Ensemble average

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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)

1 Canonical ensemble average
2 Ensemble average in other ensembles
- 2.1 Micro-canonical ensemble
- 2.2 Macro-canonical ensemble

[edit] Canonical ensemble average

[edit] 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,

β

is $\frac {1}{kT}$ , 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 τ

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.

[edit] 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: