Computer Studies of Phase Transitions and Critical Phenomena

Inhaltsverzeichnis

• 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.