Free Energy

For a given Boltzmann distribution, the free energy is defined as

\[F(\beta) := - \frac{1}{\beta} \log Z(\beta)\]

Why is the free energy useful? It contains all (most of?) the useful information in a Boltzmann distribution.

Thermodynamic Potentials

Internal Energy

Define a function \(U(\beta) := \beta F(\beta)\) called the internal energy. Differentiating with respect to the inverse temperature \(\beta\), we discover the expected energy:

\[U(\beta) := \partial_{\beta} (\beta F(\beta)) = \langle E(x) \rangle\]

Proof:

\[\begin{align*} &U(\beta)\\ &:= \partial_{\beta} (\beta F(\beta))\\ &= F(\beta) + \beta \partial_{\beta} F(\beta)\\ &= F(\beta) + \beta \big(-\frac{1}{\beta^2} \log Z(\beta) - \frac{1}{B} \partial_{\beta} \log Z(\beta) \big)\\ &= F(\beta) - F(\beta) - \partial_{\beta} \log Z(\beta) \\ &= - \frac{1}{Z(\beta)} \sum_{x \in X} e^{-\beta E(x)} (-E(x))\\ &= \sum_{x \in X} \frac{1}{Z(\beta)} e^{-\beta E(x)} E(x)\\ &= \sum_{x \in X} p_{\beta}(x) E(x)\\ &= \langle E(x) \rangle \end{align*}\]

High Temperature Limit: What happens as \(\beta \rightarrow 0\)?

\[\begin{align*} \lim_{\beta \rightarrow 0} U(\beta) &= -\sum_x p_{\beta}(x) E(x)\\ &= \sum_x \frac{1}{Z(\beta)} e^{-\beta E(x)} E(x)\\ &= \sum_x \frac{1}{Z(\beta)} (1 - \beta E(x) + \Theta(\beta^2)) E(x)\\ &= \sum_x \frac{1}{Z(\beta)} E(x) - \Theta(\beta)\\ &= \langle E(x) \rangle_{\beta=0} - \Theta(\beta)\\ \end{align*}\]

We find that the internal energy becomes the average of each state’s energy, weighed uniformly.

Low Temperatuer Limit: What happens as \(\beta \rightarrow \infty\)?

\[\begin{align*} \lim_{\beta \rightarrow \infty} U(\beta) &= \end{align*}\]

Canonical Entropy

The canonical entropy is defined as:

\[S(\beta) := - \sum_x p_{\beta}(x) \log p_{beta}(x)\]

High Temperature Limit: What happens as \(\beta \rightarrow 0\)?

Properties

• \[F(\beta) = U(\beta) - \frac{1}{\beta}S(\beta)\]