hand, at $v_jx_j\gg 1$ the scaled variance of the VDW gas goes to zero as $\omega_j\cong
(v_jx_j)^{-2}$. In this limit the particle number density is close to its lower limit, $n_j\cong 1/v_j$. Thus, there is no ’free space’ for particle number fluctuations.

 Multi-Component VDW Gas
========================

The partition function of the multi-component VDW gas equals to [@vdw; @vdw1; @crit]: $$\begin{aligned}
\label{Zvdw_HRG}
Z~=~\prod_{j=1}~\sum_{N_j=0}^{\infty}~\frac{1}{N_j!}~\left[(V~-~
\sum_i v_iN_i)~
\phi_j\right]^{N_j}
~\theta(V~-~\sum_i v_iN_i)~.\end{aligned}$$ Similar to Eq. (\[pvdw\]) one finds the equation for the pressure function in the multi-component VDW gas, $$\begin{aligned}
\label{p-mvdw}
p~=~ \sum_j~~T~x_j~,\end{aligned}$$ where $x_j$ is given by Eq. (\[x-j\]).

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

The average multiplicity of $j$th particle is: $$\begin{aligned}
\label{Nj-mvdw}
\langle N_j\rangle ~=~\lambda_j~\frac{\partial}{\partial
\lambda_j} \ln Z~=~\frac{V}{T} ~\lambda_j~\frac{\partial
p}{\partial\lambda_j}~.\end{aligned}$$ Similar to Eq. (\[var\_vdw\]) one obtains: $$\begin{aligned}
\label{NjVDW}
\langle  \Delta N_j \Delta N_k \rangle ~=~ \frac{V}{T} \left[
  \delta_{jk} ~\lambda_j  \frac{\partial p}{\partial
      \lambda_j}~ + ~\lambda_j\lambda_k~ \frac{\partial^2p}{\partial
      \lambda_j \partial \lambda_k}  \right]~.\end{aligned}$$ Calculating the derivative from Eq. (\[p-mvdw\]), $$\begin{aligned}
\label{dp}
\lambda_j~\frac{\partial p}{\partial\lambda_j}~=~
\frac{T~x_j}{1~+~\sum_i~v_i~x_i}~,\end{aligned}$$ and using Eq. (\[Nj-mvdw\]) one finds: $$\begin{aligned}
\label{Nmvdw}
\langle N_j\rangle~=~\frac{V~x_j}{1~+~\sum_i~v_i~x_i}~,~~~~~\end{aligned}$$ which extends Eq. (\[Nvdw\]) to the multi-component VDW gas. Using Eq.(\[dp\]) for the first derivatives of the pressure one finds: $$\label{pjk}
 \lambda_j\lambda_k~\frac{\partial^2 p}{\partial \lambda_j \partial \lambda_k }
~ = ~ - \frac{T~ x_j
  x_k}{\left(1+ \sum_i 