this solution might include the main effects of polydispersity. This is in fact what happens, as recently shown [Fantoni06]{} by exploiting a general perturbation theory due to Evans [Evans01]{}. The main idea is that, for weak polydispersity, size-distributions are narrowly peaked around a mean reference value ($%
\sigma _{0}$ in the present case), and hence all quantities such as$$\delta _{i}=\frac{\sigma _{i}-\sigma _{0}}{\sigma _{0}}\ll 1\text{,}
\label{eq10}$$are small. Therefore one might expand both $\epsilon _{ij},$ and all quantities appearing in $\Lambda _{ij},$ in powers of $\delta _{i}$. Similar expansion can be performed in the free energy, and hence all thermodynamic quantities can be computed. The entire procedure is described in detail in Refs. [@Sollich02; @Fantoni06; @Evans01]. Here we just summarize the main results.

The approximate range of validity of the perturbation expansion can be envisaged by considering the polydisperse HS case where the ‘exact’ BMCSL approximation [@Mansoori71] can be compared with the corresponding perturbative solution based on the one-component ($s=0$) counterpart. This is reported in the top-left part of Figure \[fig3\], where the quantity $\beta P v_0$ ($v_0=\pi \sigma_0^3/6$) is plotted against the packing fraction $\eta$ for increasing values of polydispersity. It is apparent how the perturbative solution remains rather close to 