2017/06/30 by Vasili Baranau, Ulrich Tallarek
Engineering · Materials Science · Physics and Astronomy · #Boltzmann's entropy formula #Configuration entropy #Entropy (arrow of time) #Ideal gas #Limit (mathematics) #Material Dynamics and Properties #Phase Equilibria and Thermodynamics #Phase space #SPHERES #Statistical Mechanics and Entropy #Well-defined #Work (physics) #cond-mat.stat-mech
paper · pdf · doi:10.1063/1.4999483
published as J. Chem. Phys. 147, 224503 (2017)
openalex created_date 2017/07/14 · openalex publication_date 2017/12/11 · arxiv created 2017/12/19 · arxiv updated 2017/12/20 · openalex updated_date 2026/08/05
Ozawa and Berthier [J. Chem. Phys. 146, 014502 (2017)] recently studied the configurational and vibrational entropies Sconf and Svib from the relation Stot = Sconf + Svib for polydisperse mixtures of spheres. They noticed that because the total entropy per particle Stot/N shall contain the mixing entropy per particle kBsmix and Svib/N shall not, the configurational entropy per particle Sconf/N shall diverge in the thermodynamic limit for continuous polydispersity due to the diverging smix. They also provided a resolution for this paradox and related problems—it relies on a careful redefining of Sconf and Svib. Here, we note that the relation Stot = Sconf + Svib is essentially a geometric relation in the phase space and shall hold without redefining Sconf and Svib. We also note that Stot/N diverges with N → ∞ with continuous polydispersity as well. The usual way to avoid this and other difficulties with Stot/N is to work with the excess entropy ΔStot (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 ΔStot = Sconf + ΔSvib. This form has flaws as well because Svib/N does not contain the kBsmix term and the latter is introduced into ΔSvib/N instead. Here, we suggest that this relation shall actually be written as ΔStot = ΔcSconf + ΔvSvib, where Δ = Δc + Δv, while ΔcSconf = Sconf − kBNsmix and ΔvSvib=Svib−kBN1+lnVΛdN+UNkBT 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 ΔcSconf/N instead of Sconf/N.