= Thermal light
For a single mode in <thermal equilibrium>, its <quantum harmonic oscillator> energy ladder gives a <geometric distribution>
$$
p_n=(1-q)q^n,\qquad q=e^{-\hbar\omega/(k_{\rm B}T)},\qquad
\mu=\frac{q}{1-q},\qquad 0<q<1.
$$
Its <probability generating function> is $F(z)=(1-q)/(1-qz)$. Differentiating twice gives $F''(1)=2q^2/(1-q)^2=2\mu^2$, hence $g^{(2)}(0)=2$. At the chosen mean $\mu=2$, $q=2/3$, $p_0=1/3$, $p_1=2/9$, $p_2=4/27$, and $\operatorname{Var}N=6$. Rare high occupations increase the pair count even though the mean matches the other preparations. The geometric weights also arise from a Gaussian mixture of <coherent states>.<One Hundred Years of Light Quanta by Roy Jay Glauber>[2005 individual Nobel lecture, pp. 91–92, equations (31)–(38)]{ref}
\Image[/photon-number-statistics-comparison.png]
{title=Three original one-mode photon-number models with the same mean two and different pair correlations}
{description=The bars are calculated probabilities, not measured data. Thermal occupations above fourteen are omitted without renormalization. The displayed second-order correlations use the complete distributions.}
Single-mode selection is essential to this factor of two. If $M$ independent equally populated thermal modes are collected together, let $N=\sum_{j=1}^M N_j$ with $\langle N_j\rangle=\mu/M$. Adding their <variances> gives $\operatorname{Var}N=\mu+\mu^2/M$, hence $g^{(2)}(0)=1+1/M$. Averaging over many unresolved modes can make thermal pair counts approach the independent-count reference.
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