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Computer Studies of Phase Transitions and Critical Phenomena


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Table of Contents

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.

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