Spin models

A spin model is a simple mathematical model designed to demonstrate important physical principles such at magnetism and “lattice gases”. One interesting direction for research on these systems is to develop a rigorous understanding of what happens when temperature changes induce large-scale changes in the system. Examples include the freezing transition of a liquid: as the temperature falls the system changes from one in which liquid particles are zipping around without much long-range structure into one in which the particles line up and form a frozen crystal.

The basic mathematical starting point is to model the space in which our physical system lives as graph. The nodes or vertices represent particles or sites where particles may go, and the edges between nodes represent the sites being close enough together that particle interaction can happen along edge. This framework gives us a discrete model of the geometry of the space in question, and it’s a good fit for certain problems like when gas particles occupy available sites in a crystal lattice. It’s less clear that there are suitable graphs to model e.g. 3-dimensional space, but it is certainly useful at time to simplify the smooth world of Euclidean space into a lattice graph.

To model the particles we let each node of the graph take on a “spin”. Given an assignment of spins to every node, we compute the “weight” of the assignment according to the spins themselves and interactions between spins along edges of the graph. Then we convert this weight into a probability and this constitutes a distribution to the states of the system. At this point we haven’t done any real mathematics, but we already have a system with interesting behaviour.

Model options

Glauber dynamics