Thermodynamic ensemble

A probability distribution over microstates representing a macroscopic thermodynamic system.

Thermodynamic ensemble

A thermodynamic ensemble is a probability distribution over the microstates of a system, used to compute macroscopic observables as statistical averages.

Main ensembles

Ensemble	Fixed quantities	Fluctuating	Characteristic function
Microcanonical	$E, V, N$	—	Entropy $S$
Canonical	$T, V, N$	$E$	Free energy $F$
Grand canonical	$T, V, \mu$	$E, N$	Grand potential $\Omega$

Ensemble equivalence

In the thermodynamic limit , different ensembles give the same values for intensive quantities and the same thermodynamic relations. This is ensemble equivalence.

Statistical averages

For an observable $A$ , the ensemble average is

\langle A \rangle = \sum_i p_i A_i \quad \text{(discrete)} \quad \text{or} \quad \int A(q,p) \rho(q,p)\, dq\, dp \quad \text{(continuous)}.

Fluctuations

Ensembles also predict fluctuations around mean values:

\langle (\Delta A)^2 \rangle = \langle A^2 \rangle - \langle A \rangle^2.

These typically scale as $1/N$ relative to the mean.