# Ensemble average (statistical mechanics)

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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. The grand canonical ensemble is an example of an open system.

## Canonical ensemble averageEdit

### Classical statistical mechanicsEdit

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:

- where:

- is the ensemble average of the system property A,

- is , known as thermodynamic beta,

- H is the Hamiltonian of the classical system in terms of the set of coordinates and their conjugate generalized momenta , and

- 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 mechanicsEdit

In 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:

## Ensemble average in other ensemblesEdit

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 ensembleEdit

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

### Canonical ensembleEdit

The canonical ensemble 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 ensembleEdit

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

## See alsoEdit

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