Statistical ensemble

Remarks

Example: for a system with finitely many microstates $\Omega = \set{1, \dots, n}$, any probability vector $(p_1, \dots, p_n)$ with $p_i \ge 0$ and $\sum_i p_i = 1$ is an ensemble, and $\langle A \rangle = \sum_i p_i A(i)$. Non-example: the weights $p_i = 1$ for all $i$ with $n \ge 2$ do not form an ensemble, since they sum to $n \ne 1$; dividing by $n$ gives the uniform ensemble. In classical mechanics, let an ensemble have a density $\rho$ with respect to Liouville measure on phase space. It is stationary when $\rho$ is constant along trajectories. Any density of the form $\rho = g(H)$, with $H$ the Hamiltonian, is stationary.