Residual Multiparticle Entropy for a Fractal Fluid of Hard Spheres

Abstract

The residual multiparticle entropy (RMPE) of a fluid is defined as the difference, $Δs$, 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.; López 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 $ϕ$. It is observed that, for any given dimensionality $d$, the RMPE takes negative values for small densities, reaches a negative minimum $Δs_{\text{min}}$ at a packing fraction $ϕ_{\text{min}}$, and then rapidly increases, becoming positive beyond a certain packing fraction $ϕ_0$. Interestingly, while both $ϕ_{\text{min}}$ and $ϕ_0$ monotonically decrease as dimensionality increases, the value of $Δs_{\text{min}}$ exhibits a nonmonotonic behavior, reaching an absolute minimum at a fractional dimensionality $d\simeq 2.38$. A plot of the scaled RMPE $Δs/|Δs_{\text{min}}|$ shows a quasiuniversal behavior in the region $-0.14\lesssimϕ-ϕ_0\lesssim 0.02$.