Ising Model

Problem

Suppose we had a sample space $S=\{-1,+1{\}}^N$. and $|S|=2^N$¹. Let $\sigma \in S$ where

$$\sigma =({\sigma }_1,{\sigma }_2,\dots ,{\sigma }_N)$$

a chain of ${\sigma }_i=\pm 1$. Clearly, $|S|$ grows exponentially large the more spins we have. The probability of finding a state $\sigma \in S$ is

$$p(\sigma )=\frac{1}{Z}f(\sigma )$$

where $f$ is easy to compute, and

$$Z=\sum _{\sigma \in S}f(\sigma )$$

the normalization factor. $Z$ is hard to compute.

This sets the stage for Markov Chain Monte Carlo (MCMC). Since we cannot calculate $Z$, we cannot compute $p(\sigma )$ directly.

Footnotes

1. This is a problem from statistical physics. The values are the spin states of electrons. ↩