# Residual Multiparticle Entropy for a Fractal Fluid of Hard Spheres

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## Abstract

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**2016**, 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 $\varphi $. It is observed that, for any given dimensionality d, the RMPE takes negative values for small densities, reaches a negative minimum $\Delta {s}_{\mathrm{min}}$ at a packing fraction ${\varphi}_{\mathrm{min}}$, and then rapidly increases, becoming positive beyond a certain packing fraction ${\varphi}_{0}$. Interestingly, while both ${\varphi}_{\mathrm{min}}$ and ${\varphi}_{0}$ monotonically decrease as dimensionality increases, the value of $\Delta {s}_{\mathrm{min}}$ exhibits a nonmonotonic behavior, reaching an absolute minimum at a fractional dimensionality $d\simeq 2.38$. A plot of the scaled RMPE $\Delta s/|\Delta {s}_{\mathrm{min}}|$ shows a quasiuniversal behavior in the region $-0.14\lesssim \varphi -{\varphi}_{0}\lesssim 0.02$.

## 1. Introduction

## 2. Methods

#### 2.1. General Relations

#### 2.2. Fractal Hard Spheres

## 3. Results and Discussion

## 4. Conclusions

## Author Contributions

## Funding

## Acknowledgments

## Conflicts of Interest

## Abbreviations

RMPE | Residual Multiparticle Entropy |

MC | Monte Carlo |

PY | Percus–Yevick |

## References

- Wong, P.Z.; Cao, Q.Z. Correlation function and structure factor for a mass fractal bounded by a surface fractal. Phys. Rev. B
**1992**, 45, 7627–7632. [Google Scholar] [CrossRef] - Kurzidim, J.; Coslovich, D.; Kahl, G. Single-Particle and Collective Slow Dynamics of Colloids in Porous Confinement. Phys. Rev. Lett.
**2009**, 103, 138303. [Google Scholar] [CrossRef] [PubMed] - Kim, K.; Miyazaki, K.; Saito, S. Slow dynamics, dynamic heterogeneities, and fragility of supercooled liquids confined in random media. J. Phys. Condens. Matter
**2011**, 23, 234123. [Google Scholar] [CrossRef] [PubMed] [Green Version] - Skinner, T.O.E.; Schnyder, S.K.; Aarts, D.G.A.L.; Horbach, J.; Dullens, R.P.A. Localization Dynamics of Fluids in Random Confinement. Phys. Rev. Lett.
**2013**, 111, 128301. [Google Scholar] [CrossRef] [PubMed] - Heinen, M.; Schnyder, S.K.; Brady, J.F.; Löwen, H. Classical Liquids in Fractal Dimension. Phys. Rev. Lett.
**2015**, 115, 097801. [Google Scholar] [CrossRef] [PubMed] [Green Version] - Santos, A.; López de Haro, M. Radial distribution function for hard spheres in fractal dimensions: A heuristic approximation. Phys. Rev. E
**2016**, 93, 062126. [Google Scholar] [CrossRef] [PubMed] - Nettleton, R.E.; Green, M.S. Expression in Terms of Molecular Distribution Functions for the Entropy Density in an Infinite System. J. Chem. Phys.
**1958**, 29, 1365–1370. [Google Scholar] [CrossRef] - Baranyai, A.; Evans, D.J. Direct entropy calculation from computer simulation of liquids. Phys. Rev. A
**1989**, 40, 3817–3822. [Google Scholar] [CrossRef] - Giaquinta, P.V.; Giunta, G. About entropy and correlations in a fluid of hard spheres. Phys. A
**1992**, 187, 145–158. [Google Scholar] [CrossRef] - Giaquinta, P.V. Entropy and Ordering of Hard Rods in One Dimension. Entropy
**2008**, 10, 248–260. [Google Scholar] [CrossRef] - Krekelberg, W.P.; Shen, V.K.; Errington, J.R.; Truskett, T.M. Residual multiparticle entropy does not generally change sign near freezing. J. Chem. Phys.
**2008**, 128, 161101. [Google Scholar] [CrossRef] [PubMed] - Krekelberg, W.P.; Shen, V.K.; Errington, J.R.; Truskett, T.M. Response to “Comment on ‘Residual multiparticle entropy does not generally change sign near freezing’ ” [J. Chem. Phys. 130, 037101 (2009)]. J. Chem. Phys.
**2009**, 130, 037102. [Google Scholar] [CrossRef] [Green Version] - Giaquinta, P.V. Comment on “Residual multiparticle entropy does not generally change sign near freezing” [J. Chem. Phys. 128, 161101 (2008)]. J. Chem. Phys.
**2009**, 130, 037101. [Google Scholar] [CrossRef] [PubMed] [Green Version] - Saija, F.; Pastore, G.; Giaquinta, P.V. Entropy and Fluid-Fluid Separation in Nonadditive Hard-Sphere Mixtures. J. Phys. Chem. B
**1998**, 102, 10368–10371. [Google Scholar] [CrossRef] - Costa, D.; Micali, F.; Saija, F.; Giaquinta, P.V. Entropy and Correlations in a Fluid of Hard Spherocylinders: The Onset of Nematic and Smectic Order. J. Phys. Chem. B
**2002**, 106, 12297–12306. [Google Scholar] [CrossRef] [Green Version] - Saija, F.; Saitta, A.M.; Giaquinta, P.V. Statistical entropy and density maximum anomaly in liquid water. J. Chem. Phys.
**2003**, 119, 3587–3589. [Google Scholar] [CrossRef] - Banerjee, A.; Nandi, M.K.; Sastry, S.; Bhattacharyya, S.M. Determination of onset temperature from the entropy for fragile to strong liquids. J. Chem. Phys.
**2017**, 147, 024504. [Google Scholar] [CrossRef] [PubMed] [Green Version] - Santos, A. A Concise Course on the Theory of Classical Liquids. Basics and Selected Topics. In Lecture Notes in Physics; Springer: New York, NY, USA, 2016; Volume 923. [Google Scholar]
- Lemson, G.; Sanders, R.H. On the use of the conditional density as a description of galaxy clustering. Mon. Not. R. Astron. Soc.
**1991**, 252, 319–328. [Google Scholar] [CrossRef] [Green Version] - Barker, J.A.; Henderson, D. What is “liquid”? Understanding the states of matter. Rev. Mod. Phys.
**1976**, 48, 587–671. [Google Scholar] [CrossRef] - Percus, J.K.; Yevick, G.J. Analysis of Classical Statistical Mechanics by Means of Collective Coordinates. Phys. Rev.
**1958**, 110, 1–13. [Google Scholar] [CrossRef] - Henderson, D. A simple equation of state for hard discs. Mol. Phys.
**1975**, 30, 971–972. [Google Scholar] [CrossRef] - Wertheim, M.S. Exact solution of the Percus-Yevick integral equation for hard spheres. Phys. Rev. Lett.
**1963**, 10, 321–323. [Google Scholar] [CrossRef] - Thiele, E. Equation of state for hard spheres. J. Chem. Phys.
**1963**, 39, 474–479. [Google Scholar] [CrossRef] - Alder, B.J.; Wainwright, T.E. Phase Transition in Elastic Disks. Phys. Rev.
**1962**, 127, 359–361. [Google Scholar] [CrossRef] - Thorneywork, A.L.; Abbott, J.L.; Aarts, D.G.A.L.; Dullens, R.P.A. Two-Dimensional Melting of Colloidal Hard Spheres. Phys. Rev. Lett.
**2017**, 118, 158001. [Google Scholar] [CrossRef] [PubMed] - Alder, B.J.; Wainwright, T.E. Phase Transition for a Hard Sphere System. J. Chem. Phys.
**1957**, 27, 1208–1209. [Google Scholar] [CrossRef] - Fernández, L.A.; Martín-Mayor, V.; Seoane, B.; Verrocchio, P. Equilibrium Fluid-Solid Coexistence of Hard Spheres. Phys. Rev. Lett.
**2012**, 108, 165701. [Google Scholar] [CrossRef] [PubMed] - Robles, M.; López de Haro, M.; Santos, A. Note: Equation of state and the freezing point in the hard-sphere model. J. Chem. Phys.
**2014**, 140, 136101. [Google Scholar] [CrossRef] [PubMed] [Green Version] - Vannimenus, J.; Nadal, J.P.; Martin, H. On the spreading dimension of percolation and directed percolation clusters. J. Phys. A: Math. Gen.
**1984**, 17, L351–L356. [Google Scholar] [CrossRef] - ben-Avraham, D.; Havlin, S. Diffusion and Reactions in Fractal and Disordered Systems; Cambridge University Press: Cambridge, UK, 2016. [Google Scholar]

**Figure 1.**(

**a**) Plot of ${s}_{\mathrm{ex}}(\varphi )$ (solid lines) and ${s}_{2}(\varphi )$ (dashed lines) for dimensions $d=1$, $1.5$, 2, $2.5$ and 3. The circles indicate the points where ${s}_{\mathrm{ex}}(\varphi )$ and ${s}_{2}(\varphi )$ cross; (

**b**) Plot of $\Delta s(\varphi )={s}_{\mathrm{ex}}(\varphi )-{s}_{2}(\varphi )$ for $d=1$, $1.5$, 2, $2.5$ and 3. The triangles indicate the location of the minima and the circles indicate the packing fractions ${\varphi}_{0}$ where $\Delta s=0$.

**Figure 2.**(

**a**) Plot of $\Delta {s}_{min}$ as a function of d. The circle and the arrow indicate the location of the minimum at $d\simeq 2.38$; (

**b**) Plot of ${\varphi}_{0}$ (solid line), ${\varphi}_{min}$ (dashed line), and the difference ${\varphi}_{0}-{\varphi}_{min}$ (dotted line) as functions of d. The horizontal solid line signals the value ${\varphi}_{0}-{\varphi}_{min}=0.109$. The circles represent the values $\varphi =0.68$ at $d=2$ and $\varphi =0.49$ at $d=3$ corresponding to the fluid-hexatic [25,26] and fluid-crystal [27,28,29] transitions, respectively.

**Figure 3.**(

**a**) Plot of the scaled RMPE $\Delta s/|\Delta {s}_{min}|$ as a function of the difference $\varphi -{\varphi}_{0}$ for dimensions $d=1$, $1.5$, 2, $2.5$ and 3; (

**b**) Magnification of the framed region of (

**a**). The light thick line represents the formula given by Equation (27).

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**MDPI and ACS Style**

Santos, A.; Saija, F.; Giaquinta, P.V.
Residual Multiparticle Entropy for a Fractal Fluid of Hard Spheres. *Entropy* **2018**, *20*, 544.
https://doi.org/10.3390/e20070544

**AMA Style**

Santos A, Saija F, Giaquinta PV.
Residual Multiparticle Entropy for a Fractal Fluid of Hard Spheres. *Entropy*. 2018; 20(7):544.
https://doi.org/10.3390/e20070544

**Chicago/Turabian Style**

Santos, Andrés, Franz Saija, and Paolo V. Giaquinta.
2018. "Residual Multiparticle Entropy for a Fractal Fluid of Hard Spheres" *Entropy* 20, no. 7: 544.
https://doi.org/10.3390/e20070544