Overview
Ising Model Unit

Contents

Objectives

The Ising Model describes a toy world in which little "spin vectors" live on sites in a 2 Dimensional grid. The spins are free to flip up or down, and the amount of flipping increases as the "temperature" of the system increases.

By itself, this may not seem to be very interesting. Indeed, the real value of the Ising Model comes as a simple, hands-on laboratory for exploring much bigger game:

The Physics Of Phase Transitions
Just as ordinary water exists in three different phases (vapor, liquid and solid), there are two distinct phases for a collection of Ising spins: all ligned up ("magnetic") and disordered ("non-magnetic"). The existence of one state or the other can be regarded as a "struggle" between Energy Minimization (trying to line the spins up) and Entropy Maximization (trying to scrable everything). The "winner" in this struggle is determined by the overall system temperature.

While none of the details are precisely correct, this is a good model of how permanent magnetization actually happens in iron.

Basic Concepts In Applied Statistics And Statistical Physics
The Ising Model is an example from Statistical Physics. This means that properties such as energy and magnetization on a macroscopic scale are inferred from probabilistic averages of associated microscopic quantities.

This unit uses the fundamental Boltzmann distribution function, in which probability of an individual microscopic state is determined by the energy of that state and the overall system temperature. The translation from the microscopic to macroscopic worlds is done by way of statistical averaging. Simple correlation measurements are used to quantify the qualitative notion of "clumping", as the system freezes into a magnetized state.

Computers and Computer Simulations In Science
In spite of all the simplifying assumptions, exact "solutions" to the Ising model are very difficult. However, the system is easily investigated using computer simulations.

The "hands on" components of this unit use computer simulations of Ising model dynamics (i.e., probabilistic spin flips at individual spin sites). The entire state of the system is simulated, allowing direct measurements of quantities like magnetization and correlations for various temperatures.

The calculations are done through a "Monte Carlo Simulation" - a technique which has become increasingly important in the sciences, the social sciences, finance, and a number of other areas.

Important "Indirect" Lessons
In addition to these specific content items, there are two important qualitative lessons to be drawn from studies of the Ising Model:

  1. Modeling and "Thinking Like A Physicist"

    The very simplified world of the Ising Model is far removed from the world of magnetic phenomenon in real chunks of iron. However, without the simplifications, the calculations would be impossible, and nothing would be learned.

    This is typical of the way in which physics (or other sciences) actually work. Simplified models are often the only way to gain insight into complicated systems. In many respect, the real "art" of physics lies in designing models which are simple enough to be solved, but not so simple that the essential concepts have been discarded.

  2. The Role of Quantum Mechanics In Macroscopic Phenomena

    The system and analysis within the Ising Model are all from the domain of "Classical Physics" with one important exception: the assumtion that an individual spin can exist in only one of two positions (up or down). This is contrary to normal classical physics (a little magnet can point any direction it pleases). However, the non-zero energy difference associated with "quantized" directions for the spins is critical for the formation of large-scale correlations. Put differently, without the "all or nothing" constraints, the little spin vectors in the Ising model could just "ooze" around, slowly washing away all lage scale magnetization.

    The same thing happens in the real world. Permanent magnets are a "human-scale" manifestation of Quantum Mechanic effects on atomic scales.

Components Of The Ising Model Unit

There are two main components of the Ising Model unit:

  1. A set of WWW pages describing aspects of the Ising Model and other tidbits from Statistical Physics. These are intended to provide background material for the computational components.

  2. Two "interactive" WWW pages allowing students to generate and view spin configuration pictures and to submit longer simulation runs in order to estimate physical parameters.

These components are described in a bit mode detail below.

Supporting Material WWW Pages

Introduction To The Ising Model

A fairly general discussion of background material, including some basic elements from Statistical Physics and the notion Magnetization can be associated with a transition between "ordered" and "disordered" phases in matter. The Ising Model is introduced providing a simple picture for magnetization, and the basic concepts of Monte Carlo simulations to study the system are presented.

Monte Carlo Methods and the Metropolis Algorithm

These pages use the (impossible) problem of exact probability sums in Statistical Physics to motivate the general notion of "random simulations" of physical systems - the so-called Monte Carlo approach. The method is justified (in a thoroughly hand-waving manner) by way of the Law of Large Numbers in ordinary statistics. A brief description of the particular (Metropolis) Monte Carlo procedure used in the Ising Model code is given.

Correlations: Quantifying "Clumpiness"

Two individual spins are said to be "correleted" if they tend to point in the same direction. As the Ising system "freezes out" into a magnetized phase, the correlations between distant spins are found to increase. This page introduces a family of simple correlation measures which are used in the computational unit to measure the onset of the magnetization phase transition.

(This page remains to be written).