T/#l/~').

(4.18)

Unlike in equilibrium, it is far from obvious how to relate the indices u, z and ~7 to the exponents governing the correlation length, the characteristic relaxation time, as well as the small-momentum behaviour of the critical structure factor. In particular, due to the anisotropy, there are two correlation lengths, etc. In the next section, we will see that these indices can be associated with the anomalous dimensions of the critical parameter T, time t and the ordering field ~b, which can be computed. Here, let us regard (4.18) as purely phenomenological and explore its consequences. To see how momenta scale with ( T - T~), we follow standard scaling procedures and choose # = r ~. In stark contrast to weak anisotropic systems, we need to define two u-like exponents: k•

~•

and

k l i ~ r ~11,

(4.19)

Ull=U(l+A).

(4.20)

with u•

and

It is tempting to use (4.19) and conclude that there are two correlation lengths which diverge as (a o( r -"l and ~11 o( r -~ll. However, we caution that other definitions for such a pair of correlation lengths, e.g. via appropriate second moments of G(~), will not necessarily lead to the same exponents. In the same way, setting # = t -1/z, we obtain two dynamic critical exponents z• and Z l l = Z / ( l + A ) , (4.21)


55

indicating that small perturbations with k• = 0 decay differently from those with kll = 0. Next, we investigate the small momentum behaviour of the structure factor. Restricting ourselves to criticality and t - 0 (i.e. the steady state structure factor) for convenience, we may define two r/-like exponents, r/• and r/iI, depending on which component of k is extracted from (4.18). Suppressing t and ~-, we write

S(k~, k~l) - k~2+nis• (kil/k~+a),

(4.22 a)

S(k~,k~l ) = k~2+~'lSil(k•

(4.22 b)

and

where we have assumed isotropy in the separate subspaces and denoted (]k• lklll) by (k• so that s• and sll are two functions of a single variable. From these, we find r/• = r/

and

r/iI = (r/+ 2A)/(1 + A).

(4.23)

Before continuing to investigate G(Y), several remarks are in order. Since k i scales linearly with #, as in equilibrium cases, it is not surprising that the transverse exponents u• and z• are determined by the anomalous dimensions of ~- and t, with no trace of A. The longitudinal indices, however, differ from their transverse counterparts by a contribution involving A, so that setting A = 0 restores the usual isotropic scaling. Apart from their relation to u, z and r/, these two sets of exponents are not independent. They are related through simple scaling laws involving the anisotropy exponent A, i.e. YlI:~•

(4.24 a)

Zll=Z•

and - 2 + r/II -- ( - 2 + r/•

(4.24 b)

+ A).

From our experience with isotropic systems, we might expect the scaling form of the two-point correlation function to be the same as S(k), apart from a simple factor of #a, which can be traced to the volume element ddk for the Fourier transform. However, due to the anisotropy, the longitudinal component scales with #I+A, whence

G(s177, s , t, "i-) - #d+A-2+~G(~'•

.~][#l+zx , t# z ,~_/ 1/~ ).

(4.25)

As before, we restrict ourselves to t = ~- = 0 and extract various components of s Letting ri and rjj denote [s177and Is respectively, we have 6(&,xll

) -

lg•

),

(4.26 a)

and G(s177211) - rll-d+2-r/i' gtl(r•

),

where we follow (4.22) and associate ~7-1ike exponents with G(s

(4.26 b) !

!

~7• and ~Tjj.

56


Using (4.25), we see that these two r/'s differ from their momentum space counterparts: ,

,

r/_L --?7 + A

and

A(d-

r/iI =

3)

(1 + A)

"

(4.27)

Of course, these also obey a scaling law like (4.24 b): d - 2 + 7711- ( d - 2 + 7/~_)/(1 + A).

(4.28)

Thus, generically, we need to consider four different r/dike exponents, as opposed to a single r/in the equilibrium Ising model. Such distinctions are especially important when theoretical predictions are compared with simulation data, which may be presented in either momentum-space or real-space. As a result of these associations, our four ~7's are also denoted by r/•MS, r/l~s, ~/RS and r/l~s in some literature. Note that these exponents are only meaningful and measurable if the scaling functions s• Sll, g• and gll approach finite and non-zero constants as their arguments vanish. For the standard model, at the Gaussian or treeapproximation level, this is true only for s• and gll" Both Sll(0) and g• vanish, as we will see in Section 4.2.2. While S and G are correlations of two fields: (r162 there is another twopoint "correlation" in dynamics, namely, the response function: (r162 where r is the Martin-Siggia-Rose (1973) field. For systems in equilibrium, FDT provides a relation between them, so that, for example, the susceptibility exponent 7 can be extracted from either. Mindful of the violation of FDT, we will use the latter_.in defining the susceptibility, which is the response of the ordering field r co) to a small, external "magnetic" field h(k, w)"

x(k, co, ~-) - k2 (q~(/~,co)r162 -co)),

(4.29)

where the factor k 2 is the result of conserved dynamics, preventing r from responding to a homogeneous h. In parallel to the structure factor S, let us define

Sll (k, co, T) - (r

co)r

),

(4.30)

where the subscripts stand for the presence of 1 q~- and 1 C-field. In this notation, the S discussed above is denoted by S02. Postulating scaling in analogy with (4.18), we write Sll

Sll

, w/# z, ~-/# /').

(4.31)

We remark on three differences between this form and (4.18). Here, we choose to use w instead of t, so that an "extra" factor #-z appears. Since both the magnetization and the noise are conserved, the naive dimensions of the 0-field and the 0-field differ by 2. The anomalous dimensions are, respectively,


57

defined as 1 r / a n d 89 (Bausch et al., 1976), the latter also being calculable within the framework of renormalization group. Focusing on the response to static h-fields, we set w = 0. To arrive at the exponent 7, we also take the limit k ~ 0. However, we may not set k - 0, due to the conservation law. Mindful of strong anisotropy, we should consider two separate limits: X• - x ( k •

---, 0, kll - 0)

and

X I I - x ( k • - O, kll ~ 0),

(4.32)

which we may label as a transverse susceptibility and a longitudinal one. A s s u m i n g that these are finite and non-vanishing, we may again choose # - ~-" when substituting (4.31) into (4.29). Defining two 7-like exponents, X• ~ ~--'~•

and

XII ~ T-Tll,

(4.33)

we find % - u(z- 89189

2)

and

711- u ( z - 8 9 1 8 9

2(1+A)).

(4.34)

Apart from A, these 7's may appear far from the "usual" u ( 2 - r/). The resolution of this paradox lies in the FDT. There is another definition of the "susceptibility", namely, 2 - S ( k ~ O, t - O, T).

(4.35)

Using (4.18) and making the same assumptions as before, we arrive at )~ ~ ~_-~(2-v). For A - 0 systems in equilibrium, F D T guarantees X - 2 and we obtain, in addition to the usual 7 - u ( 2 - ~7), the scaling law z - z 0 - ( ~ - r/)/2, where z0 is the naive dimension of the time scale. The model B result, z - 4 - ~7, follows from z 0 - 4 and an explicit calculation showing ~ - - ~ / (Halperin et al., 1974; Bausch et al., 1976). For nonequilibrium systems, X r 2 in general, so that we may define two 2's in the same manner as (4.32), as well as two -~'s, as in (4.33). Again, if we assume that the limits are generic constants, we find "~• - "YJl- u ( 2 - ~7).

(4.36)

However, as in the Sll case, this assumption fails for the standard model at the Gauss• level so that "~11vanishes! A standard route to arrive at the order parameter exponent/3, given the form of the two-point correlation, is to exploit G ~ r Thus, r163 t) scales a s ~(d+A-Z+r/)/2, which is consistent with equilibrium models with strong anisotropy. Using # - 7-~ leads us to: /3 - i u ( d + A - 2 + ~),

(4.37)

barring a breakdown of simple scaling. In the next section, we will find that

58


(4.37) indeed fails in the standard model, due to the presence of a dangerously irrelevant operator (Janssen and Schmittmann, 1986 b; Leung and Cardy, 1986). A careful analysis, using the scaling equation of state, produces /3 = 1/2 instead. However, (4.37) is valid in some driven systems with strong anisotropy (Section 6.1). In those cases, we may also write, as an obvious extension of an isotropic scaling law (Binder and Wang, 1989), 2/3 +-~ = vii + ( d - 1)u•

(4.38)

where we have used (4.20) and (4.36). The scaling properties of the ~b-field and the response to static h-fields, given by (4.32-4.34), allow us to determine how the external "magnetic" field scales. It is possible to apply the usual scaling law, "7 = / 3 ( 6 - 1), to each component separately, giving us the exponents 6• and 611. Finally, we consider the specific heat index a. For systems in equilibrium, both the specific heat and the internal energy are related to F, the free energy, by thermal integrations. Thus we have F ,-~ r 2-~ and, combined with F ,,~ (-d, we obtain the familiar d v - 2 - a. For non-equilibrium systems coupled to a thermal bath and an external drive, it is unclear whether a unique definition of the "internal energy" exists, in general. For the standard model we introduced, in Section 4.1.2, quantities which can be identified as internal energy and specific heat. Of course, we may also perform thermal integrations here. However, there is no reason to interpret the result as a "free energy" and postulate its scaling as (-d. To compound the difficulties here, the presence of anisotropies allows us to define, within the standard model, two energy-like quantities and two correlation lengths. An alternative route to F for equilibrium systems is a field-integration of the magnetization, using the equation of state. For our case, it is unclear if this route is equivalent, since there is no FDT to guarantee that such an integral be precisely the same as the result of the thermal integration of the internal energy. For these reasons, we refrain from writing general scaling laws for c~. Instead, we expect scaling properties to depend upon which quantity is chosen to represent "internal energy". We end this section with a generalization to systems in which strong anisotropy prevails in an arbitrary partition of the d-dimensional space into two subspaces. Let the "longitudinal" and "transverse" subspaces be of dimensions n and d - n , respectively, with n _ 1. The first modification occurs in (4.25), since ddk now involves n factors of #l+ZX, leading to the prefactor #(d-~)+~(l+A)-2+n. As a result, instead of (4.27) and (4.37), we have , rl•

, ~7- A(d - 2 - n) r/l= (I+A)

(4.39 a)


59

and fl = I u(d + n A - 2 + ~7).

(4.39 b)

In the next section, we briefly review the field theoretic renormalization group method which allows us to compute explicitly these exponents in the standard model.

4.2.2

F i e l d theoretic r e n o r m a l i z a t i o n g r o u p analysis

Systems exhibiting scaling behaviour signal scale invariant underlying structures. In case such structures are non-trivial, macroscopically many (or, theoretically infinitely many) degrees of freedom must be taken into account while such systems are said to be critical. The development of renormalization group analyses proved to be a fundamental step towards the quantitative understanding of critical phenomena. In this approach, the scale invariant properties are extracted through transformations which track the "physics" of the system under changes of a microscopic length, or momentum, scale (e.g. the lattice cutoff). Fixed points of these transformations may then be identified as scale invariant systems. If there is an associated length (e.g. the correlation length) in such systems, it must be either zero or infinite, with the latter case being "critical" or "non-trivial". The analysis of typical observables (e.g. correlation and response functions) in the neighbourhood of such a point leads to the identification of critical exponents. Finally, universality classes can be understood, in that many systems are mapped into a single fixed point under these transformations. To find fixed points and compute their associated properties quantitatively, we may use numerical or analytical techniques. Both have advantages and shortcomings. For the standard model, only the latter have been implemented and form the subject of this section. Not surprisingly, exact analytic results are rare so that, typically, progress is made via some form of perturbative treatment. Following Wilson, who discovered the key small parameter for a systematic expansion, we will set up our perturbation theory in powers of e - dc - d, where de is the upper critical dimension. Fixed points, exponents and all other quantities of interest can be computed as power series in e. To perform the actual perturbative calculations, field theory methods prove to be among the most powerful, for both static and dynamic critical phenomena. In statics, the starting point is the Hamiltonian ~ , expressed as a functional of the local order parameter field ~b(7). Thermal averages in the canonical ensemble are computed in terms of a functional integral, with

60


Boltzmann weight e-~e[~]/Z (Amit, 1984). In dynamics, the starting point is a Langevin equation of motion for field 6(Y, t), in which thermal fluctuations are introduced through the noise (Halperin et al., 1974; Hohenberg and Halperin, 1977; Section 2.3). Though a perturbation expansion can be set up by directly iterating the associated integral equation, it is technically more convenient to introduce a Martin-Siggia-Rose (1973) response field 6(Y, t) and recast the Langevin equation in terms of a dynamic functional J [ 6 , 6 ] (Janssen, 1976; de Dominicis, 1976). In this formalism, averages over the noise are given in terms of functional integrals with weight e - j [~], (3.23)" de = 4, it is identified as a dangerously irrelevant operator (Fisher, 1974 b). After a careful treatment of mixing with other irrelevant operators (Amit et al., 1977; Amit and Peliti, 1982), it is shown to play an essential role in determining both the equation of state and the critical exponent/3 (Janssen and Schmittmann, 1986 b). Finally, we quote some results for the generalized model, introduced at the end of Section 4.1.1, with n "longitudinal" dimensions. Without setting A = 1 specifically, the fields scale as q5 cx #(d+nA-2)/2 and ~ oc #(d+nA+2)/2. Meanwhile, the couplings scale as 6~ c~ #3-(d+nA)/2 and u cx #4-d-nA. Thus, dc = 6 - nA if 8 # 0, while de = 4 - nA for models without a steady current (Section 6.3). Clearly, setting ~ = n = A = 0 returns us to the familiar results of Model B.


64

Exponents and scaling functions in the tree approximation Having identified the upper critical dimension as 5, we proceed to set up an e-expansion (e - 5 - d). First, we will identify the elements constituting this perturbation theory, such as bare correlators, propagators and vertices. In the remainder of this sub-section, we will discuss the lowest order, i.e. the tree approximation. Apart from being the starting point of the loop expansion, this approximation is a "mean field" theory and provides many correct results for d > dc. Involving no loop integrals and needing no renormalization, there are no anomalous dimensions while all exponents follow from simple power counting. Also, the structure factor and two-point correlations at this level are determined by the quadratic parts of the functional alone, so that their associated scaling functions are easily calculated. In renormalization group language, the tree approximation is associated with a Gaussian fixed point. Being the Gaussian part of (4.42) and denoted by J0, it represents a critical system. Further, it is stable against the non-Gaussian parts of J~ if d > dc, as will be demonstrated in the next subsection. It should be carefully distinguished from the Gaussian fixed line associated with J f and specified by (3.24), which represents T > Tc systems. One crucial difference lies in the absence of the ~-• term, leading to anisotropic scaling here" Xll e~ x~. Let us write Jo in matrix form and in {/~,co} space: Jo[q;, q$]

i s ~

E i

,448,

[q;(-k,-~o) 4~(-k,-a~)lG -1 ~(/~'co)

The 2 • 2 matrix G -1 can be read off from (4.42)"

01 [

- i w + A(k4 + rllk~)

0

r~l

"1,449,

r02

where we wrote its elements as the (bare) vertex functions F20, F 1l, F~I and 1"02. As a reminder, r~n({k,~o}) is the one-particle irreducible part of a connected Green's function at wave vectors {k} and frequencies {w}, with the external ~- and 4~-legs removed. Note that 1-'02, as all F0n (F's with no external ~-legs), vanishes identically, due to causality. The propagator, (4~(k,,;)4~(-k, - ~) ), and the correlator, (c~(k,o;)cb(-k,-~o)), are two-point connected Green's functions, forming the matrix G. For later reference, we emphasize that the renormalized twopoint vertex functions form a similar matrix whose inversion yields the renormalized connected Green's functions. At this level, the expressions for the propagator and the correlator follow from the inverse of (4.49):

G,1 (k, ~) - .

1 tw + A(k~ + ~'llk~) '

(4.50 a)


65

and Go2(k. co) In the t-domain, these are respectively, i.e.

2)~k2

(4.50 b)

(~b(k. t)~(-/~ 0) } and

(qS(/~t)dp(-k. 0)),

Gll(k, t) - O(t) exp {-&(k 4 + "rllk~)t},

(4.51 a)

Go2(k, t) - k4 + 7.11k~exp {-A(k 4 + ~-iIk~)ltl ),

(4.51 b)

and

where O is the step function, showing the causal nature of (4.51 a). Causality also forces all G~0's to be identically zero. Note that the factor k2~ appearing in the numerator of (4.50 b) and (4.51 b) has its origin in the k-dependence of the noise correlations. While (4.48) is a fixed point associated with a critical theory, we may describe the "temperature" dependence of a near critical theory by adding the term (4.43). The sole effect, at the tree approximation, is to replace k 4 by (k 4 + rk 2) in (4.49-4.51). Its presence, sometimes explicitly indicated by, for example, G~n({k, co}, "r), will help us identify v. Next, we turn to the non-Gaussian terms in both (4.42) and (4.41). These will be treated perturbatively. Focusing on the most important two vertex functions, i.e. those associated with the couplings 8 and u, we find, at this lowest level, r~2({k.~o}) =

2~giklj

and

F13({k-',co))= ,kuk2~,

(4.52)

where both kll and k• refer to the momentum carried by the qS-leg. Higher order corrections to the vertex functions discussed so far, as well as all other F's, will involve loops, by virtue of one-particle irreducibility. Lying beyond the tree approximation, the first of these corrections - the one loop case - will be discussed in the next sub-section. In the remainder of this sub-section, we will analyse (4.49-4.52) further, and extract critical exponents and scaling functions at this level. Note that G02(k, co) is nothing but the dynamic structure factor S(k, w), defined in (3.4). Thus, by casting G02(k, t, T) in the form of (4.18), we easily identify the following exponents: A=I,

u=1/2

and

z=4,

(4.53)

so that u•

Ull=l,

z•

and

Zll=2.

(4.54)

66


Naturally, these values agree with those from naive power counting. Similarly, rewriting (4.51 b) with t = 0 in the forms (4.22 a, b) allows us to identify ~=0, (4.55 a) ~•

and

r/ll= 1,

(4.55b)

as well as the scaling functions s• and sll. Using the scaling laws (4.27) with our r/and A, the real space exponents follow: t

~TJ_= 1 and

r/iI = (3 - d)/2.

(4.56)

To appreciate such an unusual result for r/iI, let us compute the Fourier transform of (4.51 b), with t = 0, explicitly. Rescaling the longitudinal coordinates and momenta so that 7-11= 1, we have the equal-time two-point correlation function at criticality: G(~• Xll)

c~

rH(d-l)~2 exp (--r~/4rll).

(4.57)

If we insist on writing this function in the scaling form of (4.26 b), we will get a negative ~711for d > 3. Meanwhile, the scaling function gll (v) cx exp (-v2/4) is finite and non-zero as v ~ 0. On the other hand, casting it in the form of (4.26 a) leads to a more familiar equation for r/~_. The price here is that the scaling function g• (v) c~ v (1-a)/2 exp ( - 1/4v) vanishes with v! This behaviour prevents a straightforward measurement of ~7~_by analysing the large r• behaviour of G(g• Xll = 0). Indeed, it is a signal of another striking feature of our driven system, i.e. the long range negative correlation above criticality (cf. 3.17) gives way to a short range correlation as T ~ Tc! It is not known whether this "defect" remains after critical fluctuations are incorporated. Though answerable, this question has not yet been studied. Obviously, gll does not suffer from this difficulty and ~711is determined in the usual manner. By studying the propagator, Gll(k,w), which is just Sll of (4.30), we can exploit (4.31) and identify the "anomalous" dimension of the ~b-field. Not surprisingly, we find = 0. (4.58) With the help of the scaling laws (4.34) and (4.53), we obtain 7•

and

711=0.

(4.59)

Related to the unusual behaviour of (4.57), the latter is a simple consequence of rll > 0 even at Tr Finally, using (4.52) in an equation of state, more details of which will be discussed in the next sub-section, will lead to the familiar result: f l = 1/2.

(4.60)


67

Next, we will study the first approximation beyond this tree level. According to the paradigm of the renormalization group, we should expect, for systems in d < de, non-trivial corrections to these naive exponents, which can be computed as power series in e = de - d.

One-loop and exact results: the full scaling behaviour We now turn to the task of finding the singular scaling behaviour in the standard model below five dimensions. Here, taking fluctuations into account perturbatively will involve more and more severe divergences, so that the full power of renormalized field theory, as well as all its subtleties, will be needed. Presenting any calculational details is clearly outside the scope of this review. Instead, we will show some of the highlights and attempt to convey the flavour of this approach. For comprehensive treatments, see the original literature (Janssen and Schmittmann, 1986 b; Leung and Cardy, 1986). There are several remarkable features of our theory. First, starting with Jc, only a single independent renormalization is necessary. A non-trivial, infrared stable fixed point exists, giving rise to an anomalous dimension for A. Moreover, thanks to the scale invariances (4.44, 4.45), the correction to A follows exactly, to all orders in e =__5 - d. Surprisingly, to obtain this result, not even a single diagram needs to be calculated! Meanwhile, no divergences are encountered in the renormalization of )~, 4~and 4~, while only an additive renormalization (i.e. none, in dimensional regularization) suffices for T in (4.43). Thus, their associated exponents retain Gaussian values. On the other hand, since u = 0 at the fixed point Jc, critical properties below Tc such as/3, can be found only after a careful analysis of all irrelevant operators which mix with (4.47). The results for the independent exponents are: A--I+~; e

z--4;

r/--~--0;

u--l;

1 /3--~.

(4.61)

Before proceeding to the scaling functions, let us examine the extremely simple structure which leads to the exact and non-trivial A. Noting that the only vertex functions with primitive divergences are P ll and F12, we consider first the one-loop correction to the former, shown in Fig. 4.5. Recall that an 8-vertex carries a factor of kll on its 4~-leg, and that the theory is CR-invariant. Thus, we conclude that this correction is proportional to k], so that only 711

J

Fig. 4.5 The one-loop diagram for F ll.

68


needs to be renormalized to render F ll finite. Next, we consider I"12. Thanks to the "Galilean" invariance (4.45), which leads to a Ward identity relating the vertex functions F~,n+1 to F~n, we have rlz(k-", co; 0, 0) - 218kll ~~ 1~11 (/~, w)

(4.62)

in particular, where the zeros refer to the (k, w) of one of the C-legs. Thus, once 1-'11 is rendered finite, 1"12 will be finite also, since the derivative on the right-hand side will not induce any ultraviolet divergences. So, while Tll is the only coupling which needs renormalization, none is required for ~. However, the effective expansion parameter is g rather than ~, so that we should work with the renormalized couplings g and 7-11(where, for simplicity, the same notation is used). Thus we write Wilson functions for g and TI1:

~

w(g) - l z - ~ 0 g,

and

lnTii. =-

(4.63)

(4.64)

0

These describe the flow of the renormalized g and Tii, at fixed bare couplings (indicated by the subscript "0"), under changes of an external scale #, at which renormalizations are defined. Within this framework, we find, as a consequence of the definition of g (4.46), an important relationship between the Wilson functions: w(g) - - g ( e + 3((g)).

(4.65)

Therefore, we need to compute only one Wilson function through perturbation theory. Since F ll is the simpler vertex, ((g) is the only quantity to be computed. At the one-loop level, it is simply - g , so that the fixed point equation w(g*)= 0 has an infrared stable, non-trivial solution provided e > 0, namely g * - 2 e / 3 + O(e2). If we assume that, when higher order contributions are added, ((g) remains negative and is not bounded from below by -2e/3, then we may conclude that an infrared stable, non-trivial g* is always present. Furthermore, without calculating a single diagram, the fixed point value of ( is known exactly, to all orders in e: (, = ((g*) = -2e/3.

(4.66)

Since this quantity, as opposed to g*, dictates the scaling behaviour of rfl, the anomalous dimension of kll and A can be obtained exactly. To find how A depends upon ~,, we proceed to study the scaling behaviour of the renormalized vertex functions. Since neither r nor r


69

require renormalization, they are simply the bare vertex functions in which divergences are completely absorbed into the renormalized couplings. Thus, they satisfy the following renormalization group equation:

[0

0

# -~ + w(g) ~ + ((g)~-II

F,~n - 0.

(4.67)

Again, all quantities here are renormalized ones, though we do not distinguish them by using a different notation. At the fixed point, g = g* and w = 0, so that the consequence of (4.67) is that every occurrence of TIR comes with a factor #-(*. On the other hand, the scale invariances (4.44) lead to additional homogeneity properties of the F,~n as a result of, for ,. ,1/4 -(/4 example, every occurrence of r coming with a Iactor o[ T ~# * II Combining these, the full scaling behaviour of the critical theory may be summarized in the following:

F~n({k.w}) - lzPr'~n({k~•

(4.68)

where

p-(d+5-~)-(d+3-~)

h-~- ( d - 1 - - ~ )

n~.

(4.69)

Note that TII always appears with the factor k], accounting for the additional powers of # in the kll argument. Setting this power equal to 1 + A, we deduce A - 1 - ~- - 1 + ~.

(4.70)

To identify the "physical" exponents, we may use the scaling laws, e.g. (4.20), derived in Section 4.2.1, with the exception of (4.37) for/3. It may be instructive, as a check, to consider specific physical quantities and study their scaling behaviour. So, for example, the structure factor is S(k, ~) - G0z(k , w) - Fz0(k. w)/I Pll (k, ~)l 2. Using (4.68) and transforming to the t-domain, we obtain

s(k\, k l, t, 7-) =

t# 4,

(4.71)

Comparing with (4.18) and subsequent equations, we arrive at r/• - 0,

r/ll - (6 + 2~)/(6 + ~),

(4.72 a)

z•

z11-12/(6+e),

(4.72 b)

Ull--l+e/6,

(4.72 c)

U_L--1/2, and

-~=1.

(4.72 d)


70

Note that vii > v• indicating that correlations in the longitudinal direction live on a much larger scale than transverse ones. On the other hand, Zll < z z, so that longitudinal fluctuations decay much faster. Studying the correlations in real space, we have r/~_ = 1 + e/3

and

r/iI = (3 - d)(3 + e)/(6 + e),

(4.73)

beating in mind, however, the caveat associated with r/~_. We finally turn to the problem of the dangerous irrelevant coupling u and its associated operator (4.47). The main complication comes from its mixing, under the renormalization group, with other irrelevant operators of the same naive dimension. In other words, the renormalization condition appears in matrix form for this set of operators. Diagonalizing this matrix leads to eigenvectors which are used to construct eigenoperators, i.e. those linear combinations of operators that are invariant under the renormalization group transformations. The eigenvalues are functions of g. Evaluated at g*, they are the scaling powers of the associated eigenoperators. The details of this procedure, which follows standard techniques (Amit et al., 1977; Amit and Peliti, 1982), may be found in Janssen and Schmittmann (1986 b). Here, we quote two central results: (i) The dangerous irrelevant coupling u appears in only one eigenoperator. The others yield corrections to scaling which are not "dangerous" and will be ignored; (ii) Its eigenvalue n(g) can be determined exactly, thanks again to the interplay of "Galilean" and scale invariances. The fixed point value is ~, - ~(g*) = 2 - ~ - 89 = 2 ( d - 2)/3.

(4.74)

Had we included u in the argument list of Fn,, there would be an extra term in (4.67), namely, ~(g)u(O/Ou)F. Putting ~- in as well, the scaling form (4.68) now reads

r~n({k, w}, "r, u) - - lzPr~n({k_L/lZ, kll/lz(2-~*/2),od/ld4}, 7"/~2, u/z~;*). (4.75) Armed with this analysis of u, we are ready to study the exponent/3. The "naive" scaling law (4.37) fails since u = 0 at the fixed point Je. Instead, we seek fl from first principles, i.e. through the equation of state. For this, we must introduce a magnetic field h into our dynamic functional. Due to the conservation law, ~b(Y,t) will not respond to a static (w = 0), homogeneous field. To focus on ordering into strips parallel to g, we apply a magnetic field of the form h(k• Based on how an h couples to ~bin Model B, we modify our dynamic functional by adding the term f ddx f dtA~(g, t)V2h(x• Note that the transverse Laplace operator arises from the conservation law. Following, for example, Bausch et al. (1976), the equation of state results from

h-

6

r{r- r

,

(4.76)


71

where F{~, r is the vertex generating functional, and the order parameter field M ( k ) is the spatially varying magnetization in the presence of h. Note that the derivative, which originates in the conservation law, picks out the first non-zero Fourier component of M(k), which is just r in Section 4.1.1 in the thermodynamic limit. Expanding F in powers of M, we obtain

h-)k-l ~ ~n ~Mn Pln({k_L, kll -

0, w - 0, r, u)

[

.

(4.77)

k• =0

Using (4.75) in conjunction with (4.69), the scaling equation of state follows

h(M, r, u) - I.t(d+3~*/2)/2h(M/lz(d-l~*/2)/2, T//Z2, utt n*).

(4.78)

The philosophy here is to compute the P l,'S at r > 0 and then analytically continue to r < 0. Then the spontaneous magnetization will be given by the equation h(M, r, u) = 0. Had we ignored the effects of u in (4.78), the scaling behaviour of M and r would lead to the naive result (4.37): fl = 1 - e/6. However, since u scales as r -~*/2 and M 2 ~ T/U, we must modify 2fl by a factor (1 + n,/2). Using (4.74)in 2/3 = 2(1 - e/6)/(1 + n,/2), we have /3= 1/2.

(4.79)

We end our discussion of u and/3 with some remarks concerning d = 2. According to (4.74), u will be marginal, so that the foregoing conclusions may not be strictly true. However, the expected corrections are most likely logarithmic which would not be easily discernable. Thus, /3 = 1/2 should be an excellent approximation, even in d - 2. To compare with most of the data from simulations, we set d = 2, corresponding to e = 3. This procedure may seem rather hazardous, considering our theory is based on a power series in e. However, we remind the reader that our exponents do not depend explicitly upon the e-expansion of g*. Instead, they depend only upon ~,, which is given by (4.66), to all ordersin e. Of course, we should add the usual caution that applies to any perturbative approach, i.e. the possible presence of non-perturbative contributions. Barring such difficulties, we see that the latest data, summarized in Section 4.1, are in surprisingly good agreement with field theoretic results. A direct measurement of the two-point correlations in the longitudinal direction (Zhang et al., 1988) yielded rill = 2/3 in d - 2 and rill- 0 in d = 3, which compares well with (4.73). The early result of/3 _~ 1/4 (Vall6s and Marro, 1987 a) from an isotropic finite-size analysis is, we believe, superseded by Leung's (1991 a, 1992) conclusions of A = 2 and/3 = 1/2, which agrees well with (4.70) and (4.79). Also, the more recent study is fully consistent with the theoretical values uj_ = 1/2 and Ull = 3/2. Finally, our estimate of-~ _~ 1 compares well with (4.72 d). This impressive agreement between field theory and simulation results

72


gives us much confidence that the Langevin equation (2.15, 2.16) is the correct continuum limit of the discrete lattice model and that the simulation data are within the critical region. Though there is little discrepancy between all existing data and theoretical predictions for the standard model, there are many quantities which remain to be measured and/or calculated. If the agreement between data and theory continues to be as satisfactory, our understanding of the critical phenomena of this non-equilibrium steady state system will be that much the better.

Related models In this sub-section, we briefly discuss the other two candidates of critical theories, i.e. (i) ~-• > 0 with Tll -~ 0g and (ii) both z's vanish. The former, studied in some detail by Janssen and Schmittmann (1986 b) corresponds to ordering into strips transverse to the field. Though such configurations are never found to be stable in simulations of the standard model at large fields, this region of parameter space can be accessed by a suitable generalization (Bassler and Zia, 1995). Begin with a lattice gas with jumps in the transverse and longitudinal subspaces controlled by two different temperatures: T• and TII, respectively. By tuning the drive as well as the two temperatures separately, while maintaining T• > TII, we expect to reach both regimes (i) and (ii), the latter being a multicritical point. The theory corresponding to version (i) is given by the dynamic functional (4.42) with transverse and longitudinal parameters, except the 8 term, interchanged. Here, of course, only a single component kll of the wave vector k becomes critical, as opposed to the (d - 1) transverse components, so that we now have k• cx k~. Clearly, this will result in a different naive dimension for 8, leading to d c = 4.5 instead of 5. But there is no infrared stable fixed point for d M0. Otherwise, the typical configurations will consist of a region of M0 co-existing with another o f - M 0 . These regions occupy finite fractions of the entire system: 89 in the thermodynamic limit. Their shapes depend upon a number of factors, including boundary conditions and 991, in general. Determining M0(T), the co-existence curve for all T < Te, is a longstanding goal in equilibrium statistical physics, while finding the shapes of the regions appears to be only a recent interest (Rottmann and Wortis, 1984; Zia, 1988; Dobrushin et al., 1993). For the Ising model on a square lattice, we have the celebrated exact result (Onsager, 1949; Yang, 1952)" Mo(T) = {1 - [ s i n h (J/2kBT)]-4} 1/8. As for the shapes, in a system with PBC imposed, the phase-separated state makes a transition inside the co-existence curve. With T fixed, as ~ is lowered from 0 to -M0, the particle-rich region changes from being a strip to being a simply-connected droplet in order to take advantage of a lower surface energy (Leung and Zia, 1990). Once the drive is imposed, the co-existence curve is necessarily modified, with both Te and ~ being distinct from those of the Ising model. More importantly, there is a qualitative difference in the shapes of the phaseseparated region. Though no one systematically investigated these shapes, it is noteworthy that only strips parallel to E have been observed. The only information we have on Mo(T,E) comes from simulation studies, using two different methods (Vallrs and Marro, 1987; Marro and Vallrs, 1987). Though there are some unresolved discrepancies between the two sets of results, they already show several features which are absent from the equilibrium case. The first method is straightforward, using only a half-filled lattice (gJ/= 0) with T < Tc. Relying on M*, defined in (4.4), to approach the spontaneous magnetization in the thermodynamic limit, extrapolations to L ~ oo are made. The resultant data points, for the E = c~ case, are plotted (.) in Fig. 5.1. Note that all available data lie above the Onsager curve

Mo(r,c~) as given by M*(T) > Mo(r,o).

(5.1)

Compared to the equilibrium curve, the driven one resembles a parabola more closely. As a further comparison, we have included the classical co-existence curve: v / 1 - T/Tc, showing that this measure of order parameter is more consistent with an exponent less than 1/2.

Statistical

mechanics

of driven

diffuse

systems

75

The second method employs lattices with overall particle densities less than 1/2, while T is lowered from oo until phase separation occurs. Specifically, on a 100 x 100 lattice, densities are fixed at 0.05, 0.075, 0.1, 0.2 and 0.35. Driving with saturation E and monitoring the anisotropic order parameter (4.4), the transition temperature from a homogeneous state to a phase separated one is located: T* ( ~ ) . The data points, using a particle-hole symmetry to translate the densities to ~J/> 0, are also shown (O) in Fig. 5.1. It is curious that the classical co-existence curve lies between these two sets of data. We conjecture that the discrepancy between the two methods is due to finite-size effects (Leung and Zia, 1992). A careful study with various LII and L• is needed to clarify this issue. Whatever the final resolution is, the present results show that, at least for finite L, driving does not necessarily help the system order, though Tc(o~) > To(0). Figure 5.1 shows that large fields destroy an inhomogeneous equilibrium state if 9J/> 0.7 is chosen. This phenomenon can be partly understood from the other distinguishing feature of our standard model, namely, the lack of droplets. Starting with no drive and ~J/near -M0, the phase-separated state would consist of a high density droplet. However, when a large drive is imposed, the droplet is clearly unstable and evolves toward a thin strip. It is easy to conceive a thin strip being more susceptible to "evaporation" than a droplet, leading to disorder. Whether the mechanism actually consists of two successive steps is not

1.5 F

i! t

9

..."

_

..

0.5

06

|

I

!

I

1

0.1

0.2

0.9

0.4

0.5

Fig. 5.1 Co-existencecurve in the standard model, determined through M*(T) and T* (~). The data points are shown as (,) and (O), respectively. The dot-dashed line ( - . - ) is Onsager's exact result. The solid line ( - - ) is the classical curve, normalized by Tc(e~). From Marro and Vall~s (1987).

76


known. It would be both interesting and challenging to understand the evolution from an inhomogeneous equilibrium state to a homogeneous non-equilibrium state. On the theoretical front, we are aware of only one attempt to calculate the co-existence curve for our standard model, namely, the dynamic mean-field approach ofPesheva et al. (1993; Section 2.2). Since it is based on the untested notion of a generalized free energy for non-equilibrium steady state systems, it is difficult to assess how generally reliable this approach is, even though it has provided qualitatively good agreement with simulations (Pesheva, 1989). To predict more intricate details, such as the shapes of regions of different phases, this approach will need fundamental improvements, e.g. incorporating a description of inhomogeneous states. Given these difficulties, a better understanding is more likely, at present, to come from the more phenomenological, mesoscopic approach. Since we believe that criticality is well described by 7• ~ 0, one candidate for physics below criticality is to let T• take on negative values, while keeping 711 > 0. Such a choice would exclude steady states which are phase-separated in the longitudinal direction, a condition fully consistent with the observed phenomena. In terms of dynamics, having 711 > 0 and 7• < 0 in (2.15) means that a homogeneous state is stable against all fluctuations with purely longitudinal wave-vectors while being unstable only to perturbations with variations in the transverse direction. Lacking simulational evidence of this behaviour, we can only argue that this choice should be valid if E is large and T is not too far below Tc. For the opposite limit, i.e. small E and T near 0, a better candidate may be to have both ~-'s being negative. Certainly, for E - 0, a proven mesoscopic equation (Cahn and Hilliard, 1958) is (2.10) with T < 0 in (2.9). In the following, we review an application of this equation modified only by the addition of (2.14), ie. an isotrophic version of (2.15), with 7 < 0.

5.2

Dynamicsof phase separation

In this section, we make a brief detour from steady state phenomena to consider the evolution, after a rapid quench from a disordered state, toward a phase-separated state. Extensive studies of the E - 0 case (Gunton et al., 1983; Binder, 1991) found a scaling regime for which the label "quasi steady state" might be appropriate. In this regime, a large degree of universal behaviour is displayed and the time dependence can be "scaled out". To be more specific, the evolution proceeds, after a short transient, by the growth of domains of magnetization +M0. If R(t ) denotes a characteristic length for these domains at time t, then key quantities appear to depend upon t only through R(t). For example, the structure factor, or scattering intensity,


77

evolves according to (5.2)

S ( k , t) -- S ( k R ( t ) ) .

When the growth is underpinned by a conserved dynamics, an example of universality is the power law: R ~ t 1/3 (Lifshitz and Slyozov, 1961; Siggia, 1979; Kawasaki and Ohta, 1983). It is natural to ask whether such universal behaviour is modified by the drive. Three studies (Yeung et al., 1992; Puri et al., 1992, 1994) addressed this issue, using numerical techniques on the mesoscopic model. Earlier, in computer simulations of the microscopic model on large lattices, Vall6s and Marro (1987) observed relatively long-lived configurations with multiple strips before the system settles down into a single strip. A comparison between these two perspectives is not possible at present, for two reasons. First, no quantitative analysis was performed in the early study. Second, there seem to be two distinct regimes of phase separation: an early stage in which ordered domains are basically two-dimensional (Fig. 5.2) and a late stage in which thin strips spanning the longitudinal dimension of the system coalesce to form a single wide strip, so that growth takes place in essentially one (transverse) dimension. The observations of the early work clearly belong to the latter regime, while both of the recent investigations focused on the former. Our brief review here follows mainly Yeung et al. (1992). The other studies (Puri et al., 1992, 1994) used a smaller system and ran for a shorter time, arriving at essentially the same conclusions though starting with slightly different equations. For completeness, we mention the work of Lacasta et al. (1993), though the focus is quite different from ours here.

"

"I~_,.

"

n

~

~

~

~

m

~

"

~

-

I

v

a~,.._.

d

9

~

i ~ " ~ = ~ 7 -

' m

Fig. 5.2 Typical configurations after a quench from a disordered state, showing isotropic coarsening at early stages and anisotropic growth at later times. Particles (dark region) are driven from left to right. From Yeung et al. (1992).

78


Arguing that thermal fluctuations are unimportant for scaling in domain growth, the starting point is a noiseless version of (2.15). Furthermore, all anisotropies other than the driving term are dropped (Kitahara et al., 1988): et ~b(e, t) = - 89V2 [~b+ V2~b - 4~3] + o~a~b2,

(5.3)

where V stands for gradients in all directions here. All other parameters (,k, r, a, u) have been absorbed into the scales of space, time and qS. Dividing a d = 2 space into 128 x 1536 cells, this equation is solved deterministically by updating in discrete time steps. The only randomness lies in the initial state, in which the value of ~bin each cell is chosen from a Gaussian with zero mean and 0.02 variance. Several o~'s were chosen. The most extensively studied case is o~ = 0.4, with 18 runs of 28 K updates each. In Fig. 5.2, we see three typical configurations, at 250, 2 K and 16 K steps. Observe that, at early times, the pattern is essentially isotropic, similar to configurations with o~ = 0. Anisotropies are clearly present in the middle frame, while strips are well developed by the last frame. To develop a quantitative measure of this anisotropic evolution, it is essential to introduce two length scales associated with the domains: R• (t) and RII (t). Defining a "broken bond" to be a pair of nearest-neighbour cells with oppositely signed ~b's, the R's are inversely proportional to the number of broken bonds in their respective directions. An isotropic R(t) can also be defined as a measure of the density of all broken bonds, i.e. R - 2R•177 + RII). Surprisingly, R is found to obey the t 1/3 growth law throughout the entire period and for all o~, leading to the speculation that this power law is a consequence of only particle conservation and some very general restrictions. On the other hand, only for early times do R j_ and RII follow this power, a result consistent with the largely isotropic first frame. At later stages, R_L increases almost linearly with t, while RII grows much slower, close to t 1/4. Such drastic separation of scales may be the culprit behind the lack of dynamic scaling in correlation functions and scattering intensities at these later times. The crossover from isotropic to anisotropic regimes clearly depends upon the strength of o~. To estimate how such a length scales with o~, note that the left-hand side of (5.3) scales as 1/t, which is 1/R 3. On the other side, the drive term enters with g/R, giving us R r which is consistent with data. At a more qualitative level, several striking features of domain morphology were discovered. First, the "downstream" ends of the strips tend to form tips, while the other ends are flat. This tendency to form triangular structures, aligned along the field, is confirmed in a low density quench. A plausible argument for their existence is advanced, based on stability of certain interfaces and instabilities in others. Similar large triangular fingers were also ,,~


79

found at early times (Mozos and Hernfindez-Machado, 1994) when a fully ordered strip, transverse to the field, is evolved according to (5.3). Second, there exist "bridges" across the strips, moving as relatively stable entities, with velocities that depend on their widths. Regarding these bridges as a vertex of one triangle paired with a base of the next, it is believed that they reflect a d -- 1 dynamics which produces solitary waves. Surprisingly, there were no investigations of these phenomena in the microscopic model until very recently (Alexander et al., 1995). Using standard Metropolis rates, the results are, for instance, T = 0.6 and /3E =0.5, show both anticipated similarities and striking differences. Among the former are the growth laws and the lack of dynamic scaling. The most dramatic difference lies in the asymmetry associated with clusters, i.e. triangular domains in the early stages are mainly aligned opposed to the field. On the other hand, no triangular domains were observed for E > 2J, for a range for T. Qualitatively, this feature can be reproduced by splitting the first term in (5.3) into (r• 2 + TII~2)r and letting only r• be negative. It is clear that these results give us a harvest of intriguing phenomena and raise some serious questions. Isotropic Laplacians, used in (5.3), may not be sufficient to capture all the essential anisotropic properties of this system. We have seen, for example, that anisotropies besides the ~ term are responsible for generic long range correlations above criticality. Here, we see that having a negative r• is important for the absence of triangular shapes if E is sufficiently large. Thus, it is unclear whether anisotropies are irrelevant for the growth laws in general. In particular, the initial isotropic growth, as well as the familiar t 1/3 behaviour, may be absent if the fully anisotropic (2.15) is used. Of course, it may be argued that time evolution acts as coarse-graining, and automatically generates an equation like (2.15) from the isotropic (5.3). If this scenario proves to be the case, then we may regard these two equations as different levels of mesoscopic description rather than competing models. A more difficult question is the relationship between the microscopic model and the mesoscopic (5.3). Thus, it is not obvious what range of microscopic E, if any, must be chosen to reproduce the various phenomena discovered by Yeung et al. (1993), using g -- 0.4. Similarly, it is unclear what range of ~, if any exists, is needed to describe the simulation results with "E = o~". At a different level, the presence of triangular patterns is intriguing, especially since those in the microscopic model point in the opposite direction to those found with (5.3). On the other hand, triangles pointing "downstream" were also seen in a microscopic model (Section 6.2.2; Boal et al., 1991), although the system there is in a steady state, driven by both E and a density gradient. Clearly, it is important to study the competition between the attractive inter-particle forces, which tend to form round droplets, and the driving force, which elongates them. Unfortunately, so far, neither the qualitative

80


similarities nor the sharp disagreements led to any quantitative predictions. Undoubtedly, more thought-provoking new behaviour will be discovered before a comprehensive theory is established. Even more exotic phenomena were found in another study (Wickham and Sethna, 1995). Motivated by electromigration experiments on YBzCu307_6, these authors used rates which enhance the mobility of particles with few neighbours. The coarsening exponents are found to depend on the overall density (filling fraction). Further, at high filling, domains of holes tend to be highly faceted, while larger domains move more quickly and "sweep up" smaller ones on their way. At present, there is no continuum counterpart for this model, and only heuristic arguments for such behaviour have been advanced.

5.3

Interfacial properties in steady states

Returning to steady state systems, we now consider those which have completely phase separated and focus on the physics of the interface between the phases. Typically, the interface behaves as if it had an "independent existence". Regarding these excitations as "capillary waves" with a surface tension or, Buff et al. (1965) formulated the equilibrium statistical mechanics of interfaces in isotropic media by postulating a Hamiltonian: ~rA, where A is the total surface area. However, unlike a soap film, the interface does not really have an "independent existence", being a collective degree of freedom of the bulk. Instead of postulating one, an effective Hamiltonian for the interface can be derived from the bulk Hamiltonian (Diehl et al., 1980), with the leading term being erA and higher ones being curvature terms (Lin and Lowe, 1983; Zia, 1985). This derivation hinges on singling out the interface as the only soft, Goldstone mode (Wallace and Zia, 1979) and relying on the low temperature limit to "freeze out" all the massive bulk modes. In this approach, cr is identified as a free energy per unit area and, in principle, can be found as a function of T and the microscopic interactions. Though this approach is appealingly simple and successful for describing static properties, dramatic complications appear when we consider dynamics, even for cases near equilibrium which respect the FDT. The main reason is that the interface, having no "independent existence" does couple to the bulk degrees of freedom. In a system with pure relaxation (model A), all bulk modes are fast and can be "summed out" easily so that the equation of motion for the interface variables is local though non-linear (Allen and Cahn, 1976; Bausch et al., 1981). On the other hand, ifa conservation law is present (model B), all the bulk modes are slow and the resultant equation for the interface becomes non-local (Turski and Langer, 1977; Kawasaki and Ohta,


81

1982). Beyond that, the physics of interfaces in model C is even richer (Zia et al., 1988), though the static properties are still governed by erA! For a brief review, see Bausch et al. (1991). Given the intricacies of interfacial dynamics in near-equilibrium isotropic cases, we may expect that the non-equilibrium and anisotropic characteristics of the standard model will lead us to complex equations and exotic properties for the interface. So far, simulations have proven the latter correct, while a comprehensive theoretical picture is still lacking. Certainly, to arrive at an equation of motion for the interface, such as the Kardar-Parisi-Zhang (1986) equation, for this driven model would require considerable care as well as ingenuity. It is unclear if a naive postulate involving only currents within the interface (Szab6, 1994) could ever capture the essence of bulk contributions, such as long range correlations and the effects of particle conservation. Thus, we devote this section mainly to the novelties discovered through simulations, and lastly (Section 5.3.3) to a review of the initial steps towards a theory.

5.3.1

Suppression of interfacial roughness

For the Ising model in equilibrium, the surface tension is generically anisotropic and T dependent: a(h, T), where h is a unit normal of the interface (Onsager, 1944; for recent reviews, see e.g. Rottmann and Wortis, 1984; Zia, 1988). As a consequence of interfacial fluctuations, this function may display singularities in one or both variables. Probably the most celebrated example is the roughening transition, which occurs at some temperature TR. For a recent review, see, for example, van Beijeren and Nolden (1987). In the rough phase (T > TR), cr is analytic in h. All interface fluctuations behave as capillary waves, being controlled by ~ or(h)dA instead of crA. The correlations in the long wave-length limit are described by a universal power law: C(~) cx 1/~ 2,

(5.4)

~7~0

where 4is a (d - 1)-dimensional wave-vector of the capillary waves. Since w2, the square of the statistical width of the interface, is proportional to f Cdq, this infrared singularity leads to a divergence of w2 for d _< 3. On a square lattice, TR - 0, while on the cubic lattice, TR ~ 0.55Tc (Bfirkner and Stauffer, 1983; M o n e t al., 1989). Below TR, or(h) is singular at one or more values of h. Interfaces associated with these normals are called "smooth" or "flat". Correlations of the deviations from flat interfaces decay exponentially at large distances, leading to finite widths. A natural question arises: How are transitions of this kind affected by the drive? Unfortunately, due mainly to the subtle nature of roughening, neither simulations nor theoretical avenues can provide clear answers. Instead, as

82


E=0 10 w2

5

0 0

20

4O

E

0.5

E

2

E

50

6O

LII Fig. 5.3 Square of the statistical width (wE) of an interface, plotted against its length. ~o1~Note the linear dependence in the E = 0 case. In the inset, L~ is plotted against w2 the E = 2 case with various values ofp, showing that w2 is consistent with L~. From Leung et al. (1988).

simple interfacial properties are investigated, unexpected phenomena and new questions emerge. The first example is the suppression of roughness in d = 2 systems (Leung et al., 1988, 1989). Specifically, consider an L• • LII system with an interface aligned along the

2.4 2.0 t/C(q)

#

1.6-

"e~~"t

1.2-

J'""*~"

j-"

0.8

0.4

j ~

+++r+++~~. ~

~r

+ 1313' +"

-'~)[0

0[2 '

0.4 ,

.

.

.

.

0.6 ,

,

O.S 1.0

q0.67

Fig. 5.4 Inverse correlation 1/C(q) plotted against q0.67, for E = 2 (,), 50 (I; +) and LII = 100, 600. L• is 40 in all cases. From Leung and Zia (1993).


83

LII direction. Since TR -- 0 in equilibrium, w2 diverges as L~, with a universal power p = 1. We may ask if the drive affects p. Simulations were carried out with L j_ = 30, LII ranging from 10 to 60, T - 0.75 and 0.9, and four values of E. The results, displayed in Fig. 5.3, show that p is consistent with zero, as long as a drive is applied. Measurements of the correlations in this study produce a similar picture, i.e. the infrared singularity in C(4) is less severe than the equilibrium 1/q2, while finite-size effects prevent a more quantitative conclusion. However, a recent extensive study using lattices up to LII = 600 (Leung and Zia, 1993) shows a remarkable cross-over to 1/q 0"67 for both E = 2 and 50 (Fig. 5.4). If such a power persists down to the origin, then it is consistent with p _= 0. On the theoretical front, neither of the two attempts (Leung, 1988; Yeung et al., 1993) at addressing this issue succeeded in finding p ~ 0. However, in a closely related model (see Section 6.1.2), we find encouraging results such as C(q) ~ 1/q and W 2 (X gnL. Deferring discussions of all theoretical approaches until Section 5.3.3, we next explore other phenomena in the standard model.

5.3.2

Shifted boundary conditions and "tilted" Mterfaces

For interfaces in equilibrium systems with short ranged interactions, smoothness is intimately related to the existence of a discontinuity in XT~cr. Since a free energy can be obtained from an internal energy by integrating over T, this singularity will be also present in u (h), the interfacial internal energy. Indeed, the standard way to find or(h) in simulations, is to measure u (h), followed by a "thermal integration" ( M o n e t al., 1988). Motivated by this connection in equilibrium cases, Vall6s et al. (1989) carried out a simulation study of the standard model with interfaces of varying h, on L x L lattices. Instead of finding a simple singularity in the surface energy u(h), a more intriguing picture emerged. The bulk energy, U, is affected by the orientation of the interface. Beyond that, a series of new transitions, from a one-strip (of the dense phase) state to multi-strip ones, were found.

Dependence of bulk properties on interface orientation In simulations, a standard method to induce an interface which has h not aligned with one of the lattice directions is to impose shifted periodic boundary conditions (SPBC) instead of PBC. As an example, SPBC with a shift of h in ~ means that s(i + h,L) is a neighbour of s(i, 1). In an equilibrium Ising model, these conditions will lead to interfaces 'tilted' at an angle 0 - arctan (h/L) relative to 9. Strips aligned with ~ are unaffected in this example. This method is used by Mon et al. (1989) to find the anisotropic surface tension or(0, T). Clearly, imposing SPBC has no effect on the equilibrium bulk properties. In contrast, Vall6s et al., (1989) found that SPBC profoundly affects the bulk in the standard model.

84


Measuring the total internal energy per site, Utotal, as a function of both L and h, it is possible to distinguish the bulk component, U, from the interfacial one, u, through an expansion in 1/L:

Utotal(Z,h) - U(O) + u(O) sec(O)/Z -+-O(1/Z2).

(5.5)

Note that L sec (0) is the "area" of a tilted interface, and that we have implicitly assumed U to have no O(1/L) corrections in systems without boundaries (Fisher, 1971). Now, for systems in equilibrium, the interface is a manifestation of the spontaneous breaking of the Euclidean symmetry (Wallace and Zia, 1979) while the homogeneous bulk breaks only the Ising symmetry. Thus, bulk properties will not depend upon the interface orientation, i.e. U is O-independent. Thus, in a small-0 expansion, where 0 ~_ h/L any h-dependence in (5.5) should not appear before the level of h/L 2. Further, for d - 2 and T > 0 cases, u is analytic and even in 0, so that the lowest h-dependence would be O(h2/L3). Surprisingly, in the driven model, the first h-dependence is found to occur at the O(h/L) level! The only reasonable interpretation of this result is that the bulk energy is O-dependent, so that the small-0 expansion of (5.5) reads

Utotal(Z,h ) - [U(0) -+- Ut(O)(h/Z)] -+-u(O)/Z -+-O(1/Z2).

mmmmm

mmmmm

mmmmm

mmmmm

(5.6)

mmmmm

mmmmm

mmmmm mmmmo

I

mmmmmo mmmm

mumm

mmmm

mmmm

mmmm

mmmm

mnmm

mmmm

mmmm (a)

(b)

Fig. 5.5 With SPBC between the top and bottom edges, an interface must have at least one step. The rest of the bulk is not shown. If E > J, the particle at the step, marked by O in (a), will be driven downwards. Other particles will follow, but we focus on the original one. Since transverse jumps do not benefit from E, it will stay attached to the interface until, eventually, it arrives at a step of height two, as in (b). From this position, it can be driven away from the interface and into the 'bulk' of holes.


85

Specifically, shifts corresponding to 0 < 10 ~ were imposed on square lattices with L = 20, 36, 48 and 100. Setting T at 0.8 and E at essentially infinity, we measure the excess energy, U~xce~s, defined in spin language as the average number of broken bonds per site. The advantage of using U~xcess, which is clearly related to the energy, is that it approaches 2/L, for an h = 0 system as T ~ 0. In that case, the limiting stable steady state will consist of one strip with only + spins and one with only - spins, separated by two interfaces of zero width. The best fit for the excess energy, in simulations with L = 20, 36, 48 and 100, gives

Uexcess "~ [0.08 + 0.75101] + 2.68/L + . . . ,

(5.7)

where the absolute value signs are based on the h r - h symmetry. Thus we come to the rather unexpected conclusion, namely, not only does the bulk

9

I

,o

9

9

9

9~

9

9

:,:,

,

,"

9.

9

9

o

9 9

|.

I

9

9

9

E,

~:11

(a)

...

-

9

9 9

_.....

. 9

I

9

9

9

9 "

So I

9

,%

9 ~

I

9

""

| I

9

o

-.

9 ~

9 P 9

. 9

|

(b) Fig. 5.6 Two typical configurations with SPBC on a 100 • 100 lattice with h = 0 in (a) and h = 5 in (b). From Vall~s et al. (1989).

86


energy in a driven system depend on the orientation of the interface, but it does so with a kink singularity! To understand this unusual behaviour at a heuristic level, consider the T ~ 0 limit of an h - 1 system driven with E > J. Now, each interface must have at least one step. For the particle at one of these steps, there will be no neighbour to prevent it from being driven "forward" by E (Fig. 5.5 a). As it travels around the system, it will eventually wind up at a step of height two (Fig. 5.5 b). From here, the drive can take it away from the interface and into "the bulk". Once in the bulk, it will drift across the sea of holes and end up at the opposite interface. Of course, other particles will follow, setting up a steady state in which both particle-density and energy-density are positive in the bulk. Note that if E < J, a particle cannot leave the step and Uexcesswill take the equilibrium value. Thus, even in the limit of T ~ 0, Uexcesswill be a non-trivial function of E, if only one step is present. Since this dependence should exist for arbitrarily large L, we expect that the 101 term will survive in the thermodynamic limit. We end this section with some remarks. At a casual glance, the set of numerical values in (5.7) may appear somewhat scattered, but in fact they are consistent with an "eyeball estimate" of typical configurations (Figs. 5.6 a, b). (For details, see Vallrs et al., 1989.) Secondly, the dependence of the bulk density on 0 might have been expected from certain steady state solutions (Janssen and Schmittmann, 1986b; Leung, 1990) of the mesoscopic bulk equation (5.3). For example, the following solution: ~b(~) -- M~ tanh (icMJv~),

(5.8 a)

where M o - 1 + x/~e sin 0

and

~ - x j_ cos 0 + Xll sin 0,

(5.8 b)

represents a single planar interface "tilted" at an angle 0. To be precise, let h denote the normal of the interface pointing into the high-density phase, so that 0 is sin -1 (h. Xll)" Note that there is no symmetry in 0 r -0. To compare with simulation data and recover the 0 r - 0 symmetry, we need solutions periodic in finite L• (Leung, 1990), an important detail to be considered in the following section. By necessity, these solutions involve two interfaces, with 0 - 0 and 0 - -0. Indeed, a closer examination of the two interfaces in Fig. 5.6 b shows that their characteristics are quite different. The 0 > 0 edge (on the left) is sharper, with a higher bulk density in its neighbourhood. Since particles are being driven onto this edge, we will call it the "absorbing edge", with characteristics similar to that of a growing crystal. In contrast, the 0 < 0 edge has a larger width, and is associated with lower bulk density. Similar to that in Fig. 5.5, we call it the "evaporating edge". This inequivalence of the two interfaces will play an important role in the next sub-section. Meanwhile, to compare the single tilted interface solution with

87


simulations, there is an alternative, using open boundary conditions in the transverse direction. In the case of an evaporating edge with small 0, preliminary results (Rudzinsky and Zia, 1993) are consistent with (5.8). Lastly, though this study was not sensitive enough to probe the existence of a possible singularity in the interfacial energy, it arrived at a more startling conclusion, namely, there is a singularity in the bulk energy as a function of interface orientation. In the sense that it scales with the volume, this singularity is "more severe" than one in the interface energy. Without doubt,

E (a)

1 ,'j w

(b) Fig. 5.7 Two typical configurations with SPBC on a 100 x 1001attice, withh = 12(a) and h = 16 (b).

88


it is related to interfacial smoothness. However, a clear and immediate relationship, like that in equilibrium cases, remains to be found.

Domain splitting and merging When large shifts are imposed, the bulk is affected even more profoundly. There is a sequence of changes in morphology and "topology" with increasing h. First, the one-strip state is found to be unstable, "splitting" into a multi-strip state. Following that, a series of "merging" transitions occurs, where an N-strip state evolves into an ( N - 1)-strip state. Since the multiple strips are in fact connected through the SPBC, each phase is just a single strip with multiple winding around the torus. This is the motivation behind using the term "topological change". To be specific, we consider the L - 100 case, with an initial single strip state that is fully ordered, i.e. any equilibrium state at T - 0. After 200,000 MCS, the single tilted strip remains the steady state, provided h _< 11. With a shift of 12, the system evolves rapidly by finger formation on the absorbing edge (Vall6s, 1988) and eventually settles into a six-strip state (Fig. 5.7 a). On the other hand, with further shifts, the system tends to evolve to states with fewer strips. In Fig. 5.7b, we show a typical configuration of a steady state with h - 16, displaying only five strips. Note that, in both cases, the strips are "tilted" in the opposite direction from that in Fig. 5.6 b. The mechanism for splitting and merging is clearly due to a competition between the drive, which tends to align interfaces with ~11, and the interparticle attraction in a SPBC, which favours a "tilted" interface. Therefore, we should expect aligned interfaces with L/h strips, whenever the shift is commensurate with L. This feature is definitely seen in the following cases: 20/10, 36/18, 36/12, 48/16, 48/12 and 100/20. If L/h is not an integer, the number of strips, N, will be the nearest integer. When the shift is more (or less) than L/N, the strips will tilt "with" (or "against") h. Thus, as h increases from zero, the single strip tilts "with h", as in Fig. 5.6 b. After the first transition, h - 12, which is less than 100/6, leading to tilts "against h", as in Fig. 5.7 a. One, among many, interesting questions is whether a positive critical angle, 0c, for the splitting transition exists in the thermodynamic limit. With the data of Vall6s et al. (1989), the transition angles are essentially linear in l/L, extrapolating to ~6 ~ A naive application of (5.8) will certainly lead to a positive 0c, assuming that parameters like ~ exist in this limit. However, fingering instabilities are absent from (5.8), and subsequent data with L - 312 show splitting occurring at a smaller angle (Leung, 1990). If theoretical considerations are correct, 0c will vanish as L ~ c~. On the other hand, the critical shift associated with the transition cannot vanish, since there must be a minimum strip width, w0. Apart from the


89

obvious limit of one lattice spacing, thin strips are also separated from each other by thin "gaps" and, responding to the attractive interactions, will merge into wider ones. Thus, large values of L/h are also unfavourable, and only single-strip states will be allowed when h < 2w0. It seems plausible that w0 exists in the thermodynamic limit, since it may depend only upon the microscopics. If this conjecture is correct, we will have a lower limit for the critical shift and none for the critical angle. One clear signal of both splitting and merging transitions lies in the total energy Uexcess(0). In Fig. 5.8, we show the data for the four L's studied. In each case, Uexcess displays a sequence of convex curves, each of which is associated with a different N-strip state. The transition from one to another appears to be accompanied by a large drop in Uexcess. Also notable are the kink singularities associated with each commensurate shift, an understandable behaviour since the interfaces here are again aligned with ~11" Another signal lies in the average current along the field direction, a plot of which is similar to Fig. 5.8, with even sharper kinks. The nature of the splitting transitions has not been investigated in detail. A hint that it may be second order comes from a series of configurations in the time evolution of the h/L = 11/100 case, showing very large fluctuations (Vall6s, 1988). Similar snapshots of the h/L -- 12/100 case, which is just on the multistrip side of the transition, show that the initial single strip configuration evolves by finger formation, reminiscent of Mullins-Sekerka (1964) 1.0

' '''

I ....

I ....

'i'L''

; I . . . .

j

9

!

o

0.8

o

=

9

9

20

U e x c e s 8

9 A

9

A

u.t~

9

9

9 =

.. u

-

.

9 __36

9

-

9

=

__n.4

-

.

.

9

=

9

9

"

". 48

~

= ~

~

=

~

=

0.2

.

-

~176176176176

."

""

100

_

~

-

~ .

.

0.0 , , , . , 0.0

I,.,,, 0.~

I .... 0.2

! .... 0.3

! .... 0.4

i 0.5

0

Fig. 5.8 Excess total energy density, Uexcess,as a function of the shift angle 0, for L = 20, 36, 48, and 100. From Vall6s et al. (1989).

90


or Saffman-Taylor (1958) instabilities. The implication is that, as the shift increases, one mode turns soft and then unstable (Leung, 1990). In this theory, the merging transition is argued to be the result of the same mechanisms. More work on both the simulational and theoretical fronts will be helpful in settling this issue.

5.3.3

Numerical and theoretical studies

Our theoretical understanding of the physics of inhomogeneous steady states is very limited. In this section, we review some of the steps taken, and indicate some of the steps which we believe are necessary towards a more complete theory. The efforts so far have been directed towards the physics of interfaces, concentrating on deriving an appropriate equation of motion for such degrees of freedom and studying the stability properties. In an early study (Leung, 1988), starting with the bulk equation (2.15), the semi-phenomenological approach of Langer and Turski (1977) is exploited to arrive at a Langevin equation for the interface. A crucial ingredient is using the Gibbs-Thomson relation. Like its counterpart in model B, this equation is both non-linear and non-local. Focusing on small deformations of long wave-lengths (~ - . 0), this equation is linearized, leading to two predictions. First, the interface remains stable, while the dependence of the relaxation rate on ((i.e. the dispersion relation ~ (~')) is definitely affected by o~. Instead of the equilibrium ,; oc q3, the drive provides a length scale (oc 1/o~) for crossing over to a striking ~o oc q2.5. Second, tracking the effects of the bulk noise, correlations of the interface are found to be equilibrium-like, i.e. given by (5.4). Thus, for the d -- 2 case, the interface always remains rough. The source of the discrepancy between simulation data and this result is not exactly known. One possible weak link in this approach may be the use of the GibbsThomson relation, which relies on the concepts of surface tension and local equilibrium. A later study (Yeung et al., 1993), also following the semiphenomenological approach, replaces these notions by the concept of "local steady state". Though the equilibrium surface tension is still present, there is an extra term (oc 8~11. t~, with t~ being the local normal to the interface) in the Gibbs-Thomson relation. In addition, (2.13) accounts for a higher current along the interface, so that a similar extra term is also present in the relationship between the normal velocity of the interface and the discontinuity of the current across it. Focusing on the deterministic part of the problem, this study found a more dramatic crossover: to ~ oc ql.5 instead of w cx: q2.5. Of course, a conceptual difficulty now arises, namely, can the bulk modes still be neglected, now that they relax slower (with ,; oc q2) than the interfacial degrees of freedom? Putting this issue aside, the extra-strong damping is presented as the signal of roughness suppression. Unfortunately,


91

it is not possible to make a direct comparison with the simulation results, C ~ 1/q 0"67 and p ~ 0, since the computation of either C(t]) or w2 requires adding proper noise terms. At present, it is unclear how to postulate such terms based on sound principles. To be completely free from using Gibbs-Thomson-like relations, a less ambitious program would be to expand the bulk equation about the inhomogeneous steady state solution and to study all perturbations at the linear level. Identifying the interface mode as that perturbation associated with the breaking of translational invariance, it is possible to extract, for example, C(q). So far, technical difficulties prevented a successful pursuit of this program. On the other hand, it is feasible to follow this recipe in a related model, in which particles experience a random drive with zero mean (Section 6.1.2). In that case, wE is found to diverge as s (for d - 2), so that it may be a stepping-stone toward a quantitative theory of roughness suppression in the standard model. Returning to the investigations of the standard model, we consider next properties of an interface "tilted" at an angle 0. Since both studies (Leung, 1988; Yeung et al., 1993) concentrated on a single interface in an infinite system, and pursued essentially the same route, it is instructive to compare the drastically different predictions. Their main differences lie in the GibbsThomson relations. The earlier work found that all interfaces are stable at the linear level, though the dispersion relation is strongly dependent on the tilt angle. In particular, as 0 varies from - 9 0 ~ to 90 ~ we have the gamut of ~ cx q2 to w ~ q4. In stark contrast, using the generalized Gibbs-Thomson relation, Yeung et al. (1993) predicted ~ cx _q2 to ~ cx q2! Note that the negative sign means that an interface with 0 - - 9 0 ~ is unstable in this theory, though crossover to the stable ~ cx q3 is found for large q. In addition, this same mechanism is argued to stabilize the opposite interface (0 - 90~ Numerical studies with (5.3) and r - 0.1, along the lines of previous work (Yeung et al., 1992), largely confirm this scenario. On the other hand, simulation studies with " E - c~" and SPBC show quite the opposite, namely, 0 < 0 interfaces seem to be stable while fingers grow on interfaces with sufficiently large positive 0 (Vall6s, 1988), consistent with a Mullins-Sekerka (1964) type instability (Leung, 1990). There are several factors behind the apparent disagreement. We discuss the 0 > 0 and 0 < 0 cases separately. At a casual glance, the prediction of a stable 0 > 0 interface contradicts the usual Mullins-Sekerka result. However, on closer examination, the mechanism of Yeung et al. (1993) would produce a "ramp" in the density profile near the interface. A sharpening of this kind is also observed in simulations, for example, at the left interface in Fig. 5.6 b. When its effects are taken into account, the Mullins-Sekerka instability is expected to re-emerge (Jasnow et al., 1981), and fingering should destabilize such an

92


interface. However, the development of fingers is a relatively slow process, and we should regard the stabilizing mechanism as effective only at early times. As for the 0 < 0 interface, the stability seen in simulations (Vall6s, 1988; Vall6s et al., 1989) may be due to the restriction of small ]0] and large q, in the following sense. Recall that there is a crossover to stable behaviour for q ~ qs, with qs a decreasing function as 0 rises to 0. In simulations (Vall6s et al., 1989), splitting occurs before ]0] becomes very large. Thus, the predicted instability can be seen only for very small q. On the other hand, q >_ 27r/L, which may be too "large". To draw quantitative conclusions, we clearly need further studies. Another likely source of the disagreement lies in the strengths of the drive. The mesoscopic theory is expected to be valid for small E, but the simulations are performed with "E - c~"! To understand the former remark, note that the foundation of the theory, (5.3), is isotropic, apart from the o~ term. Thus, for sufficiently small r a stationary solution corresponding to a 0 - - 9 0 ~ interface can exist always, allowing for a meaningful discussion of its stability properties. Further analysis relies on the existence of sharp interfaces at all angles and an equilibrium surface tension, leading us to believe that the conclusions are most likely valid only for E 3J, say, a state with a sharp 0 - - 9 0 ~ interface will break up in the first MCS. Since it evolves without any perturbation, such a state is simply not stationary. To prevent the very existence of such stationary solutions in a dynamical equation, we must modify (5.3). One possibility is to let ~-II > 0 with ~-• < 0 in (2.15). Such an equation can never be rescaled into an isotropic form like (5.3). In the absence of stationary states with a 0 - - 9 0 ~ interface, the question of stability is irrelevant. In an attempt to understand domain splitting and merging, Leung (1990) studied the stability properties of a pair of 'tilted' interfaces in afinite system. The importance of considering a pair (with positive and negative O's) lies in the dependence of the bulk magnetization on 0 (5.8). In a finite system with both interfaces, how does the bulk behave? Numerically obtained stationary solutions to (5.3) show that the lesser M 2 fills most of the system, while a boundary layer of higher M 2 accompanies the 0 > 0 interface. That the smaller M 2 dominates is consistent with the increase of U, the bulk energy density, with 0. Also, the boundary layer is definitely observed (Fig. 5.6 b). Further, the existence of such a layer, together with an equilibrium GibbsThomson relation, leads to the Mullins-Sekerka instability and finger formation. Qualitatively, this prediction agrees with the main feature of the evolution of the h / L - 12/100 simulation (Leung, 1990). Clearly, it is premature to conclude that this theory is definitive, since there are


93

many lingering doubts about the validity of equation (5.3), sharp interfaces, Gibbs-Thomson relations and surface tension. We expect that significant progress will be possible when the differences between this approach and that of Yeung et al. (1993) are reconciled. Before concluding, we should mention that there are other interesting studies on interface instabilities in driven diffusive systems. Involving additional drives such as density gradients and open boundary conditions, these will be discussed in Section 6.2.2. Indeed, the next section will be devoted to the many variations of the standard model, leading to even richer phenomena. From this section, we conclude that the physics of the standard model below criticality, in contrast to its counterpart above or near To, is much richer but much less well understood. Many questions concerning interfaces remain to be answered. Most urgently needed is a reliable bridge between the microscopic lattice model and the mesoscopic continuum version. Nearly all simulations impose "infinite" E, where the effects of non-equilibrium dynamics are most pronounced. To describe such systems, we believe that it is more appropriate to use the fully anisotropic bulk equation (2.15), with ~-• < 0 and ~ll > 0. As we have argued, if E > 3J, there can be no stationary state with a particle-rich phase "upstream" from a hole-rich one. Further, applying a large but random E with zero mean (Section 6.1), we would arrive at the same conclusion about both interfaces (~. ~Jl - +1). For this model, there would be no g term. Thus, to account for the absence of such inhomogeneous steady states, the only reasonable phenomenological choice appears to be ~-Jl > 0. At the same time, it would be most interesting to perform simulations with E T• In the remainder of this section, we will focus on the disordered state. For temperatures fixed above Tc, the collective behaviour of our model, at small enough k and co, can be adequately predicted by a linearized version of (6.1), i.e. (3.1). Thus, all conclusions of Section 3.1.1, concerning generically singular structure factors and power law dominated correlations, follow. Simulations of the random drive model confirm such decays (Hwang,

98

B. S c h m i t t m a n n

and R. K. P. Zia

400

5'(1,0)

200-

It

0 0.6

Inl

1'.4

i

1.8

Fig. 6.1 Structure factor S(1,0) for 30 x 30 systems, driven with infinite fields, randomly (x) and uniformly (+).

1993). For the two-temperature model, both simulations (Garrido et al., 1990; Cheng et al., 1991 a) and perturbative approaches (Maes and Redig, 1991 a, b) arrive at these results. Beyond these systems, the formalism which led to ^!

G(r'~

c~) c
3. To find the statistical width of an interface in a finite system, we integrate (6.9) down to 1/L, giving us immediately the result that, in d - 2, it diverges only as In L instead of the usual x/-L. Recent simulations employing lattice sizes up to 40 x 800 (Leung and Zia, 1993) confirm a crossover from C cx 1/q 2 to Ccx 1/q as q decreases. However, a plot of 1/C vs. q does not extrapolate to the origin, posing a serious challenge to both the theoretical and simulation studies. On the theoretical front, it is conceivable that terms non-linear in X are relevant, building up further anomalous powers. Also, since bulk fluctuations are no slower than interfacial ones, their effects cannot be neglected. In either case, we believe that the approach described here is more reliable than postulating an interface model such as the KPZ equation (Kardar et al., 1986), since the long range correlations in the bulk are likely to induce non-local effective interactions in h(y,s On the simulations front, the existing data do not distinguish whether a gap is indeed present or whether a crossover occurs at even smaller q to a lower power, such as the q2/3 of the standard model (Section 5.3.1). Mediated by the power law decay of bulk correlations, unusually strong finite-size effects in both Ltl and L• present unknown hazards for data analysis. Whatever the resolution for this randomly driven case, we are hopeful that it will provide a crucial stepping stone towards a satisfactory understanding of interface phenomena in uniformly driven systems. Beyond that, it would be clearly interesting to simulate d _> 3 lattices and observe anisotropic correlations and crossover behaviour.

6.1.4

Combinations of direct and random drives

We end this section with some brief comments on natural generalizations of the standard model, which are most appropriately described by Langevin

106


equations with a combination of direct and annealed random drives (Zia, 1991). One of the most frequently raised questions is the following: What would happen if, in a simulation on a hypercubic lattice, the uniform electric field points in a generic direction, rather than along one of the lattice axes? To be more precise, jump rates in such a model would be given by (2.6) with = - ~ - E, where t~ and/~ are unit vectors in the direction of the jump and the field, respectively. Based on the isotropy of critical properties in the E = 0 Ising model, we could argue that only/~ can provide a splitting of space into "longitudinal" and "transverse" parts. The consequence is a naive conclusion, i.e. only Te, but not the universality class, will change. On closer examination, a subtle interplay appears between the lattice and the field directions, which is best illustrated by an example. Consider d = 3 a n d / ~ pointing along ~ + )3. Now, there are two types of"transverse" jumps, ones in the x - y plane vs. those in the 2-subspace. The latter are controlled by the internal energetics alone, behaving just like the transverse jumps of a standard model. Thus, we would associate a r• with ~2, as before. For jumps along either ~ or 33, however, there is an additional randomness due to the E / x / ~ terms so that we should expect transport in the x - y plane to be different. Defining x+ = (x 4- y)/v/2, we argue that the diffusive operator should be 2+

2,

(6.10)

with ?• d: T_L. The last inequality is the hallmark of a multi-temperature or random drive model. On the other hand, we must include a term like 8 ~ r 2/~x+ to account for the effects of the DC drive. The result is a combination of the two within one system. These arguments can clearly be generalized to models in which the subspace associated with ?_L has dimension #7 > 1. However, since E is a vector, the "longitudinal" subspace must remain onedimensional. Expecting rll > ?• > r• we believe that m• will vanish first, as T is lowered to some critical To, so that ~ + 1 momenta will scale as k I+A. Critical properties will belong to a new universality class (Bassler and Schmittmann, 1994 a), while de - 5 - #~. While progress on the theoretical front appears relatively straightforward, it is not simple to carry out simulations for such models, since we must work with lattices in d _> 3. Finally, recall that we discussed three possible "critical" field theories (Leung and Cardy, 1986; Janssen and Schmittmann, 1986 b; Gawedzki and Kupiainen, 1986) at the end of Section 2.3. We argued that only the first of these was acceptable for describing a critical standard model. The other two, i.e.

(b) (c)

T• > 0 with TII ~ 0, with de - 4.5 yet lacking a stable fixed point, and T• ~ 0 and TII ~ 0, with de - 8 and a stable fixed point,


107

were dismissed as being only of "theoretical" interest. We believe that it is possible to access both of these in simulations, by allowing a more general jump rate than (2.5, 6). In particular, we could add a random drive in the transverse direction or couple jumps in the transverse directions to a bath with a higher temperature, T• (Bassler and Zia, 1995). This expectation follows from the arguments (Leung and Cardy, 1986) that, in the standard model, the direct drive renormalizes TII to a higher value than 7-• so that the latter vanishes first. Thus, if we begin with T• > T, there should be a regime of sufficiently small E, in which 7-• is greater than 711,even after renormalization. On the other hand, we know that 7• < 711,if T • is close to unity. Therefore, we expect, for each E, a line in the T - T • plane on which 7• - 7-11.On one side of this line, T• is always less than TII and will vanish first as the temperatures are lowered. For these points, the critical behaviour will belong to the universality class of the standard model. On the other side of this line, TII is always smaller and should vanish first. Since there is no stable fixed point for this class of theories, we anticipate encountering a first order transition before 711 - 0 is reached (Janssen and Schmittmann, 1986 b). Surprisingly, a continuous transition is found, as TII is varied, in a simulation study of the d - 2 model with T• - ec and E / J - 0.5, 1.0 and 1.5 (Bassler and Zia, 1995). For E _> 2J, the system never orders into the appropriate state. It is possible that, like the 3- and 4-state Potts models in d - 2 (Wu, 1982), fluctuations induce a second order transition, contrary to the predictions of field theory. Lastly, case (c) should lie on the 7-• - 7-11line, where both can vanish simultaneously. Since such theories are controlled by a distinct stable fixed point (Leung and Cardy, 1986), these systems should display a new class of critical behaviour. It would be interesting to see if these theoretical expectations, from the qualitative aspects of the phase diagram to the quantitative aspects such as critical exponents, are borne out in simulations.

6.2

Chemical potential gradients and drive defects

In many physical systems, external forces or symmetry breaking fields act only locally, such as at the boundaries of the system. However, if chosen so as to induce a global current (of heat or matter) through the system, they will force the latter into a non-equilibrium steady state even though the dynamics in the bulk may be entirely equilibrium-like. The effect can be profound, as demonstrated most strikingly by fluids in temperature gradients (Section 7.6). In the following, we will investigate the effect of such local modifications of the dynamics on the statistical mechanics of our prototype models, the equilibrium and the driven Ising lattice gas. We begin by reviewing, in Section 6.2.1, the considerable amount of work that has addressed the influence of a

108


density or chemical potential gradient (CPG) on equilibrium lattice gases. Intercalation phenomena provide an intriguing, if much more complex, experimental realization of such systems. Next, we return to the standard model which, unlike equilibrium systems, carries a nonzero global current even before any gradients are imposed. Depending on whether CPG and electric field are parallel or antiparallel, the effective current will be enhanced or suppressed. In the former case, which is the subject of Section 6.2.2, the phase transition of the standard model survives, but the structure of ordered configurations is completely altered. In the latter case, even the transition itself is suppressed, as we will see in Section 6.2.3. Finally, Section 6.2.4 addresses the stability of the interface in the standard model under a transverse CPG, i.e. it explores the possibility of a Mullins-Sekerka instability of interfaces aligned with the electric field.

6.2.1

Systems with a chemical potential gradient only

Motivated by theoretical and experimental evidence for long range correlations in non-equilibrium fluids subject to temperature gradients, Spohn (1983) investigated correlations in a stochastic lattice gas driven by a CPG. Since this model system is, of course, much simpler than a real fluid, some rigorous conclusions can be drawn. In particular, the two-point correlations can be computed exactly, without recourse to the kind of hydrodynamic (i.e. coarse-grained) theories which are indispensable in the case of fluids. At a deeper level, a rigorous link between microscopic theory and fluctuating hydrodynamics can be established, thus providing stronger support for the latter approach. Earlier interest in this model focused on the densitydependence of the diffusion coefficient (Murch, 1980). Since the stochastic lattice gas which is driven by a CPG serves as prototype for much of the discussion in this section, we will describe it in some detail. Unlike fluids, it possesses only a single conserved quantity, i.e. particle density. Its energetics is also simple, namely, particles interact according to the nearest-neighbour Ising Hamiltonian (2.1). The equivalent of the "Rayleigh-B6nard" experiment for the lattice gas takes place in a slab geometry, in which the bottom and top edges of the slab, separated by a distance 2L, are coupled to particle reservoirs at different chemical potentials. To describe such an arrangement analytically, we need to define appropriate microscopic transition rates between different configurations, distinguishing between transitions involving only the bulk and those involving the edges. In the bulk, transitions are restricted to nearest-neighbour particle-hole exchanges, precisely those of an equilibrium system. Similarly, at the top and bottom edge of the slab (which extends to infinity in the remaining directions), particles are inserted and removed, according to rates that are


109

translation-invariant and local within the edge surfaces, but include the chemical potential differences in the associated detailed balance condition. The global effect of the two particle baths is, of course, to fix the boundary densities at two different values p+ and p_, setting up a current through the system. The temperature is chosen sufficiently high so that the system is always in the disordered phase. The usual hydrodynamic approach consists of postulating a diffusion equation for the macroscopic density p(s t)" -----+

~tp(Y,t) - VD(p(Y,t)) Vp(Y,t).

(6.11 a)

The assumption of "local equilibrium" implies that the diffusion coefficient D(p) may be defined, via the Green-Kubo formula, as a space-time integral over current-current correlations which are computed in global thermal equilibrium at density p. Letting y be the coordinate along the CPG, we further assume the existence of p~(y), a steady state solution to (6.11 a), which is homogeneous in the transverse directions and satisfies the boundary conditions: ps(+L) - p+. To study correlations and fluctuations, we linearize (6.11 a) in ~b(s t), the small deviations from Ps(Y), and supplement it with noise so that it becomes a Langevin equation:

~t(~(s t) - V2D(ps)dp(Y, t) - ~ .f(s t).

(6.11 b)

As usual, the Langevin force j has zero mean. For equilibrium systems, the FDT dictates a specific variance ( f f ) . Here, we again rely on the assumption of "local equilibrium" and postulate the same form:

(jm(Y,t)jn(Y',t')) - 2~Smn~5(t- t')~(s

s

(6.11 C)

where X takes the same functional form as in equilibrium, i.e. a compressibility, given by a spatial integral over the static density correlations. Together with the conditions ~b - 0 at the boundaries, these equations suffice for describing the disordered phase. To allow for rigorous conclusions, we consider the limit J / T ~ O, in which case the lattice gas reduces to a simple exclusion process. Particles hop randomly on the lattice, subject only to the excluded volume constraint and the imposed boundary condition. Since non-equilibrium effects appear already in this limit, we expect that the results for an interacting system will be qualitatively similar. Here, D - 1, so that the steady state density profile, Ps(Y), is simply 89 + y / L ) + p _ ( 1 - y / L ) ] , corresponding to a homogeneous gradient which vanishes in the thermodynamic limit. The steady state current through the system, given by - X7Ps, is just a constant: J s - (P- -p+)/2L. While these results are obvious, the equal-time density correlation, C ( s (~(s is rather more striking. Recall that, in equilibrium, C decays exponentially with r - 1s reducing to a g-function

110


in the J / T ---, 0 limit. Here, using X = ps(1 - Ps) in (6.11 c), it can be written as the sum of the two terms

C(Y) = X6(Y) + J~G(Y),

(6.12)

where G is the well-known Dirichlet Green's function of the Laplacian in a slab. While the first term is just the local equilibrium piece, the second represents the important non-equilibrium contribution. For r 1 cases. There are two major conclusions of this analysis. Firstly, the microscopic density field converges, in the hydrodynamic limit, to the solution of the nonlinear diffusion equation (6.11 a), with appropriate boundary conditions. Secondly, any local non-equilibrium steady state average converges to the associated equilibrium (Gibbs) average, provided the latter is calculated at the correct local density Ps. Since all density gradients vanish in the limit L ~ c~, our conjecture follows, at least for those averages. These statements, and the expectation that they will also hold in d > 1, still leaves plenty of scope for explorations. What is the effect of a CPG on the onset and the nature of phase separation? What shapes do the density profiles take, as functions of temperature and chemical potential, especially in ordered phases? Since the CPG introduces a spatial anisotropy, how is the thermodynamic limit approached as we let longitudinal or transverse system sizes, LII or L• go to infinity?


111

So far, there has been no detailed or systematic effort to answer these questions. Here, we highlight a few studies toward this goal. Considering an Ising lattice gas with attractive interactions J > 0 in d - 2, Andersen and Mouritsen (1989 a) carried out Monte Carlo simulations and measured the density profile. Their results are quite consistent with the intuitive picture that, in the disordered phase, the inter-particle attraction tends to gather the particles at the top edge, thus increasing the density there compared to the linear, J - 0 profile. Similarly, the density in the vicinity of the bottom edge is reduced. Thus, the curvature of the density profile is similar to that in an equilibrium Ising system with an interface below To. An analytic investigation of the profile also confirms this qualitatively, showing also that the total current is reduced, compared to the J 0 case (Fogedby and Svane, 1990). These effects become more pronounced as J / T increases toward criticality, beyond which phase segregation sets in. Now, the profile develops two plateaux, corresponding to the equilibrium bulk values of the density, with the usual interface in the centre and two crossover regions near the boundaries. As long as the longitudinal dimension of the system is large enough to allow for two well separated plateaux, this type of phase-segregated profile persists for all choices of 0 _< p• 1, while generically anisotropic, is not unique, and two versions have been considered. On the one hand, one may expect that the randomness in the hopping rates affects all spatial directions,

146


leading to t) -

+

v

ll

t) +

t) -

9

-

,

(6.22 a) where the notation of (3.24) is used. Clearly, the noise also becomes anisotropic, so that ~ and ( are distributed according to Gaussians of different widths. Note again that these are t-independent functions. On the other hand, one might characterize the landscape of barriers between sites by a coarse-grained potential lI(~), in which case, the Langevin force ~ should take the form of a gradient, ~ = - ( V , ~)11. Averaging over 11 should then lead to V2((s + ~2((s in (6.22 a) instead. However, in the presence of the drive, the longitudinal component suffers renormalization, so that this potential condition is preserved only for ~. Thus, we write -.#

~t 4~= )~((V 2 + P~z)4~(Y, t) + ~o~bz(Y, t) + V2~(Y) - ~((Y)}.

(6.22 b)

For both equations of motion, (6.22 a, b), power counting gives an upper critical dimension de - 4 , as opposed to de = 2 for the pure case. Further, both lead to the "super-diffusive" spreading of density fluctuations along the field, in the sense that the dynamic structure factor obeys the scaling form: S(kll,k• o c f ( ~ t I+A, k2xt), with a positive anisotropy exponent A. In an e-expansion around d~, A is found to be e/5 + O(e 2) and 5e2/36 + O(e 3) for the two versions, respectively. Given the intuitive expectation that quenched randomness should slow down diffusive processes, there is as yet no satisfying physical argument as to why the half-filled random system should exhibit anomalous diffusion in spatial dimensions 2 < d < 4 where the pure system does not. While the previous analysis was concerned with the effect of quenched disorder on long-time behaviour in the standard model far away from criticality, the Monte Carlo study of Lauritsen and Fogedby (Lauritsen, 1991; Lauritsen and Fogedby, 1992) focused on critical properties, in the presence of a small concentration of randomly distributed immobile impurities. The particles interact with these impurities only via the excluded volume constraint. Otherwise, the dynamics and boundary conditions are identical with those in the standard model. Using infinite E and a half-filled square lattice, the data are averaged over 10-20 different impurity configurations of fixed concentration, c. For attractive inter-particle interactions, the system orders into a single strip, at a critical Tc(c) which decreases linearly with small c. The data for the order parameter and its cumulant (Section 4.1), measured on rectangular lattices with L loc L 3, are analysed using anisotropic finite-size scaling (Leung, 1991 a, 1992; Section 4.1.4) and the exactly


147

known critical exponents of the pure system. For c--0.01, the data for different system sizes collapse sufficiently well so as to validate the scaling ansatz, and to arrive at a credible estimate of T~. For c = 0.05, the data begin to scatter more strongly, and are therefore less conclusive. Additional information is gained from the power spectrum of the current-current correlations. At criticality, this quantity decays algebraically with frequency, as ~-~. For the pure system, ~ = 1 (Leung, 1991 b). The same exponent captures the behaviour of the disordered system quite well, even up to c = 0.05. Systems with large impurity concentrations have not yet been investigated. However, it is clear that the transition cannot survive if 1 - c is below the percolation threshold. For repulsive inter-particle interactions, we saw that the continuous transition of the pure system, where it exists, belongs in the universality class of the Ising model with nonconserved order parameter (Section 6.5). In the presence of small impurity concentration, the system still orders in a checkerboard pattern. However, the ground state can be highly degenerate here. Not surprisingly, Tc again decreases with E and c. Employing the same isotropic, Ising-type finite-size analysis as for the pure system, the disordered driven system is also found to belong to the Ising universality class. Dynamically, the introduction of quenched impurities into the standard model poses at least two intriguing problems. As noted by Lauritsen and Fogedby (1992), particles can get trapped behind small clusters of impurities, such that relaxation times to steady state increase rapidly with c. Of course, at least in principle, the data can be improved by increasing the length of runs. The second issue, however, is much more fundamental. In a finite system with periodic boundary conditions, a uniformly driven particle will revisit the same impurity configuration periodically. Thus, the randomness average is plagued by strong correlations. Moreover, it is highly doubtful whether the limit t ~ c~ commutes with the thermodynamic limit Ltl, L• -~ c~. In the simulations, we clearly let t ~ c~ first, while the physical steady state is obtained through LII, L• ~ c~ before t ~ c~, i.e. particles are driven through an impurity environment which should not repeat itself. The unresolved problem that we are facing here is very basic, namely, how to simulate quenched randomness in finite systems that carry a global current. Finally, we note in passing that this system can be formulated as a two-species problem (Section 6.4), if we allow the species to obey different dynamics. Specifically, we may model the impurities by, say, the negative particles, if they are assigned a vanishing mobility. However, none of the avenues opened up by this recognition have been explored at this time.

148

6.7


Special limits

In the study of complex physical systems, the analysis of special limits and model simplifications plays a considerable role. Their choice is motivated by the search for a greater degree of solubility, even though this may carry the price of leading further away from the original system. Nevertheless, special limits often raise, and answer, questions of generic relevance, in addition to being of interest in their own right. Moreover, exact solutions, if possible, always exert their own particular attraction. In this section, we review two such limits. The first of these, based on a separation of time scales, effectively decouples the dynamics along the field from that in the transverse directions so that a d-dimensional model can be reduced to a ( d - 1)-dimensional effective theory. In the second part of this section, we present an overview of (predominantly) exact results for strictly one-dimensional driven lattice gases.

6.7.1

Extreme anisotropic rates

First suggested by van Beijeren and Schulman (van Beijeren and Schulman, 1984; Krug et al., 1986), this "fast rate" limit involves two key components. First, one introduces two different microscopic time scales, so that particlehole exchanges in the field direction are attempted at a much greater rate than exchanges in the transverse directions. Second, the drive is chosen to be infinite in order to decouple the dynamics within a given parallel lattice column from its neighbours. Both of these features are required to ensure that the particle distribution in each column can (at least approximately) equilibrate, independently of the configuration of all other columns, before the next transverse jump takes place. Given a simple column distribution function, only the ( d - 1)-dimensional transverse dynamics remains to be solved. To implement this program, we modify the jump rates of the d-dimensional standard model, in particular (2.6), as follows. So far, the anisotropy induced by the drive affected only the argument of the characteristic rate function w(x). Now, we introduce an inherent anisotropy into the rate W[~ ~ ~'] itself, by allowing for two different microscopic time scales, Ell and E• controlling jumps in the parallel and transverse directions, respectively. Thus, (2.6) is amended to W[(~ ~ (~'] = V~w[fl(A~ + gE)]

(6.23)

where c~ = I[ (2_) if the configurations ~ and (~' are separated by a parallel (transverse) jump. Equation (2.5) (i.e. the definition of g) remains unchanged. For generality, we also allow for anisotropy in the exchange couplings JII


and J•

149

of (2.1). Specific choices of w that were considered include

w ( x ) - exp ( - x / 2 ) which we will refer to as vBS rates (van Beijeren and Schulman, 1984; Krug et al., 1986) and Metropolis rates (Krug et al., 1986). The first step consists in taking the limit Fii/Fj_ -+ co, so that "infinitely many" parallel jumps take place for each transverse jump. However, the rate at which parallel exchanges are actually accepted is still strongly influenced by the local neighbourhood as long as E and the exchange coupling JII, J• are of the same order. One way of decoupling neighbouring columns consists in taking E - o c uniformly, so that jumps along the field always take place, being entirely independent of the local environment, and jumps against the field are completely suppressed. As a result, the distribution of particles in each column relaxes independently, on a time scale 1/I'll, to its steady state P*, before being perturbed by the next transverse jump. For a periodic chain of LII sites with a fixed number n of particles, P* is determined by n alone, in that it gives equal weight to each configuration of n particles, irrespective of how they are distributed along the chain (Spitzer, 1970). Thus, the steady state is homogeneous. With regard to the following discussion, the relevant feature here is the "trivial" form of P* (n), as opposed to the details of the dynamics that generate it. Thus, the choice of uniform, infinite drive is sufficient but not necessary: for instance, an annealed random drive (cf. Section 6.1), distributed bimodally with sufficiently large amplitude, or a two-temperature model (cf. Section 6.1) with T I I - o c (Krug et al., 1986) will lead to the same P*. Thus, the following results will also apply to these cases. Transverse jumps between columns take place on the slow time scale 1/s177 perturbing the stationary column distributions, which, however, return rapidly to the steady state. Thus, on this time scale, the set of single column occupation numbers {nj}, j - 1,..., L~ -1, fully specifies the microscopic configurations of the whole lattice. Since the nj are defined on a ( d - 1)-dimensional lattice, the dimensional reduction is apparent. To describe the transverse hopping of particles between neighbouring columns, one writes a master equation for the time evolution of the probability distribution P({nj }, t ),

~t P({nj}, t) - Z

{~'}

{ I~[{n~} ---+ {nj}Je({n}}, t) - l/l/[{nj} ---+{n}}]P({n}}, t)} (6.24)

where the jump rates W[{n} -+ {n'}] are formally obtained by averaging the microscopic rates W of (6.23) over the single column distribution P*. As an example (van Beijeren and Schulman, 1984), we consider four neighbouring columns in a two-dimensional system, with occupation numbers p, n, m and q, and ask for the rate with which a particle could hop from the second to the

150


third column. If we assume that the particle adjoins an empty site in the first column and two occupied sites in the second column, while the receiving site has two empty nearest neighbours in the third and an occupied one in the fourth, the rate for this process is given by: (L• L•

(n-1)(n-2) (L•177 (L• - 1)(L• - 2) (L• - 1)(L• - 2) x n (L• L•- m) r•177

- 4Jii)].

q L•

(6.25)

The first factor here is just the probability to find a hole in a column with p particles. Of course, (6.25) is only the contribution from a specific nearestneighbour configuration of the central particle-hole pair. To obtain the total rate gz for a transverse exchange, we need to sum o v e r 2 6 s u c h terms corresponding to all local environments. This completes the sequence of steps that constitute the fast rate limit, the end point being the "coarse-grained" master equation (6.24), whose primary invariance is particle-hole symmetry. The remaining task is to study its steady state properties for a variety of rate functions. In particular, one is interested in the average column occupation number (nj) and its fluctuations ~j as functions of temperature. In general, (6.24) can only be approximately solved. While differing in detail, both approaches that were suggested (van Beijeren and Schulman, 1984; Krug et al., 1986) rest on a decomposition of the column occupation number, nj -- (nj) -+- ~j. At sufficiently high temperatures, configurations are disordered, so that (nj) is independent ofj and the relative fluctuations ~j/(n) are of O(L~l/2), in the spirit of van Kampen's (1981) mean-field like fl-expansion. Taking the limit LII --. oo and keeping only terms to second order in ~j, which is equivalent to approximating P* ({nj}) by a Gaussian, one can compute, for instance, the structure factor S(k)- ~ j (~j(o)e ikj. A divergence of S(k) at a critical k~ signals an instability of that mode as temperature is lowered. For attractive interactions J• > 0, the instability is driven by homogeneous fluctuations with k~ - 0, while, for repulsive J• < 0, kc - 7r, corresponding to staggered fluctuations. The associated temperature, Ts(p), defines the spinodal and depends on the average particle density p of the system. To locate the critical temperature of the system, we recall that Ts (p) marks the temperature at which the disordered phase becomes unstable against infinitesimalfluctuations. Conventionally, the instability againstfinite fluctuations defines the phase co-existence curve To (p), which lies above the spinodal, coinciding with Ts (p) only at the critical temperature. By virtue of particle-hole symmetry, both functions are symmetric around p - 1/2,


151

taking their maximum there, so that Ts( 1 ) = To( 1 ) _ Tc defines the critical point. The interesting question here is whether a transition exists and how Tc depends upon the exchange couplings JII, J• and the rate functions w(x). We summarize only the main features, cautioning that all statements are restricted to those choices of w(x) that were studied. First, for repulsive couplings JII < 0, J• < O, the structure factor does not diverge, so that no transition is found. If JII, J• are both attractive, a transition exists for all w(x) that were investigated. For mixed couplings, such as JII > 0, J• 0, one finds that, in d - 2, the fast rate limit critical temperature Tc lies above the Onsager yet below the standard model results, for any w(x). For d _> 3, Tc can lie on either side of the equilibrium critical temperature, depending on the rate function. Before extracting critical exponents, we need to recall that the preceding discussion is based on an expansion in small fluctuations, retained up to quadratic order. This approximation is valid provided L l l - . oc at fixed T-r For T ~ Tc it leads to mean-field exponents, since it does not account for large, critical fluctuations. Thus, we obtain the classical values r/• - 0 , "~j_ - 1 and ~,• - 1/2, in the notation of Section 4.2.1. While these actually agree with the exponents of the standard model, they do not describe the randomly driven system (cf. Section 6.1) correctly, even though both models possess the same fast rate limit. Moreover, none of the "parallel" exponents exist, since the fast rate limit destroys all longitudinal correlations. We now turn to a discussion of behaviour below Tc, where the main task is to determine the co-existence curve. Most efforts have concentrated on an "equilibrium-like" analysis, arguing that one can define the equivalent of a Gibbs (van Beijeren and Schulman, 1984) or a Helmholtz (Kruget al., 1986) free energy functional, whence the co-existence curve can be determined. A special case of (6.24), which can be exactly solved, provides some support for this approach. For J• - 0 and vBS rates, the exact steady state distribution (van Beijeren and Schulman, 1984) can be written in the form

P* ({nj}) - Z -1 exp [-LllF({p})],5 ( ~;]~pj LilP)

(6.26)

with pj- nj/Llt and { p } - Pl, P2, . . . . Since J• - 0 , the columns decouple completely so that F takes the simple form F ( { p } ) - ~jf(pj), and f is exactly known. In the general case J• > 0, it is assumed that P* is still of the form (6.26), but F is no longer explicitly known, apart from being

152


kB T / J

r

0

1

Fig. 6.11 Co-existence and spinodal curves for vBS (upper pair) and Metropolis (lower pair) rates. The spinodal lies within the associated co-existence curve. From Krug et al. (1986). symmetric under exchange of any pair pg, pj. In fact, given the constraint on p, an arbitrary function r y'~j pj) can be added to F without violating the form (6.26). It is physically reasonable, however, to expect that P*({nj}) couples only columns within some local neighbourhood, so that ~ can be at most linear. Then, assuming that fluctuations are small in the thermodynamic limit, the addition of a linear r to the "Helrr.holtz free energy" F corresponds to a Legendre transform to the "Gibbs potential" G. To compute F, one defines a "chemical potential" # ( p ) - [~-l~F/~pi , evaluated at Pi =- P for all i. In the homogeneous phase, (6.26) leads to ~ # ( p ) / O p - S -1 ( k - 0; p). Two integrations yield, firstly, #(p), and hence F({p}) for a homogeneous state. If we analytically continue F to temperatures below T~ (where, strictly speaking, the homogeneous phase is unstable), we find that it develops a double well structure, symmetric around p = 1/2. The minima p+ (T), p_ (T) of F are then identified with the co-existing densities. Figure 6.11 (Krug et al., 1986) shows the spinodal and co-existence curves for vBS and Metropolis rates. For the latter, both the spinodal and the co-existence curve intersect the T = 0 axis away from p = 0 and p - 1. Thus, a sufficiently dilute or dense homogeneous phase is stable at all temperatures, down to T - - 0 . This is not the case for vBS rates. Given the mean-field character of the analysis, it is not surprising that


153

p+ - p_ cx (Tc - T)1/2 for T < Tc, so that the order parameter exponent/3 also takes its classical value 1/2. To test the validity of the arguments leading to the co-existence curve, Krug et al. (1986) performed a numerical integration of the spatially discrete equation of motion for pj(t), following in leading order from the f~expansion of (6.24). At any T < Tc, the co-existence curve is dynamically defined through the limiting values, limj_~ + ~ p] - p+, of a stationary, stable kink solution p]. Since the numerical effort is considerable, only vBS rates were investigated. For temperatures within 10% of Tc, the agreement between the numerical and analytic values of p+ is excellent. At lower temperatures, the kink solution becomes extremely difficult to locate, since it lies within the chaotic regime of the associated dynamical system. To test the predictions of the fast rate limit more directly, Vall6s and Marro (1986) performed simulations of the d = 2 standard model with anisotropic rates of the form (6.23). Thus, nearest-neighbour pairs in the field direction were sampled r = FII/F • times more often than those in the transverse direction. By studying the behaviour of typical observables such as Tc, the order parameter or nearest neighbour correlations, with increasing I', one hopes to gain insight into the limit F - . co. Unfortunately, the results are inconclusive, for two reasons: firstly, the relaxation of the system into steady state occurs on a time scale set by 1/I', so that it becomes extremely slow for I' > 20. As a result, only few data are available for those I"s, plagued by large error bars. Secondly, to arrive at reliable estimates for critical exponents, a careful anisotropic finite-size scaling analysis, along the lines of Section 4.1.2, should be performed. With those words of caution in mind, two features emerge: firstly, the data for r = 1 (the standard model itself) and I' = 5 are easily scaled onto one curve if the temperature scale is shifted by roughly 12%. For larger I', the data deviate more and more from the scaling curve. Secondly, the estimates for Tc(I') appear to decrease monotonically with increasing F, consistent with the observation that, in d = 2, the fast rate limit leads to a critical temperature below that of the standard model. To conclude, we remark that two basic questions still remain unresolved: firstly, is the transition in the fast rate limit really mean-field like, or is this an artefact of the approximations involved? As a first step towards an answer, one should derive the nonlinear terms in the Langevin equation associated with (6.24); exponents can then be computed by R G methods. This route may also shed some light on the second issue, namely, the relationship between the standard model and the fast rate limit. Clearly, they are not identical, since the fast rate limit, in contrast to the standard model, destroys all correlations along the field direction. Studying the two-temperature model of Section 6.1, Maes (1990) gained much insight, by appropriate rescaling of all longitudinal distances. It would be interesting to pursue a similar program for the standard

154


model and "derive" a fast rate limit from the Langevin equation (2.15, 2.16), by averaging over all fluctuations 4~(kll, kx, t) with kll r O. 6.7.2

M o d e l s in one dimension

In this section, we will consider driven stochastic lattice gases with vanishing nearest-neighbour coupling, J -- 0, so that the only inter-particle interaction is given by the excluded volume constraint (the only exception being (6.31)). This corresponds to the "free" gas limit E, T ~ c~, at E / T fixed, of the standard model (Section 3.1.2), restricted to one spatial dimension. In the simplest case, known as the asymmetric simple exclusion process (ASEP) of probability theory (Liggett, 1985), M identical particles move along a chain of N sites, hopping to the right with probability (1 + B)/2, and to the left with probability (1 - B ) / 2 , IBI _< 1, provided the receiving site is empty. Much interest has been generated by this model, since it is not only the simplest prototype of a driven system, but it also maps onto a surface growth model, known as the single-step model (Meakin et al., 1986; Plischke et al., 1987; Liu and Plischke, 1988; see also Krug and Spohn, 1991): Particles of"height" 2 are deposited on a surface (a line, in d = 1), under the constraint that neighbouring columns can differ only by Ah = + 1 in height. If we associate a lattice gas particle (hole) with a surface step of Ah -- + 1 ( - 1), then a particle hop to the right (left) corresponds to the removal (deposition) of a substrate particle. Thus, the two-point correlation function of the driven lattice gas is associated with the height-height correlation function of the single-step model. Furthermore, when taking the appropriate hydrodynamic limit (Rost, 1981; Liggett, 1985; Benassi and Fouque, 1987; Andjel and Vares, 1987; Lebowitz et al., 1988; Spohn, 1991; DeMasi and Presutti, 1991), the density profile obeys the one-dimensional Burgers equation (Burgers, 1974; Forster et al., 1977; Kardar et al., 1986; cf. also Section 7.3). The first issue of interest is, of course, the exact determination of the steady state distribution and of stationary correlation functions. In systems with translational invariance, the steady state is homogeneous, so that the associated equal-time correlations are "trivial", namely short-ranged. The focus here is therefore on time-dependent, or transport, properties, such as the nontrivial dynamic scaling of correlation functions, or the diffusion of a tagged particle. However, as soon as translational invariance is broken, by choosing open boundary conditions or by introducing a blockage, the non-equilibrium character of our model also manifests in the equaltime correlations, in that they decay algebraically with distance from the "defect". The time-dependence of fluctuations in such systems can be probed more deeply by investigating the motion of shocks. Finally, we review the exact steady state solution of the two-species


155

lattice gas of Section 6.4, in one spatial dimension, where it is closely related (Shnidman, 1991) to the Rubinstein-Duke reptation model (de Gennes, 1971; Rubinstein, 1987; Duke, 1989; Widom et al., 1991) for gel electrophoresis (cf. Section 7.4).

Systems with translational invar&nce We begin with the simplest case, namely, the ASEP with periodic boundary conditions. Its time evolution, in one space and one time dimension, can be expressed through its generator H, whose role resembles a quantummechanical Hamiltonian. Using a Bethe ansatz, its ground state and first excited state can be computed in the limit of large system size (Gwa and Spohn, 1992a, b; Dhar, 1987). Since H commutes with the transfer matrix of the six-vertex model (Lieb and Wu, 1972; Kasteleyn, 1975; Baxter, 1982), provided the vertex weights satisfy certain conditions, the stochastic dynamics of the ASEP in (1 + 1) dimensions maps onto the equilibrium statistical mechanics of the six-vertex model in two spatial dimensions (Gwa and Spohn, 1992 a, b; Kandel et al., 1990). To give the reader a flavour of this work, we follow Gwa and Spohn (1992 a, b) and rewrite the ASEP as a one-dimensional probabilistic cellular automaton which maps a spin configuration (sj(t)}, where sj = +1 and j = 1 , . . . , N , onto a new configuration {sj(t+dt)}, characterizing the system a short time interval dt after the initial time t. Letting sj = + 1 (-1) correspond to a particle (hole), we update pairs of spins to respect the conservation law: two neighbouring spins sj(t), Sj+l(t) exchange their positions with probability p - 1[1 + Bsj(t)] dt, or remain stationary with probability 1 - p. Given the dynamical rule, it is straightforward to obtain the master equation for the time evolution of the probability distribution P(t, {sj}). Next, we denote each spin configuration {sj} by a basis vector Is), so that P(t, {sj }) translates into a state vector IP) = ~-~s P(t, s) Is), and introduce the usual Pauli matrices ~rj - (a~, ~ry, crf) at site j, satisfying [crf, or{]- 2i6j,crf plus cyclic permutations. Then, the generator H of the master equation, defined as the (normalized) transition probability (s I exp ( - t H ) Is') in time t, is given by the non-symmetric real matrix 1

H -

m_

N

4 ~

[~ ~

9

--

x

y

1 +tB(crj cr)+1 - crfcr~+1)]

(6.27)

j=l

which reduces to the ferromagnetic Heisenberg chain if B = 0. All eigenvalues, which come in complex conjugate pairs since H is real, lie in the positive half-plane. The ground state has eigenvalue zero and is (N + 1)-fold degenerate, in the sense that each subspace is associated with

156


a fixed number, say M, with 0 < M 1/2, respectively; by particle-hole symmetry, an analogous statement holds for the low-density phase. Both high-density phases possess the same boundary and bulk densities and carry equal current ~(1 - ~ ) . To distinguish between them, we need to consider how the bulk density 1 - / 3 is approached, coming from the boundary layer. Asymptotically, the slope of the profile can be written in the form tN(p) ~ pZ exp (-ply), where ( is an appropriate length scale and z is called the decay exponent. Deep within high-density phase AI, ~-1 _ ~-1 _ ~-1 and


161

z = 0, so that the profile approaches the bulk density exponentially, on a length scale ~-1. At the critical a = 1/2, ~ diverges and z changes to 1/2. On the other hand, in high-density phase All, ~ = ~ and z = 3/2. Similar results hold, for the low-density phase, if we exchange the roles of a and /3 and replace p by N - p . Finally, in the maximal current phase, tN(P) ~PZ(1-P/N) z with z - 3 / 2 so that the density decays as p-l~2 towards its bulk value 1/2. Along the co-existence curve a = 3 < 1/2 between low- and high-density phases I, both ~ = ~ are finite, but ~-1 = ~ - 1 _ (~-1 vanishes. The density profile is linear, with positive slope (1 - 2a)/N. The phase diagram is shown in Fig. 6.12. Given that the steady state is known, in principle all higher equal-time correlation functions can be computed exactly. Derrida and Evans (1993) calculated the two-point function explicitly, and conjectured the form of three- and four-point correlation functions based on the structure of the one-point (i.e. the density profile) and two-point functions, in excellent agreement with numerical checks. Induced by the boundary conditions, long range correlations persist in the bulk. In particular, the two-point function of the lattice gas model provides information about the fluctuations in the total number of particles, M - y'~U=1 ni, since (M 2) -- 1N + 2~i>j(ninj): the importance of correlations is reflected in the variance of the total mass, (M 2) - (M) 2, which, for N ~ co, approaches the value N/8, in contrast to the value N/4 which would result if correlations were ignored.

BII

1/2

ai A11

I/2

oc

Fig. 6.12 Exact phase diagram of the ASEP with open boundary conditions in the c~-3 plane. From Schfitzand Domany (1993).

162


For a wider perspective, we recall the mapping from the lattice gas onto the single step surface growth model. Now, the parameters c~ and fl describe inhomogeneities in the deposition process at the edges of the sample (Wolf and Tang, 1990; Cook and Wolf, 1991; Kandel and Mukamel, 1992; Janowsky and Lebowitz, 1992; Derrida and Evans, 1993). The surface height h(p) at distance p from the left edge is related to the occupation number n i of the lattice gas according to h(p) ~-~/P--1[1 - 2ni]. In the case of a = fl = 1, the boundaries are, on average, at the same height, while the bulk is lower. The results for the density profile translate into (h(p)) cr ~ at the boundary, in agreement with Cook and Wolf (1991; see also Wolf and Tang, 1990), and ( h ( N x ) ) cx [Nx(1 - x ) ] 1/2 in the bulk. Thus, in contrast to a simple random walk, the surface exhibits curvature in the bulk region. This is borne out further by the behaviour of the height-height correlations, ([h(Nx) - h(Ny)] 2) - ( [h(Nx) - h(Ny)] )2" on distances that are well separated microscopically but small on the scale of the interface (i.e. for small x - y), the surface still behaves like a random walk; however, over larger distances the weak lattice gas correlations sum up, leading to a smoothing of the surface. A question of great theoretical interest concerns the microscopic nature of shocks (e.g. see Wick, 1985; Ferrari, 1986, 1992; Andjel et al., 1988; Lebowitz et al., 1988; DeMasi et al., 1989; Boldrighini, 1989; Dittrich, 1990; G~irtner and Presutti, 1990; van Beijeren, 1991; Ferrari et al., 1991; Cheng et al., 1991 b; Ferrari, 1992; Janowsky and Lebowitz, 1992, 1994; Derrida et al., 1993 d, e). It is known that solutions to the inviscid Burgers equation can develop singularities after finite times, even if the initial condition is smooth. Given that the particle density in the ASEP converges to a solution of the Burgers equation, in an appropriate scaling limit of vanishing lattice spacing and jump time (Rost, 1981; Liggett, 1985; Benassi and Fouque, 1987; Andjel and Vares, 1987; Lebowitz et al., 1988; Spohn, 1991; DeMasi and Presutti, 1991), one is naturally led to explore the microscopic equivalent of the hydrodynamic shock. The key question concerns the sharpness of the front: is it an artefact of the hydrodynamic scaling limit, in the sense that one would observe a smooth density profile on some intermediate length scale, short compared to typical hydrodynamic lengths, but still large compared to the lattice spacing? Or does the front retain its sharpness essentially all the way to the microscopic level? One of the basic problems here is a precise definition of the microscopic location of the front, in order to be able to distinguish between its instantaneous microscopic width and the fluctuations of its position. One possibility consists in fixing the (initial) particle density to the left of the front to be zero, so that one can follow the left-most particle. Alternatively, one can track the shock position by introducing a "second class" particle (Andjel et al., 1988) into the system. -


163

A second class particle behaves like a regular ("first class") particle in exchanges with holes and like a hole in exchanges with first class particles, namely, it gives way whenever a first class particle attempts to jump on it. Thus, a second class particle will predominantly move to the right if it finds itself in a region of low density, and to the left in high-density regions, so that its location can be identified with the shock position. Based on considerable analytic work (Wick, 1985; Ferrari, 1986; DeMasi et al., 1989; Ferrari et al., 1991; Ferrari, 1992; Derrida et al., 1993d, e) and numerical work (Boldrighini et al., 1989; Cheng et al., 1991b; Janowsky and Lebowitz, 1992), it has been established that shocks are microscopically sharp and stable, extending only over a few lattice spacings. The microscopic shock location performs a random walk around its average position if randomness in the initial configurations is allowed (Wick, 1985; DeMasi et al., 1989). If, on the other hand, the initial condition remains fixed, the shock fluctuations are reduced to t l/3-behaviour (Gfirtner and Presutti, 1990; van Beijeren, 1991). These results appear to be generic, in the sense that they also apply to a variety of other one-dimensional stochastic lattice models with serial or parallel dynamics, even if ferromagnetic or antiferromagnetic interactions are included (Alexander et al., 1993). Returning to the ASEP, the microscopic shock profile can in fact be determined exactly, following from the exact solution for the steady state of a system of first and second class particles on a ring (Derrida et al., 1993 d, e), with given densities Pl and P2. Since this steady state does not allow for inhomogeneities in the density distribution of first class particles, one might not expect to "see" a shock. However, if such a uniform system is viewed from a specific second class particle, then all second class particles to its right behave like first class particles, while all second class particles to its left behave like holes. Thus, the "tagged" second class particle sees exactly the same shock profile as a single second class particle in a system in which the density approaches the limiting values p+ - Pl + P2 and p_ - Pl, at + ~ , respectively (Ferrari et al., 1991). The exact shock profile approaches p+ exponentially, implying that the shock is sharp, over a characteristic length that diverges as (p+ - p_)-2 as p+ ~ p_. To study the scaling of shock fronts in finite systems, Janowsky and Lebowitz (1992, 1994) introduced a blockage into the ASEP on a ring of N sites, by reducing the transmission rate across a given bond from 1 to r, with 0 < r < 1. For a fixed overall density p and r < ( 1 - 1 2 p - 1 1 ) / (1 + [ 2 p - 11) the system phase segregates into two regions, separated by a sharply defined shock front. The asymptotic densities on either side, Pl and P 2 - - 1 - Pl, depend only upon r and the asymmetry parameter B. The fluctuations of the shock position are then found to scale as N 1/2 in the generic case and as N 1/3 if p - 1/2. Some intuitive arguments indicate that the former behaviour is caused by the random motion of particles and holes

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across the blockage, while the latter is due to the dynamical randomness of motion in the bulk. In this sense, these exponents reflect the t ~/2 vs. t 1/3 behaviour of shocks in the infinite system. Simulational results concerning shock fluctuations in a two-dimensional system, where a whole column is blocked, are discussed by Alexander et al. (1992). We note, in conclusion of this discussion, that systems with blockages exhibit phase transitions controlled by the transmission rate r, rather similar to those found in systems with open boundary conditions. Extrapolating exact results for a finite system, Janowsky and Lebowitz (1994) estimated the maximal current j ~ ( r ) carried by an infinite system with a blockage of strength r. j ~ ( r ) is an increasing function of r, approaching the limiting value 1/4 only as r ~ 1 (where j ~ appears to exhibit an essential singularity). Given that the density p~ of a homogeneous state is related to j ~ via j ~ = p~ (1 - p~), it is clear that, for each r < 1, there is a range of overall densities (symmetric around 1/2) where the system cannot exist in a homogeneous state. Instead, it must phase segregate into two regions of different densities, separated by an interface, which is, of course, our familiar shock front. A qualitatively similar phase diagram was found by Schfitz (1993 a), who obtained the exact solution of a deterministic version of the ASEP on a ring, with a single blockage. In this model, particle motion in the bulk is fully deterministic (using a two-step parallel dynamics which updates even and odd lattice sites separately), while hops across the blockage take place with probability r. Again, three phases emerge: a low- and a highdensity phase, separated by a co-existence region which is also symmetric around half-density. The shape of the regions for the deterministic and the stochastic models differ, of course. Finally, we note that surface growth models with localized defects exhibit similar phase transitions (Kandel and Mukamel, 1992). Two-species models In the following, we review two generalizations of the ASEP, corresponding to a one-dimensional version of the two-species model encountered in Section 6.4: two types of particles, carrying opposite charge, jump to nearestneighbour holes on a one-dimensional periodic chain, driven by an external "electric" field E acting along the chain. If no charge exchange is permitted, the excluded volume constraint constitutes a severe dynamical constraint in one spatial dimension: the motion of holes can change only the relative spacing, but not the sequence of positive and negative charges along the chain. The latter is an invariant of the dynamics, and parametrically enters the steady state distribution of holes which can be obtained exactly. Thus, d = 1 is the lower critical dimension for the transition between homogeneous and inhomogeneous phases (cf. Section 6.4). On the other hand, if charges are


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allowed to exchange positions, the background charge distribution evolves as well, coupling dynamically to the hole distribution. Both versions lead to intriguing conclusions, which we will discuss in the following. Beginning with the case of frozen background, the one-dimensional model is closely related to the gel electrophoresis problem (Section 7.4). If we assume that the head of a diffusing polymer creates a "tube" in the gel, acting as a one-dimensional, rigid channel for the motion of segments ("reptons") of stored polymer length, then the hopping dynamics of reptons within the tube is precisely that of our holes in the invariant charge background. In fact, the exact steady state distribution of reptons (i.e. of holes) with periodic boundary conditions was first obtained in that context (van Leeuwen and Kooiman, 1992; Kooiman and van Leeuwen, 1993). A key feature is a factorization of the multiple-hole distribution in terms of the single-hole distribution (Derrida, 1983). In the following, we summarize the mathematical results, deferring the discussion of their consequences for the electrophoresis problem to Section 7.4. To specify a configuration of particles and holes on a ring of N sites, we first ignore the location of the holes and let q - { q l , q 2 , . . . , q m } , qi - + 1 , qm+a =- ql, denote the (invariant) sequence of M positive and negative charges (Fig. 6.13). The overall charge ~ - )--~/M=1 qi, need not be zero, so that the mass current here, in contrast to Section 6.4, does not vanish. We characterize the number of holes between charges qi-1 and qi in terms of an occupation number, n i = 0, 1 , 2 , . . . , to be referred to as the occupation of the ith "cell". Of course, the total number of holes is constrained through M M + ~_, i-1 ni -- N . The field points in the direction of increasing site index i ("clockwise"). Defining the bias B = exp (E/2), a particularly convenient choice here are vBS rates (cf. Section 6.7.1), so that B - q i ( B qi) gives the

Fig. 6.13 A configuration of four positive (O) and four negative (O) charges on a ring of 16 sites. Counting clockwise from the top, ~ - ( - 1, + 1, - 1, - 1, + 1, + 1, - 1, + 1) and i f - (1,0,0,2,0, 3,0,2).

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rate with which a hole to the left (right) of a charge qi jumps to the fight (left). It is now straightforward to write down the master equation for the probability distribution P(t; ~, ~) of finding a hole distribution ~ at time t, given an invariant charge background ~. Specializing immediately to the stationary case, the steady state distribution P* (~, g) satisfies M 0 -- E { O ( n i + l ) B q ' P * ( q ' n + ) + O ( n i ) B - q i P * ( q ' n - ) ] i=l -- O(ni)[B -qi + B q i - 1 ] e * ( ~ , ~ ) }

(6.32)

where we have defined g+ - (nl,... ,ni q: 1, ni+l 4- 1,... ,riM), and O(n) is the usual step function, ensuring the presence of at least one hole before a transition can take place. The key to the solution of the multi-hole problem (6.32) rests in the recognition (van Leeuwen and Kooiman, 1992; Kooiman and van Leeuwen, 1993; Kooiman, 1993) that P*(~,g) factorizes, i.e. the ansatz M

?* (~, h') = [~M,N(~)] -1 H PT'

(6.33)

i=1

solves (6.32), provided that the Pi satisfy the equation 0 -- Pi+l g q i "[- Pi_l n-qi-1 - p i [ n -qi - n qi-l ]

(6.34)

and M

ripT'

{n} i=l

is an amplitude (the "canonical partition function") which ensures the normalization of P*. Now, (6.34) is easily recognized as the master equation governing the steady state of a much simpler problem, namely, of having only a single hole on the whole chain, located with probability Pi between charges qi-1 and qi. It is a simple recursion relation, with periodic boundary condition Pl - PM+I, which was solved by Derrida (1983). Accordingly, the steady state associated with (6.34) is given by Pi = CBqi

[

1+

]

B (qi+~'+qi+~'-~) v=l #=1

(6.35)

where C is a constant which can be absorbed into the normalization of P*. Thus, the calculation of the full steady state, P*(~,g), is now complete. Given that the total charge of the system is nonzero, it is instructive to


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compute the average net current of holes in the steady state,

(Jh(i) ) =--(O(ni)B -q~ - O(ni+l)B q~ )

(6.36)

from cell i to cell i + 1, being, by construction, the negative of the average mass current. The brackets (.) denote the average over all hole configurations {if} in a fixed background ~. With the "canonical partition function" Y'M,N(q), it is straightforward to show that

(O(ni)) ---Pi~M,N_I(q)/~.~M,N(4) ~piz(q).

(6.37)

The last equality here defines the activity z(4) of the system. Alternatively, the relation between z(4) and the canonical ~ M , N can be derived by going over to a grand-canonical formulation, lifting the constraint on the total number of holes. With the help of (6.35), the multiple-hole current (Jh (i)) is easily recognized as the single-hole current, multiplied by a (positive) factor of z(4) (Kooiman, 1993):

(Jh(i)) = z(~){B -'~ - B'~},

(6.38)

so that, for positive total charge .~ > 0, the mass current (Jm) = - ( J h ) flows along the field direction, and against it if .~ < 0. We note, in closing, that a given charge configuration 4 translates into a specific spatial configuration of the confining tube in the reptation problem. Thus, the preceding discussion provides us with an exact solution for the dynamics of a single polymer. Clearly, for quantities of interest in the reptation problem, an average over many tube configurations must be taken, so that we are faced with the serious problem of determining the associated distribution function. We will return to this question in Section 7.4. We now turn to the less restricted situation in which the charge background is no longer static, by virtue of particle-particle exchanges. In the simplest model, the rates are fully asymmetric, with positive (negative) particles jumping to the right (left) only. In addition to the usual particle-hole hopping, occurring with probability 1, positive and negative charges may exchange, with nonzero rate 7. We note, first, that this dynamics is equivalent to that of first and second class particles and holes, provided we identify positive charges with first class particles, holes with second class particles and negative charges with holes. On a ring of N sites, the associated steady state can be found exactly (Derrida et al., 1993 d, e) and has been exploited to investigate the microscopic nature of shocks in the Burgers equation (see Section 6.7.2). Our focus here rests on the case of open boundary conditions (Evans et al., 1995 a, b), i.e. positive particles enter the system at the left edge with rate a and leave it at the right with rate/3. Similarly, negative charges enter at the right and leave at the left, with rates a and/3, respectively,

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ensuring that the dynamics is invariant under the combined operation of exchanging all positive and negative charges and spatial reflection. We will refer to this symmetry as CR, noting, however, that this definition differs slightly from the earlier one (Section 2). To appreciate the key recognition here, namely the occurrence of spontaneous breaking of the CR symmetry, for a certain range of the external control parameters a,/3 and 7 (Evans et al., 1995 a, b), the following properties of the dynamics should be emphasized: (i) configurations may be characterized in terms of a local occupation variable which takes only a finite number of values, namely 0 or +1, representing the presence of a hole, positive or negative charge; (ii) the excluded volume constraint acts as an effective, short-ranged inter-particle repulsion; and (iii) the noise is small yet unbounded. It is well known that these three conditions preclude spontaneous symmetry breaking (SSB) in one-dimensional equilibrium systems. Thus, this simple model demonstrates clearly that far-from-equilibrium systems are less restrictive. In the remainder of this section, we briefly discuss the nature of the symmetry breaking, following Evans et al. (1995 a, b). Unfortunately, exact steady state solutions are available only for/3 = 1 and as a ~ c~, where no symmetry breaking occurs. However, a mean-field analysis supported by Monte Carlo data shows that, for sufficiently small /3, two phases exist in which the currents of positive and negative charges differ in the thermodynamic limit. For simplicity, we discuss the phase diagram for "~ = 1. In this case, the dynamics within the bulk reduces to that of two decoupled, fully asymmetric single-species exclusion processes, since a positive (negative) particle cannot distinguish between holes and negative (positive) charges. At the edges, however, the two processes are coupled through the boundary conditions, giving rise to effective feeding rates for the single-species dynamics. Invoking the known phase diagram for single-species biased diffusion with open boundary conditions (Derrida et al., 1992; Schtitz and Domany, 1993; see Section 6.7.2 for a discussion), the phase diagram for the two-species problem follows quite easily. Two symmetric and two non-symmetric phases are found, occupying, respectively, the large and the small /3 regions of the phase diagram. The two non-symmetric phases differ in the density profiles for the positive and negative charges: one is characterized by two (distinct) low-density profiles, while the other exhibits one low- and one-high-density profile. Also, in each of these two phases, the currents of the positive and negative charges differ from one another. Moreover, in either phase, the system can exist in two states, related to each other by CR. Clearly, a finite system of size N will flip between these two states in finite time, thus effectively restoring CR symmetry. To stabilize a state of broken


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symmetry in a conserved system, this characteristic time, 7(N), must diverge faster than N 2, the latter being the typical relaxation time of diffusive processes. A simple intuitive argument allows us to estimate 7(N). Consider, for instance, the high/low density phase, and assume that the system is predominantly filled with positive particles, associated with a small current of negative particles and holes flowing from the right to the left boundary. A droplet of the CR related state - a region of predominantly negative charges - may form near the left edge, as a result of some fluctuation. Holes can now enter the system from both boundaries, and will get trapped between the negative and positive regions. In most cases, the negative droplet will get expelled after a short time; however, if it were to survive for a time of O(N), the system would fill with holes and could thus switch to the negative phase. The likelihood of such an event can be estimated by considering the right boundary of the negative droplet which performs a biased random walk towards the left edge of the system. The probability of the droplet boundary to remain in the system after a time of O(N), if its original distance from the edge was O(1), is exponentially small in N. Thus, the characteristic life time ~-(N) of a broken symmetry state grows as exp (bN), with some constant b, stabilizing SSB in the thermodynamic limit. This picture is essentially confirmed by Monte Carlo simulations. Measuring the time dependence of the current asymmetry, expected to decay as exp [-t/T(N)] in a system of size N starting from an initial, single-phase configuration, one finds that ~-(N) grows slightly slower than exponentially; it appears, however, that the discrepancy may be due to finite-size effects, affecting the system sizes (N < 160) that were studied.

The Toom interface, a model with long range jumps In this section, we consider another system closely related to the d = 1 standard model, namely, an interface in the low noise limit of an NEC model. Known also as the "Toom model", it was introduced by Vasilyev et al. (1969), and studied extensively by Toom (1974, 1976, 1980). Consider, first, a cellular automaton based on the square Ising model, in which a configuration evolves by each spin taking on the value given by the "majority rule". Specifically, the update of a spin on a site is +1, depending only on the sign of the sum of that spin and its "north" and "east" neighbours. Dubbed the NEC model (C for centre) and generalized to include noise and bias, this non-equilibrium system is shown to have interesting bulk properties (Bennett and Grinstein, 1985; Lebowitz et al., 1990; Section 7.2). In particular, an interface running roughly from NW to SE, separating two domains of opposite spins, will drift indefinitely in the SW direction. To induce a stationary interface, Derrida et al. (1991 a, b) considered a lattice in the third quadrant, with spins at the negative x- and y-axes fixed at opposite

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+

Fig. 6.14 A typical Toom interface with low noise and bias. values. Anchored at the origin, the mean interface is oriented at some angle between 180 ~ and 270 ~ In the absence of noise, it would be exactly a staircase at 225 ~. For low noise, the configurations of this semi-infinite interface are still well approximated by the staircase, with a distribution of vertical and horizontal steps. The average number of steps will be unequal (Fig. 6.14), if there is a bias in the noise, leading to average angles other than 225 ~ Around this average, the interface fluctuates, so that a statistical width, w, as a function of L, the distance from the origin, can be defined. A primary interest is the exponent v in the scaling regime, where w c~ L ~. The connection with a driven lattice gas lies in mapping a sequence of vertical and horizontal steps to one with particles and holes. In this language, the Toom dynamics is well approximated by long range jumps, whereby a particle exchanges with the first available hole to its fight/left with rates )~+/)~_. Derrida et al. (1991 a, b) argued that the continuum version of this evolution should be governed by the KPZ equation, or one with higher non-linearities (Devillard and Spohn, 1992) if)~+ = )~_, with the results v = 1/3 and 1/4, respectively. Though these conclusions agree with the collective variables approximation, they are only in qualitative agreement with the data from simulations, where 0.285 and 0.265 are found, respectively, for the two cases: )~+ = )~_/4 and )~+ -- )~_. It is unclear whether better statistics will bring the numerical results closer to the theoretical values, or whether a non-local generalization of the KPZ equation is needed.

7

Related non-equilibrium steady state systems

While the original driven diffusive lattice gas and its variations provide simple models for studying singular collective behaviour in non-equilibrium steady


171

states, they are small windows into the much larger world of physical systems far from equilibrium. It is clearly beyond the scope of this review to catalogue all non-equilibrium phenomena. Instead, we briefly discuss those which are closely related to the driven lattice gas. Even restricted to these systems, we will not be able to provide a complete list. Rather, this section should be regarded as a collection of "selected topics", arranged in a somewhat arbitrary order. Some models will be purely "mathematical", while others will be closely related to physical systems with real experimental data.

7.1

Models with competing conserved and non-conserved dynamics

In the previous sections, only dynamics of the particle conserving type have been considered, since the particles supposedly model ions in a physical lattice. In this sense, these systems resemble the canonical ensemble in equilibrium statistical mechanics. It is natural to inquire if a system similar to the grand canonical ensemble exists, and if that system is equivalent to the standard model. A further advantage, if the constraint of half-filling is lifted, is that an ordered state below criticality would be homogeneous so that the complications of interfaces can be avoided. There have been many attempts to introduce some form of particle reservoir. Only one conclusion is clear: unlike systems in equilibrium, there is no equivalence between the ensembles. Indeed, there can be little resemblance between the original model and typical generalizations which include some form of particle non-conserving, or "spin-flip", dynamics (Glauber, 1963). For example, the steady state of a system with particle sources and sinks at the opposite boundaries .(Section 6.2.2) is quite different from that of the standard model, especially for T < Tc. Similarly, though the overall particle number is fixed in the two-layer model (Section 6.3), it is not conserved for each layer individually, leading to a surprising second phase transition. In this section, following this line of inquiry, we first review a cousin of the standard model which includes a global spin-flip dynamics at the same T. Then we list a number of related models with no external drive, in which non-equilibrium states are established through a competition between Glauber dynamics at one temperature and particle conserving dynamics (Kawasaki, 1972) at another. Wang et al. (1989) modified the dynamics of the driven lattice gas to include, with probability p, Glauber rates which are controlled by the inter-particle interactions and the same thermal bath. Monte Carlo simulations were carried out for a range of p. Though this study is not sufficiently extensive for a definitive conclusion, there is good evidence for the critical properties to "revert" to being Ising-like, for all p > 0. Such a conclusion may

172


appear trivial since, from power counting, Glauber dynamics is more relevant than Kawasaki's. However, power counting alone is not sufficient to guarantee the stability of the Wilson-Fisher fixed point. Indeed, with Glauber dynamics, there is no strong anisotropy, so that ~ 2 , the operator associated with r is of the same naive dimension as ~ 3 , which controls the Ising class. As a result, non-trivial mixing of these two couplings occurs, and an explicit computation is necessary to find the most stable fixed point in this subspace. Bassler and Schmittmann (1994 b) found, at O(e), that the WilsonFisher fixed point remains the most stable. To access the fixed point on the axis, we must "tune" the microscopic parameters until ~ 3 is absent. An example of such "tuning" exists in equilibrium systems, i.e. when accessing the tricritical point. The fixed point would be associated with ~ 5 , if the dynamics is of the Glauber type. Since ~ 2 is more relevant, it may be the fixed point controlling critical phenomena of a driven, tricritical system. Clearly, it would be interesting to find realizations of this model. For further details on universality classes of systems with non-Hamiltonian Glauber dynamics, see the next section. In the same vein, but further afield from the standard models, there are many with other forms of competing dynamics. Several authors (DeMasi et al., 1985, 1986; Dickman, 1987; Gonzalez-Miranda et al., 1987; Wang and Lebowitz, 1988; Droz et al., 1989; Tom6 and de Oliveira, 1989; Marques, 1990; Mendes and Lage, 1993) considered systems with a combination of rates at different T. An example is employing Glauber rates at finite temperature and isotropic nearest-neighbour spin exchanges at infinite T, with probability p and 1 - p , respectively. The results are: first/second order transitions for large/small p, with critical properties belonging to the Ising class. An intriguing generalization of this model (Droz et al., 1990, 1991; Xu et al., 1993) uses a Kawasaki dynamics with an infinite range. Though it is coupled to a T = c~ bath, this exchange generates an effective long range interaction, so that the system can be well understood in terms of mean field theory. Thus, a phase transition into an ordered state occurs even in d = 1, while critical singularities are classical. Systems with competing dynamics are well suited to model chemical processes which include the effects of both diffusion and reaction, from simple pair annihilation of a single species to complex catalytic processes involving many species (Schl6gl, 1972; Janssen, 1981; Grassberger, 1982; Toussain and Wilczek, 1983; Kang and Redner, 1984; Rficz, 1985; Ziff et al., 1986; Meakin and Scalapino, 1987; Grinstein et al., 1989; Browne et al., 1990; Bramson and Lebowitz, 1991; Evans, 1991; Dickman and Tom6, 1991). With discoveries of new phenomena in a variety of models, as well as connections to cellular automata (von Neumann, 1966; Wolfram, 1983), epidemics (T. E. Harris, 1974), directed percolation (Blease, 1977; Cardy


173

and Sugar, 1980; Janssen, 1985; Kinzel, 1983), multilayer adsorption (Bartelt and Privman, 1991; Evans, 1993) and Reggeon field theory (Gribov, 1967; Brower et al., 1978; Liggett, 1985), this rapidly expanding field deserves its own comprehensive review. Providing even a biography being beyond the scope here, we list some recent articles in which further references may be found: Chopard et al. (1993), Dickman (1993), Droz and Sasv/tri (1993) and Jensen and Dickman (1993).

7.2

Multi-temperature models with Glauber dynamics

The foundation underlying the dramatic effects in all driven diffusive systems is the combination of non-equilibrium and particle conserving dynamics. In the last section, we saw that introducing any form of Glauber dynamics causes the critical behaviour to revert to the Ising class. Here, we consider models with only spin-flip dynamics, but coupled to more than one thermal bath, so that non-equilibrium steady states may be established. Examples of "multi-temperature" systems abound in daily experience, e.g. a water tank with an immersion heater. However, systems subjected to such macroscopic temperature gradients are very complex, since translational invariance is absent and the thermodynamic limit is singular. Modelling such systems for simulations is also non-trivial, since variables other than Ising spins (Creutz, 1983, 1986) are needed to describe the macroscopic temperature current. In keeping with the spirit of studying the simplest systems, we only consider models which are invariant under finite translations, i.e. models in which the different baths are coupled "locally". The simplest model, where the rate of flipping each spin is coupled identically to these baths (Hill, 1982; Garrido et al., 1987), can be mapped into an equilibrium system with a single effective T. Truly non-equilibrium cases involve coupling different baths to, for example, the sites in distinct but equivalent sub-lattices. As expected, interesting phenomena such as genetic singularities are absent, though some two-point correlations in an exactly solvable d - 1 model display unusual behaviour (R/tcz and Zia, 1994). A more serious consequence is that there are no phase transitions in some d > 1 systems, bringing in question the existence of a unique lower critical dimension (Kantor and Fisher, 1989). In case a second order phase transition is present, Grinstein et al. (1985) argued that the critical properties belong to the Ising universality class. They considered the naive dimensions of operators responsible for the irreversible dynamics, and found that all are higher than that of q54.Thus, these operators are irrelevant, in an expansion around d - 4, for the Wilson-Fisher (1972) fixed point Hamiltonian. This prediction is understandable intuitively, especially if the model is isotropic as well as

174


periodic. In the language of real space renormalization, critical singularities are extracted from focusing on larger and larger "blocks". Meanwhile, the different temperature baths are coupled at the "local" level, so that the effects of non-equilibrium dynamics should be less and less significant. For anisotropic models, however, this intuitive argument clearly fails, and we must rely on more quantitative tools. On the simulations front, there are recent Monte Carlo studies of Ising models, coupled in a variety of ways to two temperature baths (B1/Ste et al., 1990, 1991; Garrido and Marro, 1992). In all cases, critical exponents are found to be identical to Onsager's, while the fourth-order cumulant (Binder, 1981) takes the same value as in equilibrium. Though no one has investigated models coupled to more than two temperature baths, no new surprises are expected. It would be a mistake to dismiss all such models, devoid of interesting behaviour except for sub-leading singularities associated with nonequilibrium dynamics. Indeed, a particular version of the two-temperature model (Heringa et al., 1992), which is a generalization of the "Toom" cellular automaton (Vasilyev et al., 1969; Toom, 1974, 1976, 1980), displays the unusual phenomenon of bistability. Unlike an equilibrium Ising model in a magnetic (H) field, where every point in H - T space, except for the co-existence line, is associated with a unique stable state, these nonequilibrium models support two stable states in a finite region of the phase diagram. Bennett and Grinstein (1985) first showed that the irreversible dynamics in a probabilistic "NEC" Toom model destroys the clusters which would otherwise nucleate the decay of a metastable state. As a result, some states with the "wrong" magnetization become absolutely stable as well. This conclusion is supported by both a mesoscopic approach based on the Langevin equation (He et al., 1990) and a rigorous proof (Lebowitz et al., 1990) using the mapping (Domany and Kinzel, 1984) of a d-dimensional probabilistic cellular automaton to an equilibrium statistical model in d + 1 dimensions. We should point out that metastable or unstable states in equilibrium can be easily stabilized into non-equilibrium steady states. A simple example in daily experience is a homogeneous water-oil mixture, which is not stable in equilibrium, being stabilized by constant stirring. Returning to critical phenomena, we note that the argument due to Grinstein et al. (1985), though based on a theory near d - 4, may still be valid for Ising models in d = 2. However, this approach clearly fails for the three-state Potts model, which displays afirst order transition in d >_ 3 and a second order transition in two dimensions (Baxter, 1973; Wu, 1982). Since ~bis dimensionless in d - 2, it is difficult to "guess" an appropriate fixed point Hamiltonian, let alone compute critical exponents associated with nonHamiltonian dynamics. We should point out that, though fixed points


175

have been found within the framework of real space renormalization group methods (Burkhardt et al., 1980; Nienhuis et al., 1980), performing systematic analyses is, unlike in field theoretic approaches, not possible. Thus, definitive conclusions about stability properties of the fixed points are lacking, since the effects of any perturbation must be checked individually and numerically. In this sense, it is interesting to ask if irreversible non-conserved dynamics is also irrelevant for the Potts critical point. Recent Monte Carlo simulation studies of a two-temperature model (Bassler and Zia, 1994) showed that, when the second order phase transition is still present, all data satisfy finite-size scaling with the exactly known equilibrium exponents. Given that the critical properties of these non-equilibrium Ising and threestate Potts models fall into their respective equilibrium universality classes, it is tempting to conjecture that this is the case for all models with second order phase transitions. A particularly intriguing testing ground for such a conjecture is the q-state Potts model for q > 4 in d = 2. Though the pure system is known to undergo a first order transition, a recent Monte Carlo study found the q = 8 model with quenched disorder to display critical singularities (Chen et al., 1992). In a broader context, it is clearly important to investigate, in addition to those described in Section 6.6, such non-equilibrium systems with quenched impurities.

7.3

Models for driven interfaces

In Section 6.7.2, we reviewed the standard model and some of its variations in d = 1. The connection to the Kardar-Parisi-Zhang equation (1986) was briefly mentioned. Here, we provide some details as well as the background in the dynamics of interfaces in cluster growth. When a crystal grows in a supersaturated or a supercooled solution, a variety of shapes is observed, from compact gems to fractal snowflakes. To understand how these shapes depend upon the ambient conditions such as concentration and temperature, a number of growth mechanisms and models have been proposed. Notable examples are dendritic growth (Langer, 1987) and diffusion limited aggregation (Meakin, 1988). Here, we focus on flux- or reaction-limited growth models (Krug and Spohn, 1991), such as the singlestep model (Section 6.7.2). These seem to be most closely related to our driven lattice gas. In this regime, it is adequate to describe the growth processes by the time dependence of an interface which is almost planar and of vanishing thickness. Thus, we consider a single-valued function, h(s t), representing the height of the interface above a reference plane, where s denotes the coordinates of this d-dimensional plane. Note that, though many models are specified for discrete heights and/or discrete space-time (Eden, 1958;


176

Void, 1959, 1963; Visscher and Bolsterli, 1972; Frank, 1974; Gilmer, 1980; Jullien and Botet, 1985; Meakin et al., 1986; Plischke et al., 1987; Krug and Spohn, 1989), their large scale behaviour is expected to be universal, so that the mesoscopic description using continuous variables h, Y and t should suffice. To capture the essence of all such models, the mesoscopic dynamic equation for h requires three basic ingredients: relaxation to planarity, forcing term for the growth and random fluctuations. The first and the last items are necessary for describing the equilibrium interface, while the second represents the non-equilibrium aspects. Putting these together gives us the KPZ equation (Karder et al., 1986): t) -

,V2h + A (Vh) 2 + ~7(Y,t)

(7.1 a)

where r/(Y, t) is a Gaussian distributed noise satisfying (~7} = 0

and

(r/(Y, t)rl(Y', t')} = 2D6(Y- Y')6(t - t').

(7.1b)

A very simple way to arrive at these equations (Grossmann et al., 1991) is to start with the equation of motion for an interface in Ising-like systems with pure relaxational dynamics. The deterministic part, proposed by Allen and Cahn (1976), states that the normal velocity, Vn, of the interface at any point is proportional to ~, the mean curvature there. For a near-planar interface in an isotropic bulk, these quantities can be expressed in terms of h. Defining g - 1 + (~7h)2, they are ~th/x/~ and t~ = X~2h/v~, to lowest order in gradients of h. The noise correlation, when properly derived from bulk dynamics (Bausch et al., 1981; Kawasaki and Ohta, 1982), is given by (7.1 b) with an additional factor of x/g" Neglecting terms of order h 3 or higher, we obtain (7.1 a, b) with A = 0. Growth of one phase at the expense of the other can be achieved by imposing a small homogeneous magnetic field on the bulk, leading to an extra term in the Cahn-Allen equation: vn (3(/~ -~-/~. Subtracting the average uniform growth (At) from h, multiplying the resultant equation by v/~ and expanding up to second order in (h, rl), we arrive at (7.1 a, b). For a one-dimensional interface, V is a scalar, which we denote by ~. Similarly, ~h is a scalar field, which may be identified as 4~. Applying ~ to (7.1 a) and defining ( - r/, we arrive at ~t~ = //~2~ -I'- (A/2)~q 52 + ~ ( . To compare with the driven system, we only need to set d - 1 in (3_22), which eliminates all terms involving V and ~. Thus, we see that the d - 1 KPZ equation is precisely the equation for the driven "free" gas in d = 1. Of course, the mapping O h - . 4~ is just the continuum analogue of S i ~ h i - h i _ 1 (Section 6.7.2). Apart from the driven system, the d - 1 KPZ equation is also identical to V n

--


177

the one-dimensional Navier-Stokes equation, which was investigated in great detail by Burgers (1974). The mapping from the KPZ equation to Burgers leads to a further connection, based on a transformation (Hopf, 1950; Cole, 1951) of the non-linear Burgers equation into a linear one. Defining Z(f, t) - exp { - A h / 2 u } , (7.1 a) becomes

8 Z('r't)-- [uV2 - 2uu A r/(f, t) ]Z ( f ,

8t

t),

(7.2)

which is Schr6dinger-like, with a random potential. Using the Feynman path (Feynman and Hibbs, 1965) formulation of quantum mechanics, an interpretation for Z emerges. It is the partition function for a directed polymer (Kardar and Zhang, 1987) in a disordered medium. For d -- 1, we may regard the polymer as an interface, so that the anomalous scaling behaviour of interfaces in this type of growth processes can be translated into the anomalous roughening in the presence of quenched bulk disorder. For a review of the latter, see, for instance, Forgacs et al. (1991). Returning to the connection between the KPZ and Navier-Stokes equations, we find it more instructive to study the general case of d > 1, which exposes clearly the nature of the mappin$ Again, we apply ~7 to (7.1 a), leading to an equation for a vector field: ~7- ~7 h. Since this field is curl-flee, ~7(~'2 ) is just 2(~7. ~)~7, so that it becomes 8t ~7- uV2~7+ A(~'. V)~' + f ,

(7.3)

---+

where we have written f for the noise ~7 ~7. Changing the sign of ~" and absorbing a factor of )~, this can be recognized as the Navier-Stokes equation, in the presence of a random force f . Of course, in hydrodynamics, ~7. ~7- 0 is the more common constraint, describing an incompressible fluid. In this sense, both this and the KPZ problem are contained, in complementary ways, in the noisy Navier-Stokes equation (7.3). The d - 1 "transverse" and the single "longitudinal" components of ~' form independent fields, appropriate for the incompressible fluid and the KPZ interface, respectively. Modern renormalization group techniques were applied by Forster et al. (1977) to both parts. Predating the study of driven interfaces, they identified the "longitudinal" part as a d > 1 Burgers equation, with conclusions which can be readily translated into the KPZ case (Kardar et al., 1986; Medina et al., 1989). The result is that, for d > 2, small nonlinearities are irrelevant while for 0 < d < 3/2, a A > 0 fixed point exists, leading to non-trivial scale invariant behaviour. In particular, for d = 1, the exact results reported in Section 6.7.2 are reproduced. Recognizing that there is only a single scalar field in both the KPZ and driven system in any dimension, we may enquire if there are multicomponent

178


generalizations of either, so that the connection to a vector field theory like the Navier-Stokes equation could be more "substantial". In the KPZ case, the simplest generalization is to consider K interfaces and postulate some coupling between them:

~t hi - vijVEhj - 89)~ijk V hj . V hk + rh

with

( rh ~Tjl oc Dij,

(7.4)

where i , j , k - 1,..., K. Barabfisi (1992) studied the d - 1, K = 2 system, using simulations on a pair of coupled single-step models and dynamic renormalization group methods on (7.4) at the one-loop level. Another application concerns the fluctuations of a stretched line, embedded in K-dimensional space and driven in a transverse direction, e.g. a vortex line in a superconductor or a drifting polymer (Erta~ and Kardar, 1992, 1993). Similarly, it is easy to write equations for driven multicomponent systems:

~-'lt~)i -- T_t_ijV2r "~- Tllij

~2

Cj + eijk~(~)jCk)

-4- " " " ,

(7.5)

where we have omitted the noise, as well as terms of higher power in either derivatives or fields. An example is the two-species lattice gas (Section 6.4), which, in d - 1, may be applicable to gel electrophoresis. Both topics will be considered in the next section (7.4). Here, we remark that such generalizations would certainly include the Navier-Stokes equation as a special case, with d = K and constraints on the couplings associated with hydrodynamics. In other problems, the appropriate couplings would be dictated by the underlying symmetries. As an example, we could consider generalizing (2.15) for the Potts model. Then the most natural quadratic term would be 8~(QijkC~j4'k), where Qijk is the unique third rank Potts invariant (Golner, 1973; Mittag and Stephen, 1974; Zia and Wallace, 1975). However, investigations along such lines seem to be a mathematical pursuit, unless there are reasonable microscopic models associated with them. Before closing, we mention several other interesting extensions of the KPZ equation, modelling more complex dynamics in driven interfaces. If crystal growth occurs on a vicinal surface, several sources of anisotropy arise. Starting from a model due to Villain (1991), Wolf (1991) studied an anisotropic version of (7.1)

~t h - u_t_V2h + ul i~2h + 1A_t_(Vh)2 + 1All(Oh)2 + r/,

(7.6)

using dynamic renormalization group techniques. A remarkable conclusion is that, if the )~'s have opposite signs, they are irrelevant in d - 2, leading to logarithmic roughness. Otherwise, they are marginal and algebraic singularities characteristic of the isotropic case re-emerge. Equation (7.6) has also been applied to the study of Toom interfaces (Section 6.7.2) in two d - 3


179

models, illustrating both of these cases (Barab~tsi et al., 1992; Jeong et al., 1993). Of course, as in the d - 1 model, the possible role of non-local terms in generalizing (7.6) remains to be explored. In (7.1), neither the deterministic nor the noise part of the equation of motion is conserved. A "conserved" KPZ equation, in which f h is held fixed, was proposed (Sun et al., 1989; Chakrabarti, 1990; Rficz et al., 1991) to describe surface diffusion" ~h

at-

--V 2

uV2h -+- (Vh) 2 + r/,

(7.7)

where ~ is a conserved Gaussian distributed noise. Another generalization considers the effects of driving on the roughening transition (Hwa et al., 1991; Rost and Spohn, 1994 a, b), by combining (7.1 a, b) with the sine-Gordon term which is responsible for the equilibrium transition (Amit et al., 1980): A (7.8) ~t Note that, due to the loss of translational invariance in h, the O(1) term in the drive, represented by F here, can no longer be transformed away. As in the pure KPZ case, there is a version of this problem with a conserved field (Sun et al., 1992). Each of these generalizations leads to a rich variety of results, not only in theory and simulations but also in comparisons with experiments on physical systems. Though the connections to the standard model grow more tenuous, it may be worthwhile to keep these crosslinks in mind for future explorations. In these few paragraphs, we have focused on the relationship between our driven bulk system and the KPZ equation for d-dimensional interfaces. Due to space limitations, it is impossible to discuss the interesting physics of driven interfaces. For that, we refer the reader to several extensive reviews (e.g. Krug and Spohn, 1991; Meakin, 1993; Halpin-Healy and Zhang, 1995; Barabf.si and Stanley, 1995) and the many articles in the proceedings of a recent workshop (Jullien et al., 1992), where further references and connections to other growth models may be found. ~h _ u V 2 h _ Y sin (2~-h) + F + ~ (Vh) 2 + r/.

7.4

Gel electrophoresis and polymers in sedimentation

When driven by a constant force through a dissipative medium, a long polymer will diffuse with a steady drift velocity. Examples are polymer sedimentation under the influence of gravity and gel electrophoresis, where an electric field acts on a charged polymer (e.g. Jorgenson, 1987). In both cases, an important question is how the drift velocity depends upon the length, or weight, of the polymer. Typically, longer and heavier polymers

180


travel more slowly, allowing for the separation of polymers by size. Other quantities of interest include the swelling exponent ~,, which characterizes how the radius of gyration of the polymer scales with its length, or the dynamic exponent z, which governs the scaling of the largest relaxation time with wave vector. Clearly, both may need to be replaced by anisotropic indices if a drive is present. For a recent review, see, for example, Doi and Edwards (1986). In the following paragraphs, we will briefly describe some of the results in driven systems, emphasizing in particular the intriguing crosslinks between the dynamics of drifting polymers and multi-species models. Beginning with the most immediate of these connections, we recall the driven, non-interacting two-species model (Section 6.4) and its d - - 1 limit (Section 6.7.2), which is closely related to the microscopic modelling of the gel electrophoresis problem (Rubinstein, 1987; Duke, 1989; Lerman and Frisch, 1982; Lumpkin and Zimm, 1982; Lumpkin et al., 1985; Slater and Noolandi, 1985, 1986; Noolandi et al., 1987; Doi et al., 1988; Jonsson et al., 1988). As the polymer drifts through the gel, its head creates a channel. Once established, this channel is assumed to be static, forming a rigid tube that confines the motion of the remaining polymer (Edwards, 1967). Parts of the latter may be coiled up inside the tube, and these pieces of stored length, called "reptons" (de Gennes, 1971), diffuse along the chain's own contour (Rubinstein, 1987), with the field acting as a bias (Duke, 1989, 1990 a, b; Widom et al., 1991). Thus, there is an intricate interplay between the motion of the endpoints and the strictly one-dimensional diffusion of the reptons. In the simplest, discrete description (Rubinstein, 1987; Duke, 1989), the gel is modelled as a regular lattice, with spacing corresponding to the average pore size, and the polymer as a flexible chain, consisting of N segments of equal length embedded in the cells of the lattice (Fig. 7.1 a). A three-state variable is associated with each pair of neighbouring segments, taking the value 0 if both segments fall into the same lattice cell, or + 1 ( - 1) if they belong to different cells, separated by an up- (down-) field step (Fig. 7.1 b). Thus, a zero represents a repton. To prevent excessive stretching of the polymer, + l's may exchange with O's only, subject to the field bias. Given appropriate jump rates, the reptons in this problem obey the same dynamics as the holes in the driven two-species lattice gas (Shnidman, 1991), with the exception of the two polymer ends whose motion is expected to follow different rules (Rubinstein, 1987; Duke, 1989). On the other hand, if we read the mapping in reverse, the periodic boundary conditions of Section 6.4 and 6.7.2 restrict the motion of head and tail in such a way as to generate periodic tube configurations with a specified length. Though the exact steady state distribution for this problem is known, it is clearly unphysical to impose such a boundary condition, which decouples the endpoint motion from internal diffusion, on the polymer. At present, it is a matter of considerable controversy whether this drastic approximation


X

181

X X X ~. X X . ..........X x

X

X

X

X

(a) 0

4-

4-

40 0

+

+

+ O

+

(b) Fig. 7.1 (a) A typical polymer chain embedded in a model gel; each monomer being denoted by 9 (from Duke, 1989); (b) the (+l, 0) configuration corresponding to the steps between successive monomers. nevertheless leads to credible results for some quantities, e.g. the drift velocity. Taking periodic boundary conditions on faith, the drift velocity can be computed from the exact steady state distribution of Section 6.7.2, provided an additional average over all tube configurations {tT} is taken. Their statistical weights, generated by the motion of the polymer head through the gel matrix, are assumed to be equal in the low-field limit. The average drift velocity, related to the diffusion coefficient via an Einstein relation (van Leeuwen, 1991), then follows as v-

wE 2N , ( 2 d + 1)N (1 - ~ E +...)

(7.9)

where the coefficient w is the product of a lattice spacing and a typical time scale for hopping (van Leeuwen and Kooiman, 1992; Kooiman and van Leeuwen, 1993). This explicitly confirms the scaling form, conjectured by Widom et al. (1991), for the E ~ 0, N ~ c~ limit at fixed EZN. We briefly mention some related work. (a) If one chooses a statistical weight for the {4} that incorporates orientational bias, v increases, through a change in the amplitudes of (7.9). (b) To account for the finite volume

182


available within the tube, the number of reptons per unit polymer length can be limited to zero or one, resulting in a "fermionic" version of the model (van Leeuwen and Kooiman, 1992; Kooiman and van Leeuwen, 1993). (c) For very large fields, loops of stored length may break out of the tube, causing "hernia" (Deutsch, 1987, 1988; Duke and Viovy, 1992) and invalidating the simple repton model. Under the mapping, this process corresponds to pair creation of charges in the two-species system. While it is difficult to realize pair creation in ionic conductors, such a process occurs quite naturally for charges on water droplets in an emulsion (section 6.4.2). (d) The length separation of polymers becomes more efficient if field pulses of unequal duration are applied periodically in opposite directions (Schwartz and Cantor, 1984; Carle et al., 1986; Deutsch, 1987, 1989; Noolandi et al., 1987, 1989; Lalande et al., 1987; Viovy, 1988a, b; Deutsch and Madden, 1989; Duke, 1989, 1990 b; Schwartz and Koval, 1989; Smith et al., 1989; Lim et al., 1990; Madden and Deutsch, 199 l; Zimm, 1991). From our perspective, it is inviting to pursue possible applications of the randomly driven models (section 6.1) to such systems. Next, we turn to the dynamics of driven polymers, drifting without obstruction in a viscous medium, e.g. a solution. The analysis should be applicable also to unpinned vortex lines in a superconductor with weak impurities. The focus here is on the swelling and dynamic exponents, rather than on the drift velocity of the centre of mass. Erta~ and Kardar (1992, 1993) proposed a set of coarse-grained equations for these problems, and studied them by simulations and renormalization group methods. The polymer is described (Rouse, 1953) by a aLdimensional position vector/~(x, t), with x being a one-dimensional continuous variable replacing the discrete monomer index. Defining relative monomer positions F(x, t ) - / ~ ( x , t ) - / ~ o ( t ) and expecting the drive to induce anisotropies in the dynamics, components parallel and perpendicular to the drive are distinguished: rlt (x, t) and F_L(x, t), the latter being (d - 1)-dimensional. Distinct equations of motion for each are introduced. Employing a mapping similar to the one relating the standard model to the KPZ equation, namely c~(x, t) - ~ r l(x, t) and ~(x, t) - ~ Fl(x, t), we cast these equations in the form of a multicomponent driven diffusive system in d = 1, i.e. t) -

~2

+

+ e•

~/~2 , (7.10)

t) -

+

Of course, fluctuations will be described by incorporating the usual Gaussian distributed conserved noise terms. Clearly, for d - 1, where there is no


183

~b-field, we recover the usual "free" driven lattice gas, or the equivalent KPZ equation. For d = 2, (7.10) describe the motion of a single, directed vortex line, driven by the Lorentz force (Erta~ and Kardar, 1992), with generalizations by Hwa (1992) to a collection of such lines. Lastly, d = 3 corresponds to the case of a drifting polymer. Working out the details of a perturbative analysis, a rich phase diagram emerges (Erta~ and Kardar, 1993). In particular, the exponents u and z and their appropriate anisotropic counterparts are calculated (Kardar and Erta~, 1995). We note in conclusion that, even though very different boundary conditions apply and d is restricted to only one here, the analysis of this problem may be exploited to study other generalizations of the standard model to a multicomponent case. Related to the study of drifting polymers is a model proposed by Alexander and Lebowitz (1990, 1994). The motivation is a generalization of the brazil nuts problem (Rosato et al., 1986, 1987), which asks why the brazil nuts rise to the top when a can of mixed nuts is shaken. A rigid "rod", or polymer, occupying N connected sites along an axis transverse to the external drive, is introduced into the "flee" gas limit of the standard model (Sections 3.1.2 and 6.7.2). Both background particles and rod obey the excluded volume constraint, and are driven by an external field. Thus, the rod can only move downstream if all N adjacent sites are empty. In a sense, it acts as an "obstruction" to the background particles. Unexpectedly, the drift velocity of the rod, as measured in simulations, is not a monotonic function of N. Beyond a critical length Nc, longer rods move faster! Clearly, this study raises many intriguing questions, such as how to incorporate rotational and bending degrees of freedom in the "rod", as well as interactions with other polymers, or with the background. As in the case of the standard model and its variants, there are far too many avenues compared to the number of available explorers.

7.5

Self-organized criticality and other models of generic scale invariance

Fractal or anomalous dimensions occur rarely in equilibrium systems, typically only at critical points. To arrive at such points, the control parameters must be "tuned" carefully. At the same time, within the subset of critical systems, scale invariance and renormalization group ideas allow us to understand the existence of universality classes, so that widely differing physical systems share the same behaviour. In contrast, fractal power laws are so abundant in nature that we are reluctant to believe that their origin lies in some careful "tuning". For example, a long standing puzzle is the presence of 1If noise in a great variety of dissipative dynamical systems (Weissman,


184

1988). Attempting to explain such ubiquitous behaviour and exploiting the connection between criticality and universality, Bak et al. (1987, 1988) introduced the notion of self-organized criticality (SOC). In this framework, an open system, when driven to a non-equilibrium steady state, would organize itself into a "critical" system, so that it generically displays universal scale invariant phenomena. In this sense, there is common ground between our driven systems and SOC, so that it is reasonable to devote some remarks to the similarities and differences here. Like driven interfaces, this topic has also attracted an extraordinary level of attention, and our remarks should not be mistaken for a critical review of SOC. The specific system studied by Bak et al. (1988) consists of a cellular automaton, modelling the dynamics of a sandpile in an open box. In the simplest d = 1 case, we define integer heights, hi, at site i, with i running from 1 to N, forming an ideal sandpile with identical square grains piled on top of one another. While the boundary at the left (i = 1) is closed, modelling a "wall" of infinite height, the boundary at the right (i = N) is open, modelling a "cliff" from which grains of sand may leave the system. Starting with any set of heights, the system evolves by toppling and, when that ceases, by adding a grain at a random site. The former, defined by moving a grain from i to i + 1, occurs when the height difference between these columns exceeds a fixed critical value K. This set of simple rules is best described in terms of Z i ~ h i - hi+l, so that the toppling rule is just zi~zi-2

and

Zi+l~Zi+l+l,

ifzi>K,

(7.11a)

while adding a grain at i is given by zi ~ zi + l

and

zi-l ~ Zi-l - 1 .

(7.11b)

The boundary conditions can be imposed by setting z0 = 0 and replacing (7.11 a) by Zu ~ ZN -- 1 and Zu-1 ~ Zu-1 + 1 when Zu exceeds K. This simple model can be generalized easily to higher dimensions. After initial transients, such systems reach steady states in which "avalanches" recur. Defining avalanche size by the number of sites toppled as the result of adding a single grain, Bak et al. (1987, 1988) found that the associated distribution displays singular scaling behaviour, i.e. there is no characteristic length scale in this system. Similarly, there is no characteristic scale in the temporal domain, leading to the label "criticality". They further argued that the presence of dissipative transport in an open system is the crucial ingredient for manifesting generically singular behaviour. Though 1/f noise was subsequently shown to be absent in both this model (Jensen et al., 1989; Kert6sz and Kiss, 1990) and physical sandpiles (Jaeger et al., 1989; Held et al., 1990), the "toy" model has spawned much research activity. Some authors concentrate on exactly solvable versions (Dhar and


185

Ramaswamy, 1989; Dhar and Majumdar, 1990; Grassberger and Manna, 1990; Lee et al., 1991; Majumdar and Dhar, 1992; Janowsky and Laberge, 1993), while others focus on the wider perspective of generic scale invariance in non-equilibrium systems (Grinstein et al., 1990; Grinstein, 1991). The similarity to our driven diffusive system is best displayed in the approach proposed by Hwa and Kardar (1989), which replaces the discrete model by a continuum Langevin equation. Before discussing this approach, we should emphasize that there is an important distinction (Grinstein, 1995) between SOC, as embodied in the sandpile model above, and generic scale invariance in continuum theories motivated by such models. The key element is the ratios of two time scales: _= (typical relaxation times) / (typical scales for the noise). It is widely believed that, for a system to show SOC, ~ must be vanishingly small. In particular, no grains are added to the model sandpile until all avalanches have ceased. Since large avalanches presumably take longer to stop and their relaxation times are not bounded by some fixed quantity, this restriction amounts to adding grains at an infinitesimal rate. On the other hand, all continuum models are based on the Langevin equation, with some Gaussian distributed noise. Inherent to this approach is a fixed rate at which the noise acts. Thus, ~ is not infinitesimal typically. Returning to our specific case, Hwa and Kardar (1989) studied the following: 8t h(Y, t) - U_LvZh + UllOZh- ~ 8h 2 + r/(Y, t),

(7.12 a)

where ~(Y, t) is a Gaussian noise of zero mean and (~7(Y, t)~7(Y', t') ) = 2 D 6 ( ~ -

Y')6(t - t').

(7.12 b)

Note that, for simplicity, transport in only one of the d dimensions is kept. Apart from the anisotropy, (7.12 a) differs from the KPZ equation (7.1 a) by the non-linear term, being ~h 2 instead of (Sh) 2. Thus, to compare this equation with the one for the standard model, there is no need to take its derivative. As it stands, (7.12a, b) may be regarded as a driven diffusive system with non-conserved noise. Several features of this equation readily reflect the process of a toppling pile of sand, i.e. a conserved and driven dynamics for the deterministic part plus the random addition of grains. Similar to the standard model (Section 4.2.2), renormalization group techniques may be exploited, though the analysis is based on an expansion in powers of 4 - d. "Critical" exponents were computed exactly (Hwa and Kardar, 1989), while universal current-current correlation and response functions were found to the two-loop level (Becker and Janssen, 1994). Given that the separation of time scales precludes meaningful comparisons

186


between this theory and sandpile models, fresh simulation studies of driven lattice gases with non-conserved noise must be designed before these predictions can be tested. On a different front, Grinstein and Lee (1991) advanced a more general theory. Arguing that (7.12 a) lacks the invariance h ~ h + integer, associated with a global shift of interface heights, they considered A0cos (27rh) instead of the quadratic term. For d > 2, the results of (7.12a, b) are retrieved, perhaps because (7.12a) is invariant under the "Galilean" transformation (4.45): h ~ h + a and Xll ~ Xll + Aat, for any a. On the other hand, since cos (27rh) introduces a degree of discreteness to the h-field, it is reasonable that there is a Kosterlitz-Thouless (1974) type transition into another phase in 1 < d < 2, albeit the singularity structures are quite distinct from the equilibrium case. It would be misleading to imply that no continuum theory could capture the essentials of ~ ~ 0 in SOC. Instead of lowering the noise rates, Carlson et al. (1990) raised the relaxation rates by introducing a density-dependent, diverging mobility into the diffusive current. Locating the singularity at the critical slope for toppling, it is possible to describe large avalanches occurring in a short time interval. Termed "singular diffusion", such an equation is only remotely related to our case. While the above model was introduced with sandpiles in mind (Bak et al., 1988), it and its variants have been applied to a range of physical phenomena, e.g. earthquakes (Bak and Tang, 1989; Carlson and Langer, 1989 a, b; Ito and Matsuzaki, 1990; Brown et al., 1991). Since all earlier models displaying SOC were based on conserved dynamics with non-conserved noise, a natural question arises (Grinstein et al., 1990): are these the necessary and sufficient conditions for a dynamical system to display "self-organized criticality"? In this context, it is significant that SOC is found in systems with a non-conserved deterministic dynamics (Christensen et al., 1992 a, b). Notable examples are "turbulent" models for forest fires (Bak et al., 1990; Drossel and Schwabl, 1992), spring-block models of earthquakes (Olami et al., 1992; Christensen and Olami, 1992a, b; Olami and Christensen, 1992), models for friction (Feder and Feder, 1991), self-organized depinning of an interface (Sneppen, 1992, 1995) and for the evolution of interacting species in an ecological system (Bak and Sneppen, 1993). From a renormalization group point of view, dynamics without conservation laws is always more relevant than conserved dynamics, so that these models are expected to display different "critical" properties. Recently, Paczuski et al. (1994) argued that the Bak-Sneppen model should fall into the universality class of Reggeon field theory (Gribov, 1967; Brower et al., 1978; Liggett, 1985). On the other hand, there are models which exhibit continuously varying exponents (Christensen, 1992), while others display macroscopic oscillations (Socolar


187

et al., 1993). Between controversies concerning the very existence of"critical" behaviour in certain models (e.g. Grassberger and Kantz, 1991) and the rapidly growing number of experiments (Grinstein, 1995), the final chapter on the nature of non-conservative SOC remains to be written.

7.6

Liquids in non-equilibrium steady states

Apart from purely theoretical motivations, the standard model was inspired by a physical system, i.e. superionic conductors. Unfortunately, it would take unreasonably large electric fields to observe, unambiguously, the dramatic effects reported here. Thus, a common criticism of the standard model is the lack of physical systems in which we may verify our predictions. In this section, we briefly review some recent studies on a related system which is readily available, i.e. liquids subject to temperature gradients or shears. Several theoretical investigations predict that such systems display one of the distinguishing features of non-equilibrium steady states, i.e. long range correlations away from criticality. Fortunately, it is possible to find liquids with parameters that allow experimental observation of these effects. However, as we will see, the price to pay here is simplicity, since these nonequilibrium systems involve coupling of several conserved densities in states without translational invariance. The study of fluctuations of fluids in equilibrium has a venerable history, dating back to Einstein (1910) and Onsager (1931). By contrast, though there were some early efforts, the systematic study of fluids in non-equilibrium steady states began less than two decades ago. In particular, the work of Onuki and Kawasaki (1978, 1979), Procaccia et al. (1979), Machta et al. (1979) and Kirkpatrick et al. (1979) sparked extensive investigations by many researchers. Our purpose here is to highlight both the similarities and differences between our model and fluids driven by temperature or velocity gradients. The major differences have been listed above. The similarities are the presence of generic singularities in the one-phase region and strong anisotropy, which modifies critical behaviour significantly.

7.6.1

Linearized hydrodynamics f a r f r o m criticality

The starting point for a theory of fluctuations in fluids is a set of phenomenological equations of hydrodynamics with random currents, i.e. Langevin equations, like (2.15). One major difference is that there are five conserved fields in hydrodynamics, i.e. mass, momentum and energy densities. If the fluid consists of a single species of particles, the first two fields are identical to particle and velocity densities. Of course, being coupled

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and non-linear, these equations are not solvable in general, even without randomness. Our interest is narrower, i.e. small fluctuations about a known time-independent state, so that it is adequate to linearize these equations. Further, we restrict our attention to the single-phase region. Thus, for equilibrium cases, the stationary state is given simply by homogeneous densities, positive for the particle and energy fields and zero for the velocities. The linearized stochastic equations can now be solved straightforwardly. To reproduce the correct equilibrium distributions for these fluctuations, the distributions of the random currents must be chosen properly, namely, in accordance with the FDT. Denoting these fluctuations by ~, the linear hydrodynamic operator by Heq, the randomness by r/, with correlation matrix ~eq, these statements may be summarized symbolically by ~t ~ -~-~eqt~-+-T] and (~7~7)- [~eq, while FDT implies that ~eq is twice the dissipative part of Heq. From here, ( ~ ) may be computed along the same lines as in Section 3.1.1, and the result of a particular matrix element can be used to compare with experimental data from light scattering. For the non-equilibrium fluids under consideration here, the timeindependent states are inhomogeneous in some of the densities, e.g. energy or velocity, induced by suitable boundary conditions. Linearizing the hydrodynamic equations about such states and assuming the system is far from criticality (except Onuki and Kawasaki, 1979), we obtain extra terms that depend upon the details of these inhomogeneities. Thus, we write ~t ~ (Heq + H)~ + ~7. For non-equilibrium systems in general, it is unclear what the appropriate form for (~rl) should be. However, considering only cases with "small" gradients, it is argued that a "local equilibrium" condition should be valid. Specifically, if L v denotes the distance over which the densities vary significantly, and Ls the linear dimension of a small volume around a point R, then we choose to focus on the regime Lv >> Ls. Within Ls, we approximate the conditions by those of an equilibrium system subjected to a set of "local" thermodynamic parameters (Grabert, 1982; Schmitz, 1988). Thus, the choice for (~7~7) is again [~eq(R). In this sense, FDT is "not violated". On the other hand, it is clear that there is no longer the intimate relationship between this ~eq and the dissipative part of (Heq + ~]). Indeed, the origins of long range correlations and singularities in ( ~ ) may be traced to this key feature. Finally, since the inhomogeneities are assumed to be small, all results are obtained only to lowest non-trivial order, e.g. in 1/kLv, where k is a typical wavevector in the study, with 1/kLs < 0(1). Focusing on liquids driven by a constant temperature gradient along the z axis, 1/L v is just d T / T d z . In these analyses, the effect of gravity is neglected, which is an adequate approximation for experiments with small gradients, especially if the sample is "heated from above" (Keiser, 1978; Ronis et al., 1979, 1980; van der Zwan et al., 1981; Tremblay et al., 1981; Kirkpatrick


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et al., 1982a, b; Schmitz and Cohen, 1985, 1987; Law and Sengers, 1989; Schmitz, 1989). Now, d T / d z appears in ~] and couples the five normal fluid modes, leading to several novel phenomena. All have been observed experimentally. For the propagating sound modes, the Brillouin doublet is modified into non-Lorentzian forms and by asymmetric amounts proportional to l~. ~ T / k 2 (Beysens et al., 1980; Kiefte et al., 1984). For the Rayleigh line, the coupling of the heat and viscous modes leads to a dynamic structure factor S(k, w) consisting of two Lorentzians. Though non-equilibrium effects appear only at O([dT/dz] 2) (Kirkpatrick et al., 1982b), they are quite dramatic for small wavevectors in the x - y plane, diverging as 1/k 4. Of course, unlike the standard model, this system is not translationally invariant, and there is no singularity at k = 0. Instead, the 1/k 4 behaviour is cut offwhen k reaches 1/Lv. In practice, it is nevertheless possible to detect intensities enhanced over their equilibrium values by a hundred-fold (Law et al., 1990; Segr6 et al., 1992). Turning to fluids under shear, we assume isothermal systems and consider only mass and velocity densities, studying small fluctuations around a theoretically arbitrary, inhomogeneous steady state ~70(/~) (Machta et al., 1979, 1980; Tremblay et al., 1981; Dufty and Lutsko, 1986). In practice, however, only uniform shears are used so that X7if0 is constant. Following the same methods as above, the main conclusions are similar. Due to the coupling of the density and longitudinal velocity modes, the Brillouin lines are modified by amounts proportional to k. V ~o "lc/k2, to lowest order. At the most simplistic level, we may interpret such a factor as the correction to the vis~sity term uk 2 in (7.3), as a result of linearizing the hydrodynamic term (~'. X7)~7. Similar to the anisotropic singularities discussed in Section 3.1.1, both of these effects also translate into long range correlations in real space. Lastly, there are also investigations of drives by a combination of temperature gradient and shear (Tremblay and Tremblay, 1982; P6rezMadrid and Rubi, 1986) and, recently, a combination of gravity and temperature gradients (Segr6 et al., 1993). In addition, simulations have been performed (Mansour et al., 1987; Garcia et al., 1987). Readers interested in the details of non-equilibrium fluids should consult one of several reviews (Cohen, 1981; Fox, 1982; Tremblay, 1984; Schmitz, 1988; Dorfman et al., 1994). Before ending this section, we mention several theoretical and experimental studies (Helfand and Fredickson, 1989; Onuki, 1990; Milner, 1991; Wu et al., 1991) of the structure factor in polymer solutions under shear. The usual Ornstein-Zernike form gives way to a double peaked pattern, reminiscent of Fig. 3.1b. More strikingly, the alignment of these peaks rotates as the shear rate increases.

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7.6.2


Phase transitions under shear

In this section, we turn to investigations of the effect of shear flow on a system near Te (Onuki and Kawasaki, 1978, 1979). Since large fluctuations are present, the linearized equations of the last section are inadequate. Expecting simple perturbation theory to fail at d < 3, renormalization group methods are essential (Graham, 1975). Approximating the fluid as incompressible (Kawasaki, 1970), and restricting to a spatially uniform shear, v-'0 = cry2, the starting point is similar to our (2.10) and (7.3) 6ar

atq~ ~- )~V 2 ~

--

--+

cryOxq$- p V" (c>~) + r (7.13)

--, -g$~o~ + f , i~tg = ~V2~7- p~ V where t9 is the mode coupfing strength and acg'is (2.9) with an additional term ~v9/2. Both (fie) and ( f f ) are assumed to take the equilibrium isotropic forms, despite the presence of the anisotropic cr term. Remarkably, this theory is translationally invariant, albeit in a limited sense. In particular, all equal-time correlations are precisely invariant. It is possible to study the thermodynamic limit here, because a system with shear, unlike one subjected to a temperature gradient, does not necessarily involve both the high- and low-temperature phases. Since er introduces a time scale, we can divide fluctuations into ones which decay faster or slower than or. Associated with this is a length scale, L,, so that the system undergoes a cross-over from a regime of weak shear to strong, as the correlation length, {, exceeds L,. If { < Lo, shear does create generic anisotropic singularities, though they are softer than those in the standard model. The dramatic effects occur as we approach criticality. First, note that the shear appears at the linear level in (7.13), so that, similar to our results in Section 3.1.2, even the Gaussian fixed point differs from the equilibrium case (Onuki and Kawasaki, 1978). A consequence is that, even away from Tr S(k) is anisotropic and singular at k - 0, e.g. I / S = r + k8/Sk2x/5+ k 2, where k,~,, 1/L,, for strong shear. Unlike our (3.8), this S is continuous but not differentiable, so that the effect of shear is less pronounced, numerically. From this exPression, we easily see the presence of strong anisotropy. For d > 4, where non-linearities are irrelevant, we have kx ~ kS. Next, to analyse critical properties, Onuki and Kawasaki (1979) employed the technique of approximate recursion and showed that the critical temperature is lowered from the equilibrium value and the upper critical dimension is shifted to 2.4 with a new, non-trivial fixed point. Thus, in the physical dimension d = 3, the critical properties show a crossover, from the


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equilibrium Ising universality class to mean-field like behaviour. Most of these predictions have been observed (Beysens and Gbadamassi, 1980) and carefully analysed (Onuki et al., 1981). More recently, there have been studies of many other critical fluid systems. To review them would be beyond our scope. Here, we only mention the cases of random stirring (Onuki, 1984; Satten and Ronis, 1986; Chan et al., 1987), freezing of colloids under shear (Bagchi and Thirumalai, 1988), and the effects of shear on the isotropic-nematic transition in liquid crystalline fluids (Olmsted and Goldbart, 1991, 1992). Finally, we report an interesting Monte Carlo study of the effects of "shear" on phase transitions in the Ising lattice gas (Chan and Lin, 1990). Using a square lattice with periodic boundary conditions only for the rows, the "shear" is imposed as follows. After a fixed number of updates using the "normal" Kawasaki nearest neighbour exchange, all spins in the nth row are displaced, to the right, by n lattice spacings. Spins in the top and bottom rows are quenched, pointing randomly up or down. Intriguingly, the critical temperature is also found to be higher than Onsager's by < 50%, when shear is imposed, with a critical exponent /3 ~ 0.4 and a co-existence curve very similar in shape to Fig. 5.1. During phase separation, striped patterns of ordered phase, reminiscent of Fig. 5.7, are also observed though these may merge during longer runs. Since velocity fields are missing from this model, it is unclear whether such a simplified version still captures the physics of phase transitions in liquids subjected to shear. A decisive answer would certainly help our understanding of how to identify the key features characterizing phase transitions in non-equilibrium steady states. To conclude, we recapitulate the main differences between the standard model and liquids in non-equilibrium steady states. The former distinguishes itself by having (i) only one relevant mesoscopic degree of freedom, the particle density, to which the drive is directly coupled, and (ii) translationally invariant stationary states. In contrast, the latter consists of five conserved fields, with mode-coupling being a crucial ingredient for the external drive to produce "singular" behaviour. Further, translational invariance is spoilt by gradient drives, so that true singularities are probably absent. In the standard model, FDT violation is built in phenomenologically, while in nonequilibrium fluids, it manifests, paradoxically, as a result of imposing the FDT in the specification of the noise spectrum. The latter is based on the local equilibrium hypothesis which, to quote Tremblay (1984), "may not be valid ... when the external source is ... an electric field". Finally, thanks to translational invariance in our models, it is possible to study critical phenomena in a straightforward manner. In contrast, we foresee both conceptual and technical difficulties when correlation lengths exceed Lv,

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though it is possible to regard systems with gradients as a highly complex crossover phenomenon (Onuki and Kawasaki, 1979). Clearly, it is important to study physical systems, no matter how complicated they are. On the other hand, it is equally important to study simple models which facilitate theoretical analyses and allow us to probe the bare essentials needed for singular behaviour in non-equilibrium systems.

8

Summary

and outlook

In this book, we have reviewed some of the progress toward understanding driven diffusive systems, i.e. systems evolving under a conserved dynamics, coupled to both an external drive and a thermal reservoir. For simplicity, our primary focus is the behaviour of non-equilibrium states which share an important characteristic of equilibrium states, namely, being translationally invariant in both space and time. For this purpose and, partly motivated by the physics of superionic conductors, Katz, Lebowitz and Spohn (1983, 1984) introduced a particularly simple model. Starting with the Ising lattice gas, whose equilibrium properties are well understood, they made a "minor" modification to model a uniform external drive. The consequences are, however, far from trivial. Apart from the expected, i.e. a non-zero average particle current, this non-equilibrium steady state displays a rich variety of novel phenomena, including a new universality class of singularities near the critical point and long range correlations at all temperatures. Partly because of the utter simplicity in its microscopic rules, and partly due to historical precedence, this model, among the many driven diffusive systems, has enjoyed the most attention in the past decade. For these reasons, we dubbed it the "standard model" and, with pedagogical intent, devoted Sections 2-5 to a fairly detailed review. The collective behaviour, found in computer simulations, can be reasonably well understood in terms of a Langevin equation for the particle density, with noise terms which do not respect the fluctuation-dissipation theorem. This approach, being entirely equivalent to the use of Landau-Ginzburg Hamiltonians for describing large scale co-operative phenomena in equilibrium systems, may be termed "meanfield theory". Valid especially for temperatures far above T~, it is suitable for predicting the behaviour of small fluctuations, such as generically singular, long range correlations (Section 3). Near criticality, used in conjunction with field theoretic renormalization group methods, such a continuum theory yields a sizable body of analytic results, e.g. the upper critical dimension being 5 and, surprisingly, exact critical exponents in 2 _< d _< 5 (Section 4). A subtle reason for the change in critical dimension, apart from the different underlying symmetries between the Ising and this model, lies in strong


193

anisotropic scaling. Here, even at the mean-field level, lengths transverse and longitudinal to the drive scale with different powers. Though some equilibrium systems display such behaviour, they rely on microscopic Hamiltonians with long range interactions, in sharp contrast to the purely local microscopic dynamics in the standard model. It might be argued that, since conservation laws are long ranged by nature, such behaviour is to be expected. However, we emphasize that violation of the fluctuationdissipation theorem is also a crucial ingredient, without which timeindependent thermodynamic quantities are necessarily equilibrium-like. The presence of strong anisotropy calls for a more sophisticated finite-size scaling analysis of computer simulations, leading to critical exponents which are entirely consistent with field theoretic predictions. While macroscopic properties for T > Tc are quite well understood, most of the physics below criticality, especially the behaviour of interfaces, remains puzzling (Section 5). For example, there is still no quantitative explanation of the suppression of interfacial roughness and, associated with this, the anomalous height-height correlations: C(q) cx 1/q0.67; nor is there a good theory of the profound effects of boundary conditions on bulk properties. After the introduction of the standard model, it became clear that there exist many other simple modifications of the Ising lattice gas which lead to novel behaviour in non-equilibrium steady states. These "variations", briefly reviewed in Section 6, are mostly motivated by physical considerations. Thus, the difficulty of imposing a uniform drive on a periodic lattice naturally prompted the investigation of AC or random fields. Since the former introduces a microscopic time scale, the effects of a random drive were first studied (Section 6.1). Similarly, it is natural to consider systems with DC drives but with open boundaries (Section 6.2), such as a household wire hooked to a battery. In Section 6.3, we reviewed an investigation of multilayer lattice gases, with possible applications to intercalated compounds. Systems containing more than one driven species of particles or quenched impurities are abundant in nature, and motivated considerable work on these variations (Sections 6.4 and 6.6). Meanwhile, since the effective interaction between ions on a lattice is a combination of Coulomb and elastic forces, it is natural to consider Ising lattice gases with repulsive inter-particle forces (Section 6.5), though the relevant density here is actually not conserved. In most cases, more surprising phenomena were discovered. For example, it is remarkable that several key thermodynamic quantities for the uniformly and randomly driven systems appear to be essentially identical, though both the underlying dynamics and symmetries differ drastically. Theoretically, these systems are predicted to belong to different universality classes. Indeed, while the fixed point of the standard model is an FDT violating non-equilibrium theory, the fixed point of the randomly driven model is

194


identical to a Hamiltonian system of uniaxial magnets with dipolar interactions! The critical regions seem to be quite small, and only detailed investigations can discern the differences. No doubt, more refined data are needed to arrive at sharper conclusions. Another totally unexpected phenomenon is the presence of two phase transitions in the bilayered lattice gas, from disordered layers to a pair of standard phase segregated planes and then to planes with distinct, homogeneous densities! At present, there is no viable explanation for these transitions. Finally, we devoted Section 6.7 to some special limits of the standard model, where exact solutions are available. Though these exact results are helpful, they provide only limited insight into the mysteries of driven diffusive systems, since all these limiting cases lack some of the crucial ingredients which make the standard model both interesting and intractable. An analogous situation is the pre-Onsager period, when the exact solution of a d = 1 Ising model failed to quell the debate over the existence of a phase transition. Apart from the standard model and its variations, there is a plethora of systems driven far from equilibrium into steady states. A number are close cousins of the driven diffusive system. Each is extremely interesting and merits separate reviews. While it is clearly beyond the scope of this book to delve into these systems, we believe that it is worthwhile to provide some "bridges" here (Section 7). We hasten to add, however, that even this abbreviated listing of non-equilibrium steady state systems is far from comprehensive. Returning to driven diffusive systems, the best summary of the present status appears to be: though considerable progress was made in the last decade, open questions abound. The latter range from the mundane to the fundamental, and from the specific to the general. Below, we offer a partial list, and invite the reader to extend it. Since the phase transition depends strongly upon the dynamics, it is desirable to develop some intuitive notion for the qualitative features of the phase diagram. Thus, we would like a simple and reliable heuristic argument for why the transition remains second order, and why Tc increases with E, for both the standard and the randomly driven model. If we think of the drive, especially for large E, as an extra source of noise, we would argue that a lower temperature is needed for the system to order. A further twist is that Tc is lower, provided long range jumps in the transverse direction are allowed (Bassler, 1993). Without some intuitive ideas, we will always encounter such surprises when the microscopic dynamics is altered in apparently "trivial" ways. On the other hand, it remains an outstanding puzzle how little the driven system appears to be affected by the seemingly major change from DC to AC or random fields. These surprises indicate how poorly we understand these driven lattice gases, though their microscopic dynamics is deceptively simple. For equilibrium systems, real space


195

renormalization group methods, while not always systematic, have served well in giving such intuitive pictures. Perhaps dynamical versions of these methods (Deker and Haake, 1980, Mazenko and Valls, 1982), suitably generalized to deal with non-equilibrium systems, can fill the vacuum here. The investigation of critical properties so far relies on only two approaches, field theoretic renormalization group techniques and Monte Carlo simulations with finite-size scaling. Other techniques, such as the Monte Carlo renormalization group (Swendsen 1982), proven extremely successful in equilibrium cases, should be devised for non-equilibrium systems like ours. Also, the present agreement between theory and finite-size scaling for the critical exponents is based on a phenomenological approach. Hopefully, a systematic treatment, along the lines of what has been accomplished for critical dynamics in equilibrium systems (Niel and Zinn-Justin, 1987; Niel, 1988), will be pursued. Related to this issue, but at the fundamental level, we may question the very existence of thermodynamics, since bulk properties appear to depend sensitively on the boundary conditions (Section 5.3.2). We have seen that some non-equilibrium critical phenomena are controlled by Hamiltonian fixed points while others are truly non-Hamiltonian. So far, a case-by-case analysis is necessary to determine which type of critical behaviour will be displayed. An intriguing question naturally arises. Is there a simple criterion, based on the microscopic dynamics, for determining if a critical non-equilibrium steady state belongs to the same universality class as an equilibrium system? We emphasize that the term "equilibrium system" means more than finding an effective Hamiltonian for P*, the steady state distribution (Browne and Kleban, 1989). The additional ingredient is that the associated dynamics must satisfy detailed balance. Perhaps the unified approach based on closed time paths (Chou et al., 1985) can provide a framework for answers. Beyond the standard model, it is natural to ask how other types of equilibrium transitions are affected by "slight" modifications so that nonequilibrium steady states are established. We have in mind, for example, various multicritical points (Lawrie and Sarbach, 1984; Knobler and Scott, 1984), roughening (van Beijeren and Nolden, 1987; Kosterlitz and Thouless, 1974), wetting (Sullivan and Telo da Gama, 1986; Kroll, 1987; Dietrich, 1988) and surface transitions (Diehl, 1986). Extending this line further, we can ask if and how non-equilibrium dynamics affects metastable, or even unstable states. Certainly, external forces can stabilize such states. As a trivial example, constant stirring will keep a homogeneous mixture of oil and water from evolving towards the stable inhomogeneous state indefinitely. At the more quantitative level, the effects of shear on phase separation have been measured (Min et al., 1989; Min and Goldburg, 1993). Other examples abound, from mathematical models displaying bistable states (Toom, 1980;

196


Bennett and Grinstein, 1985; He et al., 1990; see also Section 7.2) to physical systems such as lasers. We have studied driven lattice gas models in which the inter-particle interactions are purely attractive or repulsive. An obvious generalization is to include a mixture of both J > 0 and J < 0 terms. If these are competing, a rich variety of phases as well as commensurate-incommensurate transitions exist already in equilibrium (den Nijs, 1988). Surely, interesting behaviour will arise if such systems are driven far from equilibrium. A specific example is an extension of the two-layer system (Section 6.3), stacking many planes of the J > 0 standard model, forbidding interlayer particle exchanges yet allowing a repulsive interlayer interaction. This seemingly contrived model is, in fact, motivated by the physics of intercalated layered materials (Safran and Hamman, 1979). Such anisotropic interactions would be responsible for staging phenomena (Dresselhaus and Dresselhaus, 1981; Kaluarachchi and Frindt, 1983; Levi-Setti et al., 1985; Solin, 1986; Kirczenow, 1988) where guest atoms form layers of ordered domains during intercalation. Recently, in both experiments (Carlow et al., 1990) and simulations (Carlow and Frindt, 1993) of Ag/TiS2, finger-like domains are found, reminiscent of those in our model (Section 6.2). Surely, it is worthwhile to explore an extension of the standard model to multilayer structures with mixed interactions. Beyond these structures, competing interactions are known to produce frustrated or glassy states (e.g. see, M6zard et al., 1987). The effects of drive on these systems are clearly prime targets for future investigations. To venture further, we contemplate applying our driven lattice gas model to the study of flux creep in type II superconductors (Anderson and Kim, 1964; Doniach, 1990) or Josephson-junction arrays (Li and Teitel, 1990). The layered structure of the high-Tc materials provides effectively twodimensional systems, within which regions surrounding a flux line would play the role of "particles". The inter-"particle" interactions would arise from fluctuations of the vortex lines (Nelson, 1988, 1991). In the presence of an electric current, the Lorentz force induces a biased drift on the flux lines and plays the role of our "external drive". This drift results in a positive resistivity (Feigel'man et al., 1989; Nattermann, 1990), whose dependence on various control parameters is clearly important. It would be interesting to explore the applicability of our findings, especially in the model with repulsive interactions (Section 6.5), to this problem. Certainly, lattice gas models have been exploited successfully in modelling transitions in the resistivity (Li and Teitel, 1990), as well as 1/f noise (Jensen, 1990; Fiig and Jensen, 1993) observed in flux flow (Yeh and Kao, 1984). The list of physical systems in non-equilibrium steady states, to which we may turn for sources of insight and possible applications, seems endless. Closely related to the flux creep problem are sliding charge density waves


197

(Gorkov and Griiner, 1989) and fluid flow driven in random media (Narayan and Fisher, 1994). To name a few others, we mention the transport of hot electrons (Thornber and Feynman, 1970; Su et al., 1991), biased diffusion of atoms on surfaces (Wang and Tsong, 1982; Whitman et al., 1991), dynamics of fracture (Herrmann and Roux, 1990; Freund, 1990), and flow patterns in granular materials (Hong, 1993) and traffic (Leutzbach, 1988; Biham et al., 1992; Cuesta et al., 1993). Though our primary focus is steady states, these are by no means the only result of time-independent drives. Other consequences are much more complex phenomena, from periodic states such as Rayleigh-B6nard convection to turbulence and chaos. In turn, these can be viewed within the larger context of periodically driven systems (Jung, 1993). It is clear that the standard model and its variations proved to be fertile grounds for the study of driven diffusive systems. Though we have witnessed much progress in recent years, every step forward also opened new vistas into novel puzzles. We hope that the investigations of these systems provided a small but stimulating step in the long journey toward a viable theory of non-equilibrium steady states, which could serve as a tiny window into the vast world of dynamical phenomena in interacting many-body systems.

9

Acknowledgements

This work is supported in part by grants from the National Science Foundation through the Division of Materials Research, the Jeffress Memorial Trust and the Cornell National Supercomputer Facility. We gratefully acknowledge the hospitality of Heinrich-Heine-Universit~it, Simon Fraser University, the Technical University of Denmark, the Isaac Newton Institute, Forschungszentrum Jiilich, Instituut-Lorentz, and Academia Sinica (Taipei), where part of the research and writing of this book was performed. We are indebted to our collaborators, F. J. Alexander, K. E. Bassler, H. B1/Ste, T. Blum, D. Boal, K. Hwang, H. K. Janssen, G. Korniss, C. A. LaBerge, H. Larsen, K.-t. Leung, K. K. M on, N. Pesheva, E. L. Praestgaard, Z. R/tcz, M. S. Rudzinsky, Y. Shnidman, Z. Toroczkai, L. Vall6s and I. Vilfan, who patiently taught us many aspects of this subject. One of us (RKPZ) would like to thank Y. Shnidman specially, for insisting that he gets involved in this topic. We have benefited from numerous illuminating discussions with many colleagues, including but not limited to, J. V. Andersen, P. Bak, R. Bausch, V. Becker, A. D. Bruce, G. Carlow, K. Christensen, B. Derrida, R. Desai, R. Dickman, H. W. Diehl, S. Dietrich, C. Doering, H. C. Fogedby,

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G. Foltin, D. Forster, R. F. Frindt, P. L. Garrido, C. Godr6che, R. Graham, M. Grant, G. Grinstein, J. D. Gunton, H. Guo, A. Hern/mdez-Machado, T. Hwa, S. A. Janowsky D. Jasnow, M. Kardar, J. Kert6sz, T. Kirkpatrick, J. Krug, K. B. Lauritsen, T. K. Lee, P. A. Lindggtrd, A. J. McKane, O. G. Mouritsen, D. Mukamel, K. Oerding, M. Plischke, S. Puri, J. V. Sengers, H. Spohn, Z. B. Su, S. Teitel, A. Toom, H. van Beijeren, J. M. J. van Leeuwen, L. K. Wickham, R. Workman, M. Wortis, C. Yeung and L. Yu. Last, but far from the least, we give special thanks to J. L. Lebowitz, who is responsible partly for starting this research topic and entirely for starting us off on this review. Finally, we thank Jenny for her moral support and gentle encouragements, without which this publication might have never come to fruition.

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INDEX Note: Figures are indicated by italic page numbers

Absorbing edge [of interface], 86 Anisotropic growth, 77-80 Anisotropic rates, see Extreme anisotropic rates

Anisotropic structure factors, 42, 43 Anisotropy, see also Strong...; Weak anisotropy Anisotropy exponent, 42, 44, 51, 52, 54-59, 65, 67, 100-103, 146 Annealed random drive, effects, 94, 149 Anomalous diffusion, 34, 146 Antiferromagnetic order, 139, 142 Ashkin-Teller model, 8, 127 Asymmetric simple exclusion process (ASEP), 154 blockage introduced, 163-164 with open boundary conditions, 159-161 with periodic boundary conditions, 155

Backgammon-pattern configurations, 113, 114 Bak-Sneppen model, 186 Barber-pole configurations, 130, 131 Bistability, in two-temperature model, 174 Blocking transition,see Two-species models

Blume-Emery-Griffiths model, 8, 127 Boundary conditions for standard model, 12 see also Open...; Shifted periodic boundary conditions Boundary-induced phase transitions, 159-160

Brazil nut problem, 183 Bulk properties effect of interface orientation, 83-87, 92 phase transitions in critical region, 99-103 Burgers equation and driven-interface models, 177 and microscopic nature of shocks, 162, 167 one-dimensional, 154

Cahn-Allen equation, 80, 176 Cahn-Hilliard equation, 76 Capillary waves, 81, 105 Catalytic process, 172 Chemical potential gradients (CPGs) effects, 108-112 in combination with electric fields, 112-118 transverse CPG, interface stability affected, 119-121 Chemical processes, modelling of, 172 Cluster variational method, 16 dynamic version, 16-17 Co-existence curves, 18, 74-76 for standard model, 76 fast rate limit, 152, 153 Competing [conservation/nonconservation] dynamics, 171-173 Competing [repulsion/attraction] interactions, 195-196 Configurations backgammon-board pattern, 113, 114 barber-pole pattern, 130, 131 for polarized lattice gas, 136-137 during phase separation, 77

216

with shifted periodic boundary conditions, 85, 87 Correlations, long-range, 25--40, 93 see also Power law decays Critical dimension, upper, 61-63, 94, 100 Critical exponents randomly driven and multi-temperature models, 101, 103 standard model from simulations, 44--46, 50 relations, 54-59 from theory, 65-66, 67-71 Critical phenomena, 6-7, 40-73 simulation studies, 40-52 theoretical investigations, 52-73 Critical temperature Onsager value, 10, 40 randomly driven and multi-temperature models, 103 standard model, 15, 18, 44 system with repulsive interactions, 141

Dangerously irrelevant operators, 36, 37, 63, 70, 100 Dipolar systems, critical properties, 102 Directed percolation, 172 Domain splitting and merging, 87-90 critical angle, 89 Driven-interface models, 175-179 Dynamic functional, 6, 35, 60 randomly driven and multi-temperature models, 103 standard model, 35, 61, 62 Dynamic mean-field theory, 5, 16-19, 141 Dynamic scaling, 78

Earthquake models, 186 'Electric' field, 11 Entropy, maximum-entropy principle, 18 Epidemics, 172 Escher, M C, 12 Evaporating edge [of interface], 86 Excess energy, 84-85

Index

as function of shift angle, 88 Exponential decays in two-point correlation functions, 32-33 Extreme anisotropic rates, 148-154

Fast rate limit [for standard model], 8, 149-154 FDT see Fluctuation-dissipation theorem Finger formation, 88, 112-118 factors affecting, 91, 92, 117 Finite-size scaling anisotropic, 44, 51-52, 103, 147 in mean-field approaches, 16 isotropic, 7, 44, 51, 52, 140, 143 Fixed-point, see also Standard model; Wilson-Fisher... Hamiltonians, 101-102, 173 randomly driven and multi-temperature models, 101 Fixed-line, Gaussian, 33-37 Fluctuation-dissipation theorem (FDT), 21, 101 violation of, 23, 25, 35, 192-193 effects, 24, 38-39, 104 Fluids, non-equilibrium steady states, 187-191 Flux creep, 196 Fokker-Planck equation, 25 Forest fire models, 186 G Galilean transformation, 63, 145 Gaussian dynamic models, 33-37 Gel electrophoresis, 9, 127, 179, 182183 Generic scale invariance, 26-37, 183186 compared with self-organized criticality, 185 Gibbs-Thomson relation, 90, 119 Glauber spin-flip dynamics, 17, 171 compared with Kawasaki dynamics, 171-172 multi-temperature models with, 173-175

217

Index

Goldstone modes, 104, 133 Green's functions, connected, 60, 64 Ground states [for half-filled lattices] 13 I-I Harris criterion, 144 Hydrodynamic approach, 109 Hydrodynamics, linearized equations, 187-189

Intercalation, 112 Interface orientation bulk properties affected by, 83-87, 92 finger formation in combined drive systems affected by, 117 Interface stability, effect of transverse chemical potential gradient, 119 Interfacial fluctuations, anomalous correlations, 103-105 Interfacial energy, 83 Interfacial properties, randomly driven and two-temperature models, 103-105 standard model, 80-93 Interfacial roughness, suppression of, 7, 81-83 Internal energy, 47, 58 fluctuations in, 49-50 Interparticle interactions attraction, 20 mixture of competing interactions, 195-196 repulsion, 18-19, 138-144 Ionic conductors, charge carriers in, 127 coarse-grained dynamics, 20

and Navier-Stokes equation, 177 Kawasaki dynamics, 11, 95, 141, 191 compared with Glauber spin-flip dynamics, 171-172 with infinite range, 172 Kosteflitz-Thouless transition, 143, 186

Langevin equation, 5-6, 19-25 and dynamic functional, 35, 60, 61 for driven interfaces, 175-178 multi-species model, 131, 177, 181 noiseless version, 78 randomly driven and multi-temperature models, 97 standard model, 22, 27, 34, 78, systems driven by chemical potential gradient, 109, 115 systems with generic scale invariance, 184 systems with quenched random impurities, 145-146 systems with repulsive interactions, 139 two-layer model, 125 Langevin force, 20, 109 Layered compounds, staging in, 112, 122, 196 Lifshitz points, 52 Line defects, effects, 118-119 Liquids in non-equilibrium steady states, 187-192 and Brillouin lines, 188, 189 and Rayleigh line, 188 and temperature gradients, 189 under shear, 190-192 Long-range correlations, 25-40, 93 see also Power law decays

M K

K models, 52 Kardar-Parisi-Zhang (KPZ) equation, 80, 170, 175, 176 and Burgers equation, 177 conserved version, 179 and driven interfaces, 9, 175, 176, 177

Martin-Siggia-Rose response field, 35, 56 Mean-field theory, 5, 16-19, 141, 160, 192 Metropolis rates, 12, 128, 135, 141,152, 153

Microemulsions, 127 model for, 128, 135-138 Model A, 52, 80, 126, 139

218

Index

Model B, 20, 52, 63, 80, 101 Model C, 80, 126, 139 Mullins-Sekerka instability, 73, 89, 91, 120 Multicritical points, 73 Multilayer models, 8, 121-126, 196 Multi-species models, 127-138 Multi-temperature models, 95, 96, 98 with Glauber dynamics, 173-175

Navier-Stokes equation, 177, 178 Nrel temperature, 139 Noise correlation matrix, 21-24, 27-30 I/f, 182, 183 North-east-centre (NEC) model, 169, 174 see also Toom model

dynamics affecting, 18, 99 standard model, 13 polarized lattice gas, 137 repulsive-interaction model, 140 two-species models, 168 Phase separation, dynamics, 7, 76-80 Phase transitions boundary-induced, 159-160 continuous, 72-73, 107 effects of shear flow, 189-191 signals of, 41 splitting and merging, 87-90 Polarized lattice gas (PLG), 135-138 Polymer sedimentation, 9, 179-183 Potts models, 8, 127, 174, 175 Power counting, 61-66, 100, 102, 126, 141,146, 172 Power law decays critical, 45 above criticality, 26-32, 97, 98

Q One-dimensional models, 9, 154-155 open boundary conditions, 158, 159-162 shocks in, 162-164 systems with translational invariance, 155-158 Toom model, 169-170 two-species models, 164-169 One-loop diagram, 67 Onsager temperature, 10, 40 Open boundary conditions and chemical potential gradients, 108-118 in 1D models, 158, 159-162 Ornstein-Zernike form [of structure factor], 29, 33, 189 O Path probability method, 16 Periodically (AC) driven systems, 94, 197 Phase boundaries, methods of locating, 17-18 Phase diagrams

Quenched impurities, effects, 144-148, 175

Randomly driven systems, 94-107 Rayleigh-Brnard experiment, equivalent for lattice gas, 108 References listed, 198-213 Reggeon field theory, 172, 173, 185 Related non-equlibrium steady-state systems, 170-191 Renormalization group analysis, 6, 21, 52 Gaussian dynamic models, 33-37 randomly driven and multi-temperature models, 99-103 standard model, 59-73 one-loop results, 67-73 systems with quenched random impurities, 144-146 Reptation models, 127, 134, 165-167, 180-182 Repulsive interactions mapping by gauge transformation, 8, 139


standard model with, 18-19, 138144 Response functions, 56 Roughening transition, 81 effects of driving, 179 Rubinstein-Duke model see Reptation models

Saffman-Taylor instability, 89 Sandpile models, 184-186 Scale invariance see Generic scale invariance Scaling hypothesis, 52 Scaling laws, with strong anisotropy, 53-59, 192 Self-organized criticality (SOC), 9, 183-186 Shear, phase transitions under, 189-191 Shifted periodic boundary conditions (SPBC), 83-93 and splitting/merging transitions, 8790 Shocks development in 1D models, 159, 162-164 microscopic nature, 162, 167 Sine-Gordon, 179 Single-step surface growth model, 154, 162 Singular diffusion, 186 Six vertex model, 155-157 Specific heat, 47, 58 Spin-flip dynamics, 17, 171 see also Glauber spin-flip dynamics Staging phenomena [in layered materials], 112, 122, 196 Standard [non-equilibrium] model, 4 boundary conditions specified, 12 with chemical potential gradient, 108-112 in combination with electric field, 112-118 interface stability in transverse CPG, 119-121 co-existence curve for, 76 collective behaviour, 95-98, 192 with combination of direct and random

219

drives, 105-107 criticisms/limitations, 5, 187 driving field introduced, 11 and dynamic mean-field theories, 5, 16-19 fast rate limit, 148-153 finite-size effects, 51-52 fixed point, 68 interface fluctuations suppression, 81-83, 103-105 lack of droplets in ordered states, 75 master equation, 11 mesoscopic approach, 5-6, 19-25, 115 microscopic dynamics, 11-13 and multi-layer models, 121-126 and multi-species models, 127-138 and multi-temperature model, 95, 96, 98 one-dimensional models, 9, 154-170 phase separation, 76-80 with quenched impurities, 144-148 and randomly driven systems, 95-98, 193 rates, microscopic, 11-12 with repulsive interactions, 18-19, 138-144 scaling behaviour, 67-73 Strip ordering, 72, 74, 147 Strong anisotropic scaling, 53-59, 192 Strong anisotropy, 42 implications, 44 Structure factors, 27-28 above-criticality, 86, 1 O0 contour plots, 28 fluctuations in, 49-50 Ornstein-Zernike form, 29, 33, 189 see also Anisotropic structure factors Superconductors, flux creep in, 196 Superionic conductors, 4-5, 8, 187 Surface growth models, see Driveninterface models Susceptibilities, 46, 56-57 Symmetry charge conjugation (C), 15 Euclidean, 84 Galilean, see Galilean transformation Ising, 26, 38, 39, 44, 84, 95, 96, 123, 139 O(n), 96

220

Index

particle conservation, 19 randomly driven and multi-temperature models, 94-96 reflection (R), 16 reflection, in SPBC, 85-86 standard model, 15-16, 24, 38 supersymmetry, 40 systems with repulsive interactions, 139 time reversal, 25

Three-point correlation functions, 37--40, 48-49 Toom model, 169-170, 174 Traffic flow model, 134 Translational invariance, 155-158, 191 Tree approximation, 64-67 Tricritical point, 133, 140-142 Two-layer model, 121-126 Two-point correlation functions, 30-32, 53, 66, 154 Two-point correlations and exponential decays, 32-33 and power law decays, 26-32 and simulation studies, 41-48 singularities in, 26-37 Two-species models, 127 blocking transition in, 128-135 one-dimensional models, 164-169 polarized lattice gas, 135-138 see also Blume-Emery-Griffiths model; Potts model

Two-temperature model, 95, 96, 98, 171-173 see also Multi-temperature models

Universality class, 15, 52 for combined direct and random drives, 106 for quenched random impurities, 145 for repulsive interactions, 141 Ising, 10, 94, 139, 141 standard model, 59-72 two-layer model, 126 two-temperature model, 95, 99, 101 uniaxial magnets with dipolar interactions, 95, 101 V Van Beijeren-Schulman (vBS) rates, 18, 149, 151,152, 153 Vertex functions, 60, 64 Vicinal-surface in crystal growth, 178

W

Ward identity, 60, 68 Weak anisotropy, 42, 53 Wilson-Fisher fixed point, 6, 21, 141, 172 Wilson functions, 68 Winding number, 130

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