We will give the basic setup for statistical mechanics, and define the log-partition function, the Gibbs distribution, and the Ising model.

The basic problem

Imagine you have $n$ particles, each of which can have one of two states $\{-1, +1\}$. These spins can be collectively encoded as $\sigma \in \{-1,+1\}^n$, the spin vector.

We want to estimate (for $\beta > 0$) the sum $$Z = \sum_{\sigma\in \{\pm 1\}^n} e^{\beta f(\sigma)}$$ for various functions $f$.

Here is a really easy case. Suppose that $f(\sigma) = \sum_i h_i\sigma_i$. Then, we can simply factor the sum $$Z = \sum_{\sigma\in \{\pm 1\}^n} e^{\beta f(\sigma)} = \prod_{i=1}^n \left(e^{h_i} + e^{-h_i} \right).$$

The first nontrivial case for $f$ are quadratic functions of $\sigma$, which is known as the Ising model.

Basic phyiscal interpretation

In the above, $f$ is actually the (neg)energy of the spins. When the energy is lower (i.e. $f$ is higher), the system is more stable and prefers to be in that state.

In particular, for the given (neg)energy $f$, we can define the probability distribution $$p(\sigma) \propto e^{\beta f(\sigma)}$$ known as the Boltzmann distribution. (Why this? See this later section.)

The parameter $\beta > 0$ is the inverse temperature of the system. We can study the behavior of the probability distribution for extreme $\beta$:

• When $\beta \to 0$, the temperature goes to infinity and the probability distribution is uniform over all states.
• When $\beta \to \infty$, the temperature goes to zero and the probability distribution is only nonzero for states where $f$ is maximized. Over these of minimal energy (called ground states), the probability distribution is uniform.

In this framework, $Z$ is the normalizing constant for the distribution. This is important if we want to compute the probability $p(\sigma)$ explicitly.

One last thing to note is that for linear $f$ (as we saw earlier), we get a product distribution over $\{\pm 1\}^n$, which is the same as saying that each spin is independent.

Ising model

Suppose that every pair of spins can either prefer alignment or disalignment. We encode this using interaction coefficients $\{J_{ij}\}$:

• If $J_{ij} > 0$, particles $i$ and $j$ have a preference of alignment. The larger $J_{ij}$ is, the larger the preference for alignment.
• If $J_{ij} < 0$, particles $i$ and $j$ have a preference of alignment. The smaller $J_{ij}$ is, the larger the preference for disalignment.

The sum of the overall pairs of preferences form the overall (neg)energy $$f(\sigma) = \sum_{i < j} J_{ij} \sigma_i \sigma_j.$$