$4\pi r_j^3/3$, multiplied by a factor of 4. A Laplace transform of Eq. (\[Zvdw\]) reads: [$$\begin{aligned}
 \label{Zvdw1}
& \widehat{Z}(s,T,\mu_j)~\equiv~
\int_0^{\infty}dV\exp(-sV)~Z(V,T,\mu_j)\nonumber ~
=~\sum_{N=0}^{\infty}~\frac{
\phi_j^N}{N!}~
\int_{v_jN}^{\infty}dV\exp(-sV)~\left(V~-~v_jN\right)^N~\nonumber \\
 &=~\left[~s~-~\exp(-v_js)~\phi_j\right]^{-1}~.
 \end{aligned}$$]{} The system pressure is defined by the pole-singularity, $s^*$, of the function, $\widehat{Z}$ (\[Zvdw1\]), [$$\begin{aligned}
 p~\equiv ~\lim_{V\rightarrow\infty}~ T~\frac{\ln
Z}{V}~=~T~s^*~,
 \end{aligned}$$]{} and, thus, satisfies the following equation, $$\begin{aligned}
\label{pvdw}
p~=~
\exp\left(- ~\frac{v_j~ p}{T}\right)~T~\phi_j~.\end{aligned}$$ For $v_j=0$, Eq.(\[pvdw\]) is evidently reduced to the ideal gas result, $p_{id}=T\phi_j$. The average particle number in the VDW gas equals to: $$\begin{aligned}
\label{Nav-vdw}
\langle N_j\rangle ~=~ \frac{1}{Z(V,T,\mu_j)} \left( \lambda_j
  \frac{\partial}{\partial \lambda_j} \right)~Z(V,T,\mu_j)
~=\frac{V}{T}~\lambda_j~\frac{\partial p}{\partial\lambda_j}~.\end{aligned}$$ To find the particle number fluctuations one needs to calculate, $$\begin{aligned}
\label{N2}
\langle N_j^2\rangle ~=~
\frac{1}{Z}~\left(\lambda_j\frac{\partial}{\partial \lambda_j}
\right)^2~ Z ~=~ \frac{V}{T}~\lambda_j~\frac{\partial
p}{\partial\lambda_j}~ + ~\left(\frac{V}{T} \right)^2
~\lambda_j^2~\left(\frac{\partial p}{\partial\lambda_j}\right)^2
~+~\frac{V}{T}~\lambda_j^2~ \frac{\partial^2
p}{\partial\lambda_j^2}~.
 \end{aligned}$$ The variance is therefore: $$\begin{aligned}
\label{var_vdw}
\langle \left( \Delta N_j \right)^2 \rangle ~=~  \frac{V}{T} ~
\left[ \lambda_j~ \frac{\partial p}{\partial \lambda_j}~ +~
 \lambda_j^2~ \frac{\partial^2 p}{\partial\lambda_j^2}\right]~.\end{aligned}$$ Thus, one can write the scaled variance as the following, $$\begin{aligned}
\label{omega_j_vdw}
\omega_j ~= ~\frac{\langle \left( \Delta N_j \right)^2
\rangle}{\langle N_j
  \rangle}~ =~ 1~+~\lambda_j~ \frac{\partial^2
    p}{\partial\lambda_j^2}~ \times ~\left(\frac{\partial
    p}{\partial\lambda_j}\right)^{-1}~.\end{aligned}$$ From Eq. (\[pvdw\]) one finds: $$\begin{aligned}
\label{p1lambda}
\lambda_j~\frac{\partial p}{\partial\lambda_j}~
=~T~x_j~-~\lambda_j~v_j~x_j~\frac{\partial p}{\partial\lambda_j}~,\end{aligned}$$ where the notation, [$$\begin{aligned}
  \label{x-j}
x_j~\equiv~\exp\left(-~\frac{v_j~p}{T}\right)~\phi_j~,
 \end{aligned}$$]{} is introduced to simplify 