(\[corr-id\]) simplifies to: [$$\begin{aligned}
\langle \Delta N_j \Delta N_k \rangle ~=~ \langle N_j^* \rangle
~\delta_{jk} ~+~\sum_{R} ~\langle N_{R} ~\rangle  \langle  n_j n_k
\rangle_R~. \label{corr-id-B}
 \end{aligned}$$]{} From Eqs. (\[av-id\]) and (\[corr-id-B\]) it immediately follows, [$$\begin{aligned}
  \label{omega-id-1}
\omega_j~=~\frac{\langle N_j^* \rangle ~ ~+~\sum_{R} ~\langle
N_{R} ~\rangle  \langle  n_j^2 \rangle_R}{\langle N_j^* \rangle ~
~+~\sum_{R} ~\langle N_{R} ~\rangle  \langle  n_j \rangle_R}~.
 \end{aligned}$$]{} Thus, $\omega_j\ge 1$ as a consequence of the inequality $(n_{j,R}^r)^2\ge n_{j,R}^r$ , i.e. the scaled variance of $j$th hadron exceeds 1 due to the presence of resonances decaying into more than one $j$th hadron. Using Eqs. (\[av-id\]) and (\[corr-id-B\]) for average yields and correlators one finds from (\[positive\]), [$$\begin{aligned}
\omega_+~=~\frac{\sum_{j,~q_j>o}~[~\langle N_j^* \rangle ~
~+~\sum_{k,~q_k>0}\sum_{R} ~\langle N_{R} ~\rangle
\langle n_jn_k \rangle_R~]}{\sum_{j,~q_j>o}~[~\langle N_j^* \rangle ~
~+~\sum_{R} ~\langle N_{R} ~\rangle  \langle  n_j \rangle_R~]}~,
 \end{aligned}$$]{} and similar expressions for final state $\omega_-$ and $\omega_{ch}$ with the summation over $q_j<0,~q_k<0$ and $q_j\neq
0,~q_k\neq 0$, respectively.

One-Component VDW Gas 
======================

The VDW excluded volume procedure gives the GCE partition function of one component Boltzmann gas [@vdw]: [$$\begin{aligned}
 \label{Zvdw}
Z(V,T,\mu_j)~=~\sum_{N=0}^{\infty}~\frac{1}{N!}~\left[
(V~-~v_jN)~\phi_j~\right]^N
~\theta(V-v_jN)~,
 \end{aligned}$$]{} where $v_j$ is the proper volume of the $j$th particle. For the ‘hard sphere’ particles with radius $r_j$ the volume parameter $v_j$ equals the ‘hard-core particle volume’, 