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

Ozawa and Berthier [J. Chem. Phys. 146, 014502 (2017)] recently studied the configurational and vibrational entropies S_conf and S_vib from the relation S_tot = S_conf + S_vib for polydisperse mixtures of spheres. They noticed that because the total entropy per particle S_tot/N shall contain the mixing entropy per particle k_Bs_mix and S_vib/N shall not, the configurational entropy per particle S_conf/N shall diverge in the thermodynamic limit for continuous polydispersity due to the diverging s_mix. They also provided a resolution for this paradox and related problems-it relies on a careful redefining of S_conf and S_vib. Here, we note that the relation S_tot = S_conf + S_vib is essentially a geometric relation in the phase space and shall hold without redefining S_conf and S_vib. We also note that S_tot/N diverges with N → ∞ 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 ΔS_tot (relative to the ideal gas of the same polydispersity). Speedy applied this approach to the relation above in his work [Mol. Phys. 95, 169 (1998)] and wrote this relation as ΔS_tot = S_conf + ΔS_vib. This form has flaws as well because S_vib/N does not contain the k_Bs_mix term and the latter is introduced into ΔS_vib/N instead. Here, we suggest that this relation shall actually be written as ΔS_tot = Δ_cS_conf + Δ_vS_vib, where Δ = Δ_c + Δ_v, while Δ_cS_conf = S_conf - k_BNs_mix and Δ_vS_vib=S_vib-k_BN1+lnVΛ^dN+UNk_BT with N, V, T, U, d, and Λ standing for the number of particles, volume, temperature, internal energy, dimensionality, and de Broglie wavelength, respectively. In this form, all the terms per particle are always finite for N → ∞ 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 Δ_cS_conf/N instead of S_conf/N.