Computer Studies of Phase Transitions and Critical Phenomena

This book is based on research carried out by the author in close collabora­ tion 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 Depart­ ment of Physics, University of British Columb~a, Canada. I wish to thank the Department of Chemistry at Aarhus University for providing me with splen­ did 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 provi­ ding 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 extre­ mely 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 Jubilreumsle­ gat of 1981. Their support is gratefully acknowledged.

1. Introduction.- 2. Computer Methods in the Study of Phase Transitions and Critical Phenomena.- 2.1 Statistical Mechanics and Phase Transitions.- 2.1.1 Modern theories of phase transitions and critical phenomena.- 2.1.2 Statistical mechanics, order parameters, fluctuations, critical exponents, scaling, and universality.- 2.2 Numerical Simulation Techniques.- 2.2.1 Monte Carlo methods.- 2.2.2 A Monte Carlo importance-sampling method.- 2.2.3 A realization of a Monte Carlo method.- 2.2.4 General limitations of the Monte Carlo method.- 2.2.5 Broken ergodicity.- 2.2.6 Distribution functions.- 2.2.7 Coarse-graining techniques and criteria of convergence.- 2.2.8 Finite-size effects.- 2.2.9 Determining the nature of a phase transition.- 2.2.10 Computational details.- 2.2.11 General advantages of the Monte Carlo method: Applications.- 2.3 Exact Configurational Counting and Series Expansions.- 2.3.1 A general approach.- 2.3.2 The moment method.- 2.3.3 Principles of the calculation.- 2.3.4 Step 1. Determination of all distinct graphs and their multiplicities.- 2.3.5 Step 2. Embedding of connected graphs into a lattice.- 2.3.6 General correlation function series.- 2.3.7 Capabilities and limitations of a general approach.- 3. Monte Carlo Pure-model Calculations.- 3.1 Critical Behavior of the Three-dimensional Ising Model.- 3.1.1 The Ising model and its order parameter.- 3.1.2 Numerical evidence of a phase transition in the Ising model on a diamond lattice.- 3.1.3 Finite-size scaling analysis and critical behavior.- 3.1.4 Are Monte Carlo techniques practicable in the study of critical phenomena?.- 3.2 Phase Behavior of Ising Models with Multi-spin Interactions.- 3.2.1 Higher-order exchange in magnetic systems.- 3.2.2 Ising models with multi-spin interactions.- 3.2.3 First-order phase transitions of Ising models with pure multi-spin interactions.- 3.2.4 Universality and tricritical behavior of Ising models with two- and four-spin interactions: Pair interactions as a symmetry-breaking field.- 3.3 Thermodynamics of One-dimensional Heisenberg Models.- 3.3.1 One-dimensional magnetic models.- 3.3.2 The anisotropic Heisenberg model in a magnetic field.- 3.3.3 Comparison with theoretical calculations on a continuum model.- 3.3.4 A model ofthe linear magnet CsNiF3?.- 4. Testing Modern Theories of Critical Phenomena.- 4.1 Fluctuation-induced First-order Phase Transitions.- 4.1.1 The role of fixed points in the renormalization group theory.- 4.1.2 Motivation for computer studies of fluctuation-induced first-order phase transitions.- 4.1.3 Phase transitions in antiferromagnets with order Parameters of dimension n=6 and n=3.- 4.1.4 Crossover from first-order to continuous transitions in a symmetry-breaking field.- 4.1.5 Fluctuation-induced first-order phase transitions in Ising models with competing interactions.- 4.2 Critical Phenomena at Marginal Dimensionality.- 4.2.1 The role of a marginal spatial dimension.- 4.2.2 Computer experiments of hypercubic Ising models: ?A romance of many dimensions?.- 4.2.3 Susceptibility and critical isotherm of the four-dimensional Ising model.- 4.2.4 Conclusions on critical behavior in marginal dimensions.- 4.3 Basic Assumptions of Critical Correlation Theories.- 4.3.1 Review of a critical correlation theory.- 4.3.2 Testing the basic assumption by Monte Carlo calculations.- 5. Numerical Experiments.- 5.1 Phase Transitions in Lipid Bilayers and Biological Membranes.- 5.1.1 What are biological membranes and what do they do?.- 5.1.2 Lipid bilayers are model membranes.- 5.1.3 Phase behavior of lipid bilayers.- 5.1.4 Back to biology: Are phase transitions at all relevant to the biological functions of the membrane?.- 5.1.5 Theories of lipid bilayer phase transitions.- 5.1.6 Computer simulations of lipid bilayers.- 5.1.7 Multi-state models of lipid bilayers.- 5.1.8 Computer simulations of the q-state models for the gel-fluid phase transition.- 5.1.9 Computer Simulation of the phase behavior of lipid bilayers with ?impurities?: cholesterol, proteins, and Polypeptides.- 5.1.10 Have Computer studies provided any new insight into the properties of biological membranes?.- 5.2 Nuclear Dipolar Magnetic Ordering and Phase Transitions.- 5.2.1 Nuclear dipolar magnetic ordering.- 5.2.2 The secular dipolar Hamiltonian.- 5.2.3 Perspectives in studies of nuclear dipolar magnetic ordering.- 5.2.4 Motivation for a numerical Simulation study of nuclear dipolar magnetic ordering.- 5.2.5 Monte Carlo studies of systems with truncated classical secular dipolar interactions.- 5.2.6 Nature of the spin structures: ?Permanent? structures or the devil’s staircase?.- 5.2.7 Double-layered spin structures in CaF2-like systems: Continuous transitions and critical behavior.- 5.2.8 Multi-layered spin structures in CaF2-like systems: Firstorder phase transitions.- 5.2.9 Can series expansions provide information on the nature of the phase transitions?.- 5.2.10 Nuclear antiferrimagnetic susceptibilities of systems with two spin species: LiF and LiH.- 5.3 Phase Transitions of Adsorbed Monolayers.- 5.3.1 Two-dimensional phases of molecules adsorbed on solid surfaces.- 5.3.2 N2 physisorbed on graphite: The anisotropic-planar rotor model.- 5.3.3 The Heisenberg model with cubic anisotropy.- 5.3.4 Fluctuation-induced first-order phase transition in the anisotropic-planar rotor model.- 5.3.5 Comparison with experiments on N2 physisorbed on graphite.- 5.3.6 Phase behavior on the anisotropic-planar rotor model with vacancies.- 5.3.7 Physical realizations of the anisotropic-planar rotor model with vacancies.- 5.4 Kinetics of Growth.- 5.4.1 Growth.- 5.4.2 Computer Simulation of domain-growth kinetics.- 5.4.3 Domain-growth kinetics of herringbonephases.- 5.4.4 Domain-growth kinetics of pinwheel phases.- 5.4.5 Kinetics of growth and critical phenomena.