Partition Function and Observables

In statistical mechanics one introduces a temperature and defines a partition function as follows:

$$Z = \sum_{\{s\}} e^{-\beta H(\{s\})}$$

where $\beta = 1/kT$. The sum is over all possible choices of spins. Since each spin takes on two values independently of the other spins, the number of terms in the sum is $2^V$ where $V = n_x n_y$ is the number of sites. For a modest $64\times 64$ lattice, there are some $10^{1233}$ terms.

The quantity

$$P(\{s\}) = \exp[-\beta H(\{s\})]/Z$$ (1)

gives the probability of encountering the set of spins $\{s\}$ (also called a configuration of spins) at the inverse temperature $\beta$. Any observable ${\cal O}(\{s\})$ must be a function of the spins. For example, the average spin

$$m = \sum_{\bf u} s_{\bf u}/V$$

is an observable that measures the degree of magnetization. Statistical mechanics tells us that the thermodynamic average of any observable is given by

$$\langle {\cal O} \rangle = \sum_{\{s\}} {\cal O}(\{s\}) e^{-\beta H(\{s\})}/Z$$

that is, it is just the expectation value of the observable according to the probability $P(\{s\})$ above.

A real ferromagnet can be magnetized at low temperature, but when it is heated it loses its magnetization. At the ``Curie'' temperature there is a phase transition from a magnetized to a nonmagnetized system. The Ising model imitates this behavior. In the limit of (first) infinite volume and (then) zero external field $h$, the average spin $m$ should be nonzero at low temperature but should decrease abruptly to zero at the Curie temperature and remain zero at higher temperatures. Onsager's solution gives $\beta_c = \ln(1+sqrt{2})/2$.