= Translational and two-level energy variance
{title2=$\sigma_E^2=3N(k_BT)^2/2+N\epsilon^2p(1-p)$}
For a dilute classical <ideal gas> whose atoms carry independent nondegenerate internal levels $0,\epsilon$, the <canonical partition function> factorizes into translation and $(1+e^{-\beta\epsilon})^N$. The upper-level <probability> is $p=(1+e^{\beta\epsilon})^{-1}$. Taking two derivatives of its logarithm gives the title sum of independent <variances>. The internal contribution freezes exponentially at low <temperature> and tends to $N\epsilon^2/4$ at high <temperature>. The translational classical result cannot be continued through <quantum degeneracy> without changing the model.
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