Canonical ensemble

Definition

Let $\Omega$ be the set of microstates of a system with a fixed number of particles, and let $H \colon \Omega \to \R$ be its energy. Let $\beta = 1 / (k_B T) > 0$, where $T$ is the temperature and $k_B$ the Boltzmann constant.

1. Discrete case: let $\Omega$ be countable with $Z(\beta) = \sum_{\omega \in \Omega} e^{-\beta H(\omega)} < \infty$. The canonical ensemble gives $\omega$ the probability $p(\omega) = e^{-\beta H(\omega)} / Z(\beta)$.
2. Classical case: let $\Omega$ be the phase space of $N$ identical particles in a box $V \subset \R^3$. Let $Z(\beta) = \frac{1}{N! \, h^{3N}} \int_{\Omega} e^{-\beta H} \, d\Gamma$ be finite, where $h > 0$ is a constant with the units of action. The canonical ensemble has density $e^{-\beta H} / (N! \, h^{3N} Z(\beta))$ with respect to Liouville measure $d\Gamma$.

The number $Z(\beta)$ is the canonical partition function, and $F = -\beta^{-1} \ln Z(\beta)$ is the Helmholtz free energy.