Springer Series in Computational Physics

Transcription

1 Springer Series in Computational Physics Editors: H. Cabannes M. Holt H. B. Keller 1. Killeen S. A. Orszag V. V. Rusanov

2 Ole G. Mouritsen Computer Studies of Phase Transitions and Critical Phenomena With 79 Figures Springer-Verlag Berlin Heidelberg New York Tokyo 1984

3 Dr. Ole G. Mouritsen Aarhus University, Department of Chemistry, Physical Chemistry Division, Langelandgade 140, DK-8000 Aarhus C, Denmark Editors H.Cabannes Mecanique Theoretique Universite Pierre et Marie Curie Tour 66. 4, Place lussieu F Paris, France M.Holt College of Engineering and Mechanical Engineering University of California Berkeley, CA 94720, USA H.B. Keller Applied Mathematics Firestone Laboratory California Institute of Technology Pasadena, CA 91125, USA 1. Killeen Lawrence Livermore Laboratory P.O. Box 808 Livermore, CA 94551, USA S.A. Orszag Department of Mathematics Massachusetts Institute of Technology Cambridge, MA 02139, USA v. V. Rusanov Keldysh Institute of Applied Mathematics 4 Miusskaya pi. SU Moscow, USSR ISBN-13: DOl: / e-isbn -13: This work is subject to copyright. All rights are reserved, whether the whole or part of the material is concerned, specifically those of translation, reprinting, reuse of illustrations, broadcasting, reproduction by photocopying machine or similar means, and storage in data banks. Under 54 ofthe German Copyright Law, where copies are made for other than private use, a fee is payable to "Verwertungsgesellschaft Wort", Munich. by Springer-Verlag Berlin Heidelberg 1984 Softcover reprint of the hardcover I st edition 1984 The use of registered names, trademarks, etc. in this publication does not imply, even in the absence of a specific statement, that such names are exempt from the relevant protective laws and regulations and therefore free for general use. Offset printing: Beltz Offsetdruck,6944 Hemsbach/Bergstr. Bookbinding: J. SchiifferOHG, 6718 Griinstadt. 2153/

4 The most successful computer experiment is the one which makes itself superfluous

5 Preface This book is based on research carried out by the author in close collaboration with a number of colleagues. In particular, I wish to thank Per Bak, A. John Berlinsky, Hans C. Fogedby, Barry Frank, S.1. Knak Jensen, David Mukamel, David Pink, and Martin Zuckermann for fruitful and extremely stimulating cooperation. It is a pleasure for me to note that active interaction with most of these colleagues is still continuing. The work has been performed at several different institutions, notably the Department of Chemistry, Aarhus University, Denmark, and the Department of Physics, University of British Columb~a, Canada. I wish to thank the Department of Chemistry at Aarhus University for providing me with splendid research facilities over the years. From May 1980 to August 1981, I visited the Department of Physics at the University of British Columbia and I would like to express my sincere gratitude to members ofthe department for providing me with excellent working conditions. My special thanks are due to Professor Myer Bloom who introduced me to the field of phase transitions in biological membranes and in whose biomembrane group I found an extremely stimulating scientific atmosphere happily married with a most agreeable social climate. During the last two years when a major part ofthis work was carried out, I was supported by AlS De Danske Spritfabrikker through their Jubilreumslegat of Their support is gratefully acknowledged. The manuscript for this book has been typeset by means of the TEX system* as implemented at RECA U(The Regional Computing Center of Aarhus University). I am greatly indebted to Marianne N0mberg for her enthusiasm in piloting this project in mathematical typesetting. I wish to thank her for diligence and authority in typing the manuscript into the computer - and for her patient struggles to get it out again. Steen Larsen and Benedict L0fstedt of RECA U are sincerely thanked for indispensable guidance in using TEX, for * TEX is a typesetting system intended especially for the creation of mathematical books. The system was invented by prof. Donald E. Knuth, the University of Stanford, California. The TEX implementation used in producing this monograph is based on the TEX May 1980 version from Stanford, adapted to RECAU's CDC Cyber environment by Erik Bertelsen, RECAU. The typesetting device used is a Compugraphic MCS 8600 phototypesetterinstalled atrecau.

6 VIII Preface designing the formats, and for valuable aid in expediting the typesetting. I appreciate the technical assistance of Arne Lindahl in preparing the illustrations. Finally, my thanks are owed to Kirsten for her constant encouragement and for bearing with me. I thank her for a thorough criticism of my technicalese and other linguistic oddities. I wish to dedicate this book to my son Jonas who did his best to make it impossible. Aarhus, Denmark, June 1984 Ole G. Mouritsen

7 Table of Contents 1. Introduction Computer Methods in the Study of Phase Transitions and Critical Phenomena Statistical Mechanics and Phase Transitions Modern theories of phase transitions and critical phenomena Statistical mechanics, order parameters, fluctuations, critical exponents, scaling, and universality Numerical Simulation Techniques Monte Carlo methods A Monte Carlo importance-sampling method A realization of a Monte Carlo method General limitations of the Monte Carlo method Broken ergodicity Distributionfunctions Coarse-graining techniques and criteria of convergence Finite-size effects Determining the nature of a phase transition Computational details General advantages of the Monte Carlo method: Applications Exact Configurational Counting and Series Expansions A general approach The moment method Principles of the calculation Step 1. Determination of all distinct graphs and their multiplicities Step 2. Embedding of connected graphs into a lattice General correlation function series Capabilities and limitations of a general approach Monte Carlo Pure-model Calculations Critical Behavior of the Three-dimensional Ising Model The Ising model and its order parameter

8 x Contents Numerical evidence of a phase transition in the Ising model on a diamond lattice Finite-size scaling analysis and critical behavior Are Monte Carlo techniques practicable in the study of critical phenomena? Phase Behavior of Ising Models with Multi-spin Interactions Higher-order exchange in magnetic systems Ising models with multi-spin interactions First-order phase transitions of Ising models with pure multi-spin interactions Universality and tricritical behavior of Ising models with two- and four-spin interactions: Pair interactions as a symmetry-breaking field Thermodynamics of One-dimensional Heisenberg Models One-dimensional magnetic models The anisotropic Heisenberg model in a magnetic field Comparison with theoretical calculations on a continuum model A model of the linear magnet CsNiF3? Testing Modem Theories of Critical Phenomena Fluctuation-induced First-order Phase Transitions The role of fixed points in the renormalization group theory Motivation for computer studies of fluctuation-induced first-order phase transitions Phase transitions in antiferromagnets with order parameters of dimension n=6 and n= Crossover from first-order to continuous transitions in a symmetry-breaking field Fluctuation-induced first-order phase transitions in Ising models with competing interactions Critical Phenomena at Marginal Dimensionality The role of a marginal spatial dimension.' Computer experiments of hypercubic Ising models:»a romance of many dimensions« Susceptibility and critical isotherm of the four-dimensional Ising model Conclusions on critical behavior in marginal dimensions Basic Assumptions of Critical Correlation Theories Review of a critical correlation theory Testing the basic assumption by Monte Carlo calculations

9 Contents XI 5. Numerical Experiments Phase Transitions in Lipid Bilayers and Biological Membranes What are biological membranes and what do they do? Lipid bilayers are model membranes Phase behavior oflipid bilayers Back to biology: Are phase transitions at all relevant to the biological functions of the membrane? Theories of lipid bilayer phase transitions Computer simulations oflipid bilayers Multi-state models of lipid bilayers Computer simulations of the q-state models for the gel-fluid phase transition Computer simulation of the phase behavior of lipid bilayers with»impurities«: cholesterol, proteins, and polypeptides Have computer studies provided any new insight into the properties of biological membranes? Nuclear Dipolar Magnetic Ordering and Phase Transitions Nuclear dipolar magnetic ordering The secular dipolar Hamiltonian Perspectives in studies of nuclear dipolar magnetic ordering Motivation for a numerical simulation study of nuclear, dipolar magnetic ordering Monte Carlo studies of systems with truncated classical secular dipolar interactions Nature of the spin structures:»permanent«structures or the devil's staircase? Double-layered spin structures in CaF2-like systems: Continuous transitions and critical behavior Multi-layered spin structures in CaF2-like systems: Firstorder phase transitions Can series expansions provide information on the nature of the phase transitions? Nuclear antiferrimagnetic susceptibilities of systems with two spin species: LiP and LiH Phase Transitions of Adsorbed Monolayers Two-dimensional phases of molecules adsorbed on solid surfaces N2 physisorbed on graphite: The anisotropic-planar rotor model The Heisenberg model with cubic anisotropy Fluctuation-induced first-order phase transition in the anisotropic-planar rotor model

10 XII Contents Comparison with experiments on N2 physisorbed on graphite Phase behavior on the anisotropic-planar rotor model with vacancies Physical realizations of the anisotropic-planar rotor model with vacancies Kinetics of Growth Growth Computer simulation of domain-growth kinetics Domain-growth kinetics of herringbone phases Domain-growth kinetics of pinwheel phases Kinetics of growth and critical phenomena Bibliography Subject Index