# Boltzmann Distribution

Suppose we have a system, and let \(X\) denote the set of a system’s possible
configurations. By configuration, we mean a complete microscopic determination of the system’s
state. The **Boltzmann distribution** plays a central role in statistical mechanics,
and is defined as:

\[p_{\beta}(x) = \frac{1}{Z(\beta)} \exp(-\beta E(x))\]
where \(\beta = 1/T > 0\) is called the **inverse temperature** and \(E(x)\) is the
energy of the configuration \(x \in X\). The normalizing constant \(Z(\beta)\) is
called the **partition function** and is defined as:

\[Z(\beta) = \sum_{x \in X} \exp \big( -\beta E(x) \big)\]
## Relation to Free Energy

See free energy.

## Properties

- By embedding the Boltzmann distribution in a one-parameter continuum, by taking
\(\beta \rightarrow 0\), we get a uniform distribution and by taking \(\beta \rightarrow \infty\)
we get a distribution concentrated on the maximum. Very nice!

## Examples

### Binary (Ising) Spins

Consider a single particle, with one of two possible spin values: \(x \in \{-1, +1\}\).
In a magnetic field \(B\), the energy of the particle is defined as:

\[E(x) = - B x\]
This means that the energy is lower when \(x\) points in the direction of the magnetic
field. The Boltzmann distribution is thus

\[p_{\beta}(x) = \frac{1}{Z(\beta)} exp(-\beta E(x)) = \frac{1}{Z(\beta)} \exp(\beta B x)\]
The average value of the system (called the magnetization) is given by

\[\langle x \rangle = \sum_{x \in X} p_{\beta}(x) x = -\exp(-\beta B) + \exp(\beta B) = \tanh( \beta B)\]
For this simple system, when the temperature \(T = 1/\beta >> |B|\), the magnetization is
small, meaning the expected value is near 0. However, when the temperature is small, the
magnetization approaches \(\pm 1\), meaning the spin matches the magnetic field.

### Multiple (Potts) Spins

Consider a single particle, with one of two several discrete spin values: \(x \in \{1, 2, ..., q \}\).
In a magnetic field \(B\) pointing in direction \(r\), the energy of the particle is defined as:

\[E(x) = - B \delta_{x, r}\]
The average value of the system (called the magnetization) is given by:

\[\langle \delta_{x, r} \rangle = \sum_{x \in X} p_{\beta}(x) \delta{x, r}
= p_{\beta}(r) = \frac{\exp (\beta B)}{ \exp(\beta B ) + q - 1}\]
As with the Ising spin, \(T = 1/\beta >> |B| \Rightarrow\), the magnetization is
small, meaning the expected value is near 0. However, when the temperature is small, the
magnetization approaches \(\pm 1\), meaning the spin matches the magnetic field.