\ \ \ \ \ \ \ \ \ \
\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ elsewhere,}%
\end{array}%
\right.   \label{eq2}$$where the expressions for the parameters $a_{i}$ and $b_{i}$ may be found in [@Gazzillo00], while the quantity $$\Lambda _{ij}=t_{ij}\ y_{ij}(\sigma _{ij})\ \sigma _{ij}^{2}\ ,  \label{eq3}$$which depends on the cavity function at contact $y_{ij}(\sigma _{ij})$, must be solution of the following system of quadratic equations $$\Lambda _{ij}=\alpha _{ij}+\beta _{ij}\sum_{m}x_{m}\left[ \Lambda
_{im}\Lambda _{jm}-\frac{1}{2}\left( \Lambda _{im}\Gamma _{mj}+\Lambda
_{jm}\Gamma _{mi}\right) \right] ~,~~~i,j=1,2,\ldots ,M  \label{eq4}$$Here, $x_{m}$ is the molar fraction of the $m$-th species ($m=1,\ldots ,M$), while $\alpha _{ij}=t_{ij}\ y_{ij}^{\mathrm{HS-PY}}(\sigma _{ij})\ \sigma
_{ij}^{2}$,   $\beta _{ij}=12\rho \ t_{ij}\sigma _{ij}$ ($\rho $ is the total number density), and $\Gamma _{ij}=\sigma _{i}\sigma _{j}/(1-\eta )$, with $\eta $ being the packing fraction [@Fantoni06]. The solution of these equations for $\left\{ \Lambda _{ij}\right\} $ is the real bottleneck of the multi-component PY result, as mentioned in the Introduction: for large $M$ (and in particular for $M\rightarrow \infty $ ) this calculation is next to impossible, neither analytically nor numerically.

As a consequence, although the PY closure is commonly believed to 