Chapter 8: Statistical Mechanics, Phase Transitions, and the Ising Model

Lecture 1: Monday January 14

• The Ising Model and Statistical Mechanics
• Magnetism
• Ising Model
• Statistical Mechanics
• Mean Field Theory
• Root Finding Algorithms
• Bisection method
• Convergence rate
• Secant method
• Newton's tangent method
• Computational Physics Library

Lecture 2: Wednesday January 16

• The Monte Carlo Method
• Uniform Sampling
• Compare Deterministic Quadrature
• Importance Sampling
• Metropolis Monte Carlo Method
• Programming the Metropolis Algorithm
• The Ising Model in One Dimension
• Thermal equilibrium properties
• Energy cost and entropy advantage
• Peierl's droplet argument
• Metropolis Algorithm for the Ising Model

Lecture 3: Friday January 18

• Two-Dimensional Ising Model of Ferromagnetism
• Ferromagnetism, Paramagnetism, and Curie Temperature
• Monte Carlo Simulation
• Algorithm of Metropolis {\it et al.}
• Monte Carlo Program to Simulate the 2-D Ising Model
• Efficient evaluation of Boltzmann factors
• Taking one Metropolis step
• One Monte-Carlo step per spin
• Measuring observables
• Steering the calculation: the {\tt main} function
• Ferromagnetism below the Curie temperature

Lecture 4: Wednesday January 23

• The Ising model in two dimensions
• Peierls' droplet argument in two dimensions
• Exact solution of the two dimensional Ising model
• Magnetization in zero field
• Finite sized magnets are unstable
• Sharp phase transitions require infinite systems
• Simulating finite sized systems
• The Ising Phase Transition
• Order parameter and order of phase transition

Lecture 5: Friday January 25

• Second Order Phase Transitions and Critical Fluctuations
• Magnetic Susceptibility
• Critical divergences are due to long range correlations
• Divergence of the Susceptibility
• Scaling laws
• Finite-Size Scaling
• Scaling of the Susceptibility
• Simplified scaling analysis
• First-Order Phase Transitions

Lecture 6: Monday January 28

• Critical Slowing Down and Autocorrelation Time
• Autocorrelation time in Metropolis simulations
• Code to measure the autocorrelation time

Lecture 7: Wednesday January 30

• Basic Problem for Random Systems
• Proof of Thermalization
• Trial Step and Acceptance Probability
• Connectedness and Ergodicity
• Cluster Algorithms to Reduce Critical Slowing Down
• Swendsen-Wang Cluster Algorithm
• Efficient Cluster Decomposition Algorithms
• Backtracking algorithm
• Hoshen-Koppelman Algorithm
• Wolff Single Cluster Algorithm
• OpenGL Demo Program for Cluster Algorithms