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# 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:

$Z(\beta) = \sum_{x \in X} \exp \big( -\beta E(x) \big)$

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.