In statistical mechanics, the ensemble average is defined as the mean of a quantity that is a function of the micro-state of a system (the ensemble of possible states), according to the distribution of the system on its micro-states in this ensemble.

Since the Ensemble average is dependent of the ensemble chosen, its mathematical expression varies from ensemble to ensemble. However, the mean obtained for a given physical quantity doesn't depend on the ensemble chosen at the thermodynamic limit.

Statistical ensemble (mathematical physics)

Canonical ensemble average

classical statistical mechanics

For a classical system in thermal equilibrium with its environment, the ensemble average takes the form of an integral over the phase space of the system:

$\backslash bar\{A\}=\backslash frac\{\backslash int\{Ae^\{-\backslash beta\; H(q\_1,\; q\_2,\; ...\; q\_M,\; p\_1,\; p\_2,\; ...\; p\_N)\}d\backslash tau\}\}\{\backslash int\{e^\{-\backslash beta\; H(q\_1,\; q\_2,\; ...\; q\_M,\; p\_1,\; p\_2,\; ...\; p\_N)\}d\backslash tau\}\}$

where:

$\backslash bar\{A\}$ is the ensemble average of the system property A,

$\backslash beta$ is $\backslash frac\; \{1\}\{kT\}$, known as thermodynamic beta,

H is the Hamiltonian (or energy function) of the classical system in terms of the set of coordinates $q\_i$ and their conjugate generalized momenta $p\_i$, and

$d\backslash tau$ is the volume element of the classical phase space of interest.

The denominator in this expression is known as the partition function, and is denoted by the letter Z.

quantum statistical mechanics

For a quantum system in thermal equilibrium with its environment, the weighted average takes the form of a sum over quantum energy states, rather than a continuous integral:

characterization of the classical limit

Ensemble average in other ensembles

Micro-canonical ensemble

Macro-canonical ensemble

Statistical mechanics