the formulas below. From Eq. (\[p1lambda\]) it follows, $$\begin{aligned}
\label{p1lambda1}
\lambda_j  ~\frac{\partial p}{\partial\lambda_j}~=~
\frac{T~x_j}{1~+~v_jx_j}~,\end{aligned}$$ and one finds: $$\begin{aligned}
\label{Nvdw}
\langle N_j\rangle~=~\frac{V~x_j}{1~+~v_j~ x_j}~.\end{aligned}$$ From Eqs. (\[pvdw\],\[x-j\],\[Nvdw\]) one finds the VDW equation of state, familiar from textbooks, [$$\begin{aligned}
p~\left[V~-~v_j\langle N_j\rangle\right]~=~\langle N_j\rangle~T~.
 \end{aligned}$$]{} At small particle density, $v_jx_j\ll 1$, one finds from Eq. (\[Nvdw\]), $$\begin{aligned}
\label{n1vdw}
\langle N_j\rangle ~\cong V~\phi_j~\left(1~-~2v_j\phi_j\right) ~=~\langle
N_j^{id}\rangle~\left(1~-~2v_jn_j^{id}\right)~,\end{aligned}$$ where $n_j^{id}=\phi_j$ is the ideal gas particle number density. In the opposite limiting case of high particle density, $v_jx_j\gg 1$, one obtains, $$\begin{aligned}
\label{n2vdw}
N_j~\cong ~\frac{V}{v_j}~,\end{aligned}$$ thus, the value of $1/v_j$ is the upper limit for the particle number density in the VDW gas (\[Zvdw\]). From Eq. (\[p1lambda\]) and (\[p1lambda1\]) it follows, $$\begin{aligned}
\label{p2lambda1}
\lambda_j^2~\frac{\partial^2p}{\partial \lambda_j^2}~ =~
-~\frac{Tv_jx_j^2}{(1~+~v_jx_j)^2}~\left[2~-~
\frac{v_jx_j}{1~+~v_jx_j}\right]~.\end{aligned}$$ From Eqs. (\[omega\_j\_vdw\],\[p1lambda1\],\[p2lambda1\]) one then obtains, $$\begin{aligned}
\label{omegavdw}
\omega_j ~
=~\left(1~+~v_j~x_j\right)^{-2}~.\end{aligned}$$ From Eq. (\[omegavdw\]) it follows that the scaled variance in the VDW gas is always smaller than the ideal gas value (\[omegaid\]) $\omega_j = 1$, i.e. the non-zero proper volume $v_j$ suppresses the particle number fluctuations. At small particle number densities, $v_jx_j \ll 1$, the particle number fluctuations in the VDW gas are approximately equal to those in an ideal gas (\[omegaid\]). The Eq. (\[omegavdw\]) gives, $\omega_j~\cong~1~-~2v_j\phi_j$ . On the other 