Another resolution of the configurational entropy paradox as applied to hard spheres

Recently, Ozawa and Berthier [J. Chem. Phys., 2017, 146, 014502] studied the configurational and vibrational entropies $S_c$ and $S_v$ from the relation $S_{tot}=S_c+S_v$ for polydisperse mixtures of spheres. They noticed that because $S_{tot}/N$ shall contain the mixing entropy per particle $k_B s_m$ and $S_v/N$ shall not, the configurational entropy per particle $S_c/N$ shall diverge in the thermodynamic limit for continuous polydispersity due to the diverging $s_m$. They also provided a resolution for this paradox and related problems-it relies on a careful redefining of $S_c$ and $S_v$. Here, we note that the relation $S_{tot}=S_c+S_v$ is essentially a geometric relation in the phase space and shall hold without redefining $S_c$ and $S_v$. We also note that the total entropy per particle $S_{tot}/N$ diverges with $N \to \infty$ with continuous polydispersity as well. The usual way to avoid this and other difficulties with $S_{tot}/N$ is to work with the excess entropy $\Delta S_{tot}$ (relative to the ideal gas of the same polydispersity). Speedy [Mol. Phys., 1998, 95, 169] applied this approach to the relation above and wrote this relation as $\Delta S_{tot}=S_c+\Delta S_v$. This form has flows as well, because $S_v/N$ does not contain the $k_B s_m$ term and the latter is introduced into $\Delta S_v/N$ instead. Here, we suggest that this relation shall actually be written as $\Delta S_{tot}=\Delta_c S_c+\Delta_v S_v$, where $\Delta=\Delta_c+\Delta_v$ while $\Delta_c S_c=S_c-k_B N s_m$ and $\Delta_v S_v=S_v-k_B N[1+\ln(V/\Lambda^d N)+U/N k_B T]$ with $\Lambda$ standing for the de Broglie wavelength. In this form, all the terms per particle are always finite for $N \to \infty$ and continuous when introducing a small polydispersity to a monodisperse system. We also suggest that the Adam-Gibbs and related relations shall in fact contain $\Delta_c S_c/N$ instead of $S_c/N$.