Editing Ensemble (mathematical physics) (section)

== Ensemble average ==
In [[statistical mechanics]], the '''ensemble average''' is defined as the [[mean]] of a quantity that is a function of the [[Microstate (statistical mechanics)|microstate]] of a system, according to the distribution of the system on its micro-states in this [[statistical ensemble (mathematical physics)|ensemble]].

Since the ensemble average is dependent on the [[statistical ensemble (mathematical physics)|ensemble]] chosen, its mathematical expression varies from ensemble to ensemble. However, the [[mean]] obtained for a given physical quantity does not depend on the ensemble chosen at the [[thermodynamic limit]].
The [[grand canonical ensemble]] is an example of an [[Thermodynamic system#Open system|open system]].<ref>{{Cite web |title=Statistical mechanics of classical systems |url=http://physics.gmu.edu/~pnikolic/PHYS307/lectures/ensembles.pdf |access-date=3 November 2023 |website=George Mason University Physics and Astronomy Department}}</ref>

=== 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:
<math display="block">\bar{A} = \frac{ \displaystyle
    \int{A\exp\left[-\beta H(q_1, q_2, \dots, q_M, p_1, p_2, \dots, p_N)\right] \, d\tau}
}{ \displaystyle
    \int{\exp \left[-\beta H(q_1, q_2, \dots, q_M, p_1, p_2, \dots, p_N)\right] \, d\tau}
},</math>
where
* <math>\bar{A}</math> is the ensemble average of the system property {{mvar|A}},
* <math>\beta</math> is <math>\frac{1}{kT}</math>, known as [[thermodynamic beta]],
* {{mvar|H}} is the [[Hamiltonian mechanics|Hamiltonian]] of the classical system in terms of the set of coordinates <math>q_i</math> and their conjugate generalized momenta <math>p_i</math>,
* <math>d\tau</math> is the [[volume element]] of the classical phase space of interest.

The denominator in this expression is known as the [[partition function (statistical mechanics)|partition function]] and is denoted by the letter ''Z''.

=== Quantum statistical mechanics ===
{{unsourced section|date=November 2023}}
In [[quantum statistical mechanics]], for a quantum system in thermal equilibrium with its environment, the weighted average takes the form of a sum over [[Energy eigenvalues|quantum energy states]], rather than a continuous integral:{{clarify|reason=This is true only for discrete spectrum (bound states); continuous spectrum still involves continuous integration.|date=November 2023}}
<math display="block">\bar{A} = \frac{\sum_i A_ie^{-\beta E_i}}{\sum_i e^{-\beta E_i}}.</math>

=== Canonical ensemble average ===
The generalized version of the [[partition function (mathematics)|partition function]] provides the complete framework for working with ensemble averages in thermodynamics, [[information theory]], [[statistical mechanics]] and [[quantum mechanics]].

The [[microcanonical ensemble]] represents an isolated system in which energy (''E''), volume (''V'') and the number of particles (''N'') are all constant. 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. The [[grand canonical ensemble]] represents an open system which can exchange energy (''E'') and particles (''N'') with its surroundings, but the volume (''V'') is kept constant.