应用于硬球的组态熵佯谬的另一种解决方法
Another resolution of the configurational entropy paradox as applied to hard spheres
摘要：
最近，Ozawa和Berthier [J. Chem. Phys.，2017，146，014502]研究了来自多分散球体混合物的关系式$S_{tot}=S_c+S_v$的构型和振动熵$S_c$和$S_v$。他们注意到，由于$S_{tot}/N$应包含每个粒子的混合熵$k_B s_m$，而$S_v/N$不应包含，因此每个粒子的构型熵$S_c/N$应在连续多分散性的热力学极限中发散，这是由于发散的$s_m$。他们还为这个悖论和相关问题提供了一个解决方案-它依赖于对$S_c$和$S_v$的仔细重新定义。这里，我们注意到，关系$S_{tot}=S_c+S_v$本质上是相空间中的几何关系，并且在不重新定义$S_c$和$S_v$的情况下将保持。我们还注意到，每个粒子的总熵$S_{tot}/N$也与具有连续多分散性的$N \to \infty$不同。避免$S_{tot}/N$的这种困难和其他困难的通常方法是使用过量熵$ΔS_{tot}$（相对于相同多分散性的理想气体）。快速[摩尔。物理、1998，95，169]将该方法应用于上述关系，并将该关系写为$ΔS_{tot}=S_c+ΔS_v$。此表单也有流，因为$S_v/N$不包含$k_B s_m$术语，而后者被引入$ΔS_v/N$。在这里，我们建议这个关系实际上应该写为$ΔS_{tot}=Δ_c S_c+Δ_v S_v$，其中$Δ=Δ_c+Δ_v$，而$Δ_c S_c=S_c-k_B N s_m$和$Δ_v S_v=S_v-k_B N[1+\ln(V/Λ^d N)+U/N k_B T]$，其中$Λ$代表德布罗意波长。在这种形式中，对于$N \to \infty$，每个颗粒的所有项总是有限的，并且当将小的多分散性引入单分散体系时是连续的。我们还建议Adam-Gibbs关系和相关关系实际上应该包含$Δ_c S_c/N$而不是$S_c/N$。
英文摘要：
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 $ΔS_{tot}$ (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 $ΔS_{tot}=S_c+ΔS_v$. 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 $ΔS_v/N$ instead. Here, we suggest that this relation shall actually be written as $ΔS_{tot}=Δ_c S_c+Δ_v S_v$, where $Δ=Δ_c+Δ_v$ while $Δ_c S_c=S_c-k_B N s_m$ and $Δ_v S_v=S_v-k_B N[1+\ln(V/Λ^d N)+U/N k_B T]$ with $Λ$ 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 $Δ_c S_c/N$ instead of $S_c/N$.
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