Ensemble

In Classical Statistical Mechanics, is the probability distribution defined on phase space.

Given a Hamiltonian system with phase space coordinates $({\mathbf{r}}_1,{\mathbf{p}}_1;\dots ;{\mathbf{r}}_N,{\mathbf{p}}_N)$, we introduce a probability density $\rho ({\mathbf{r}}^N,{\mathbf{p}}^N)$ so that $\rho d^{3N}r{d}^{3N}p$ is the chance to find the system in that “cell” of phase space. A normalization condition is, of course, assumed:

Different macroscopic constraints (fixed energy, fixed temperature, etc.) lead us to choose different $\rho$, different ensembles.
Important cases:

• microcanonical ensemble,
• canonical ensemble,
• ...

This probability measures are built upon the Liouville measure, exactly in the same way that it happens for the famous probability distributions (see probability density function) with respect to the Lebesgue measure.

In Quantum Mechanics ...