# The Two-Dimensional Ising Model

## Table of Contents

### An Introduction To Monte Carlo Statistical Mechanics #

A report written in the 1st year of my Ph.D. (November 1997).

This acted as my initial introduction to the multicanonical Monte Carlo method. Here is the introduction - the full text can be downloaded in PDF format from the link at the bottom of this page.

#### Introduction #

This report will outline the theory and findings of a simple Ising model simulation, and is intended to act as an introduction to the general concepts involved in applying Monte Carlo techniques to statistical physics. The two-dimensional Ising ferromagnet model is one of the simplest examples from this field, and yet its implementation allows many of the techniques of this kind of Monte Carlo simulation to be demonstrated. Also, the 2-D Ising model is one of the few statistical physics models that can be solved analytically, and so it provides a “gold standard” against which to test numerical simulation techniques.

Firstly, the theory behind the Ising model will be examined, from the basic concepts of statistical physics to the specific implementation of the Ising ferromagnet. Also, the general behaviour expected from the Ising model (including the phase change behaviour) will be discussed.

Following the theory, the concepts of the Monte Carlo approach to statistical physics problems will be outlined, with the Ising model being used as an example. The basic methods of canonical Monte Carlo will be explained, along with the limitations of such simulations. Following this the concepts of multi-canonical Monte Carlo simulation will be introduced. The next step will be to outline the ways in which properties such as the magnetisation of a system can be extracted from the simulation, and how the errors in these results can be estimated.

Results will be presented for the canonical and multi-canonical approaches. This will lead to a discussion on the performance of the simulations, including notes on how the techniques might be improved. Finally, the wider applicability of this work will be examined, with reference to the recently developed lattice-switch MCMC technique.