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.