Ensemble average (statistical mechanics)

In statistical mechanics, the ensemble average is defined as the mean of a quantity that is a function of the microstate 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 on 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. Grand canonical ensemble is an example of open system.

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:

${\bar {A}}={\frac {\int {Ae^{-\beta H(q_{1},q_{2},...q_{M},p_{1},p_{2},...p_{N})}d\tau }}{\int {e^{-\beta H(q_{1},q_{2},...q_{M},p_{1},p_{2},...p_{N})}d\tau }}}$

where:

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

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

H is the Hamiltonian of the classical system in terms of the set of coordinates $q_{i}$ and their conjugate generalized momenta $p_{i}$, and

$d\tau$ 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:

${\bar {A}}={\frac {\sum _{i}{A_{i}e^{-\beta E_{i}}}}{\sum _{i}{e^{-\beta E_{i}}}}}$

Ensemble average in other ensembles

The generalized version of the partition function provides the complete framework for working with ensemble averages in thermodynamics, information theory, statistical mechanics and quantum mechanics.

Microcanonical ensemble

It represents an isolated system in which energy (E), volume (V) and the number of particles (N) are all constant.

Canonical ensemble

It represents a closed system which can exchange energy (E) with its surroundings (usually a heat bath), but the volume (V) and the number of particles (N) are all constant.

Grand canonical ensemble

It represents an open system which can exchange energy (E) as well as particles with its surroundings but the volume (V) is kept constant.