numbers $\left(b_j,s_j,q_j\right)$. The $j$-summation in Eq. (\[Z-id\]) is taken over all hadron species. Global chemical potentials ($\mu_B$, $\mu_S$, $\mu_Q$) and temperature $T$ regulate the systems average charges (baryonic, strangeness, electrical) and energy densities. Finally, $V$ is the volume of the system and $K_2$ is the modified Hankel function.

Primordial Fluctuations
-----------------------

In the GCE the first two moments of the multiplicity distribution of $j$th particle are: $$\begin{aligned}
\label{Nid}
\langle N_j\rangle~&=~\frac{1}{Z_{id}} \left( \lambda_j
  \frac{\partial}{\partial \lambda_j} 
\right)~ Z_{id} ~=~ V~\phi_j~,\\
\langle N_j^2 \rangle  ~&=~\frac{1}{Z_{id}} \left( \lambda_j
\frac{\partial}{\partial \lambda_j} \right)^2 ~ Z_{id} ~=~ \left(
V \phi_j \right)^2 ~+~V \phi_j~.\end{aligned}$$ Thus, $\phi_j$ is the $j$th hadron number density, $n_j^{id}$. The variance of the $j$th particle number distribution is: $$\begin{aligned}
\label{DeltaNj}
\langle \left( \Delta N_j \right)^2 \rangle ~\equiv~ \langle
\left(N_j~-~\langle N_j\rangle \right) \rangle^2~=~\langle N_j^2 \rangle -
\langle N_j \rangle^2 ~=~ \langle N_j \rangle~,\end{aligned}$$ and the scaled variance equals unity, $$\begin{aligned}
\label{omegaid}
 \omega_j ~\equiv~\frac{\langle \left( \Delta N_j \right)^2 \rangle}
{\langle N_j\rangle}~=~1~.\end{aligned}$$ For the average multiplicity and variance of, for example, positively charged primordial hadrons (i.e. before resonance decay) one obtains: $$\begin{aligned}
\label{positive}
\langle N_+\rangle ~= ~\sum_{j,~q_j>0}\langle N_j \rangle ~,\qquad
~ \qquad ~ \langle\left( \Delta N_+ \right)^2 \rangle~ =~
\sum_{j,~q_j>0} ~\sum_{k,~q_k>0} ~\langle \Delta N_j \Delta N_k
\rangle~.\end{aligned}$$ 