{ "cells": [ { "cell_type": "markdown", "metadata": { "collapsed": false }, "source": [ "# Monte Carlo Study of Ferro-magnetism using an Ising Model\n", "\n", "The goal of this project (over Exercises 1-8) is to create a statistical model simulating the evolution of magnetism as a function of material temperature. \n", "\n", "Since the emergence of magnetism is attributed to the contribution of a great many small atomic magnetic dipoles a statistical method is to be utilised:\n", "- Monte Carlo methods\n", "- Random number generation\n", "- Ferromagetism\n", "- Ising Model\n", "\n", "The subject of of this project will be statistical in nature, and hence a basic understanding of Monte Carlo methods and random number algorithms will be necessary. \n" ] }, { "cell_type": "markdown", "metadata": { "collapsed": false }, "source": [ "# Monte Carlo Methods\n", "\n", "Numerical computations which utilise random numbers are called Monte Carlo methods after the famous casino. The obvious applications of such methods are in [stochastic physics](https://en.wikipedia.org/wiki/Stochastic): e.g., statistical thermodynamics. However, there are other, less obvious, applications including the evaluation of multi-dimensional integrals.\n", "\n", "This method was popularised by physicists such as Stanislaw Ulam, Enrico Fermi, John von Neumann, and Nicholas Metropolis, among others. A famous early use was employed by Enrico Fermi who in 1930 used a random method to calculate the properties of the recently discovered neutron. Of course, these early simulations where greatly restricted by the limited computational power available at that time.\n", "\n", "Uses of Monte Carlo methods require large amounts of random numbers, and it was their use that spurred the development of pseudorandom number generators, which were far quicker to use than the tables of random numbers which had been previously used for statistical sampling.