Monte Carlo Simulation of the Ising and Skyrme Models. Steven Bornn-Gilman & Alex Do

Abstract

Statistical Mechanics provides an analytical method for evaluating expectation values for properties of fixed temperature systems. For large, or realistically sized systems, obtaining such values through a weighted average of a Boltzmann probability distribution involves many sums that require large amounts of computation time. The method of Monte Carlo simulation is the only numerical method that can give good estimations of expectation values without requiring extremely intensive amounts of computer time. The process involves simulating a system and choosing an algorithm that lets the system change from state to state in such a way that it follows the correct probability distribution, automatically weighting any averages of physical properties to fit the correct distribution. QCD theory is currently the accepted theory for particle physics. This theory effectively explains high-energy particle interactions, but mathematical calculations for the low-energy regime are nearly impossible. The Skyrme model is a low energy model of nuclear physics, and is the current area of research of our mentor Prof. Alec Schramm. We have been investigating the Skyrme model using Monte Carlo simulation on computers. The specific algorithm of Monte Carlo simulation used in our research was the Metropolis algorithm. To learn the mechanics behind Monte Carlo simulation, we also studied a simpler system, the Ising model.