Course:Statistical Mechanics/Ensemble theory/The canonical and the
microcanonical ensemble
We could now ask how the microcanonical and the canonical ensembles are
related. Since in the canonical ensemble we have removed the constraint of
having constant energy, the energy of a system will in general fluctuate
around its mean value. We can therefore ask if these fluctuations are
relevant or not. In fact if it turns out that they are negligible (at least
in the thermodynamic limit) then we can conclude that the canonical and
microcanonical ensembles are equivalent.
Let us therefore compute \left \langle \mathcal{H} \right \rangle and σ_E^2
= \left \langle \left ( \mathcal{H} -\left \langle \mathcal{H} \right
\rangle \right )^2 \right \rangle . First of all, from the definition of
the canonical partition function we have:
\left \langle \mathcal{H} \right \rangle = \frac{1}{Z} ∫e^{-β\mathcal{H} }
\mathcal{H} dΓ= -\frac{1}{Z} \frac{\partial Z}{\partial β} =
-\frac{\partial \ln Z}{\partial β}
and:
σ_E^2 = \left \langle \mathcal{H} \right \rangle - \left \langle
\mathcal{H} \right \rangle ^2 = \frac{1}{Z} \frac{\partial ² Z}{\partial
β²} -\left ( -\frac{\partial \ln Z}{\partial β} \right )^2 = \frac{\partial
^2 \ln Z}{\partial β²} = -\frac{\partial \left \langle \mathcal{H} \right
\rangle }{\partial β} = = -\frac{\partial T}{\partial β} \frac{\partial
\left \langle \mathcal{H} \right \rangle }{\partial T} = k_B T² C_V
⇒ σ_E² ⁼ k_B T² C_V
This is a fluctuation-dissipation relation, which we couldn't find using
only thermodynamics. Therefore, the relative fluctuation of energy is:
\frac{σ_E}{\left \langle \mathcal{H} \right \rangle } =\frac{√(k_B T²
C_V)}{\left \langle \mathcal{H} \right \rangle }
Both \left \langle \mathcal{H} \right \rangle and C_V are extensive
quantities, i.e. proportional to N, and therefore:
\frac{σ_E}{\left \langle \mathcal{H} \right \rangle } \propto
\frac{1}{√(N)}
Thus, if our system is macroscopic the relative fluctuations of energy are
absolutely negligible (as we have already seen, for N \sim 10^{23} this
relative fluctuation is of the order of 10^{-11})! We can therefore
conclude that the canonical and microcanonical ensembles are indeed
equivalent.