Editing Ensemble (mathematical physics) (section)

=== 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''.