Residual Multiparticle Entropy for a Fractal Fluid of Hard Spheres

TL;DR

This study investigates the residual multiparticle entropy (RMPE) in fractal hard-sphere fluids across dimensions 1<d<3, revealing nonmonotonic behavior and near-universal scaling near structural transition points.

Contribution

It applies a recent fractional-dimensional hard-sphere model to analyze RMPE behavior, uncovering nonmonotonic trends and a fractional dimension where RMPE's minimum occurs.

Findings

RMPE becomes positive beyond a certain packing fraction for all dimensions.

The minimum RMPE occurs at a fractional dimension around 2.38.

Scaled RMPE shows quasiuniversal behavior near transition points.

Abstract

The residual multiparticle entropy (RMPE) of a fluid is defined as the difference, $\Delta s$, between the excess entropy per particle (relative to an ideal gas with the same temperature and density), $s_\text{ex}$, and the pair-correlation contribution, $s_2$. Thus, the RMPE represents the net contribution to $s_\text{ex}$ due to spatial correlations involving three, four, or more particles. A heuristic `ordering' criterion identifies the vanishing of the RMPE as an underlying signature of an impending structural or thermodynamic transition of the system from a less ordered to a more spatially organized condition (freezing is a typical example). Regardless of this, the knowledge of the RMPE is important to assess the impact of non-pair multiparticle correlations on the entropy of the fluid. Recently, an accurate and simple proposal for the thermodynamic and structural properties of a hard-sphere fluid in fractional dimension $1<d<3$ has been proposed [Santos, A.; L\'opez de Haro, M. \emph{Phys. Rev. E} \textbf{2016}, \emph{93}, 062126]. The aim of this work is to use this approach to evaluate the RMPE as a function of both $d$ and the packing fraction $\phi$. It is observed that, for any given dimensionality $d$, the RMPE takes negative values for small densities, reaches a negative minimum $\Delta s_{\text{min}}$ at a packing fraction $\phi_{\text{min}}$, and then rapidly increases, becoming positive beyond a certain packing fraction $\phi_0$. Interestingly, while both $\phi_{\text{min}}$ and $\phi_0$ monotonically decrease as dimensionality increases, the value of $\Delta s_{\text{min}}$ exhibits a nonmonotonic behavior, reaching an absolute minimum at a fractional dimensionality $d\simeq 2.38$. A plot of the scaled RMPE $\Delta s/|\Delta s_{\text{min}}|$ shows a quasiuniversal behavior in the region $-0.14\lesssim\phi-\phi_0\lesssim 0.02$.