= Canonical partition function
{title2=$Z(\beta)$}
= Partition function
{synonym}
The <partition function> $Z(\beta)=\sum_s e^{-\beta E_s}$ is the sum of the <Boltzmann weights> over individual <microstates>. Dividing by it makes their <probabilities> sum to one. When summing distinct <energies> instead, include the <state degeneracy>: $Z=\sum_E g(E)e^{-\beta E}$. For two independent distinguishable <two-level systems>,
$$
Z_2=1+2e^{-\beta\Delta}+e^{-2\beta\Delta}=Z_1^2.
$$
The middle coefficient counts two <microstates>, exactly as in the preceding figure. For $N$ such units, $Z_N=Z_1^N$; this factorization requires additive <energy> and <independence>, and generally fails with interactions.
The usefulness goes beyond normalization. For fixed <energies>, <derivatives> of the <natural logarithm> of the <partition function> yield measurable quantities:
$$
U=\langle E\rangle=-\partial_\beta\ln Z,\qquad
\operatorname{Var}(E)=\partial_\beta^2\ln Z,\qquad
S=k_B(\ln Z+\beta U).
$$
Here $U$ is <internal energy>, the second expression is the <energy> <variance>, and $S$ is <Gibbs entropy>. For one <two-level system> at fixed $\Delta$, its <heat capacity> $C=\partial U/\partial T$ and <entropy> are
$$
U=\Delta p,\qquad \frac{C}{k_B}=\frac{(\beta\Delta)^2e^{\beta\Delta}}{(1+e^{\beta\Delta})^2},\qquad
\frac{S}{k_B}=\ln(1+e^{-\beta\Delta})+\beta\Delta p.
$$
At $\beta\Delta=\ln3$, these give $U/\Delta=0.25$, $C/k_B\simeq0.2263$ and $S/k_B\simeq0.5623$. For independent units, all three quantities add.
The figure shows why <heat capacity> peaks. At low <temperature>, the excited <energy> is rarely occupied; at high <temperature>, the <probabilities> are almost equal and further warming barely changes <internal energy>. The largest response lies between these limits, at $k_BT/\Delta\simeq0.4168$, where $C/k_B\simeq0.4392$. Meanwhile <entropy> rises from zero toward $k_B\ln2$, reflecting two equally likely alternatives. This high-<temperature> limit belongs to the stipulated <two-level system>; extra accessible <energy eigenstates> would change it.
\Image[/two-level-thermal-response.png]
{title=Population, heat capacity and entropy of one two-level system as temperature increases}
Back to article page
