# Fluctuations, Finite-Size Effects and the Thermodynamic Limit in Computer Simulations: Revisiting the Spatial Block Analysis Method

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## Abstract

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## 1. Introduction

## 2. Boundary and Ensemble Finite-Size Effects

## 3. Finite-Size Ornstein–Zernike Integral Equation

## 4. Mixtures

## 5. Summary and Concluding Remarks

## Acknowledgments

## Author Contributions

## Conflicts of Interest

## Abbreviations

SBA | Spatial block analysis |

TSLJ | Truncated and shifted Lennard–Jones |

MD | Molecular dynamics |

TL | Thermodynamic limit |

PBCs | Periodic boundary conditions |

OZ | Ornstein–Zernike |

KB | Kirkwood–Buff |

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**Figure 1.**Snapshot of the simulation box for a system of particles interacting via a TSLJ potential at density $\rho {\sigma}^{3}=0.1$ and temperature ${k}_{\mathrm{B}}T=1.2\u03f5$. In this particular example, a box of linear size ${L}_{0}$ has been subdivided into blocks of linear dimension $L={L}_{0}/5$ as indicated by the different color shades. The figure has been rendered with the Visual Molecular Dynamics (VMD) program [31].

**Figure 2.**Fluctuations of the number of particles ${\chi}_{T}(L,{L}_{0})$ as a function of $\sigma /L$ for systems described by a TSLJ potential with ${r}_{c}/\sigma ={2}^{1/6}$. Data corresponding to system sizes ${N}_{0}={10}^{4},{10}^{5}$ and ${10}^{6}$ are presented using red squares, blue triangles and green circles, respectively. The vertical lines indicate the limit $\sigma /{L}_{0}$ at which fluctuations become zero. The black horizontal dashed line indicates the value ${\chi}_{T}^{\infty}=\rho {k}_{\mathrm{B}}T{\kappa}_{T}=0.0295$ with ${\kappa}_{T}$ the bulk compressibility obtained with the method described in [6].

**Figure 3.**Fluctuations of the number of particles ${\chi}_{T}(L,{L}_{0})$ as a function of the ratio $L/{L}_{0}$ for systems described by a TSLJ potential with ${r}_{c}/\sigma ={2}^{1/6}$. Results corresponding to systems of ${N}_{0}={10}^{5}$ particles with densities $\rho {\sigma}^{3}=0.1,\phantom{\rule{0.166667em}{0ex}}0.2$ and 0.3 are presented using red squares, blue triangles and green circles, respectively. The theoretical prediction presented in the text is plotted using the corresponding value for ${\chi}_{T}^{\infty}$, obtained as described in [6], and solid-line curves with the same color code.

**Figure 4.**Fluctuations of the number of particles ${\chi}_{T}(L,{L}_{0})$ as a function of the ratio $L/{L}_{0}$ for systems described by a TSLJ potential with ${r}_{c}/\sigma ={2}^{1/6}$. Results corresponding to sizes ${N}_{0}={10}^{4},\phantom{\rule{0.166667em}{0ex}}{10}^{5}$ and ${10}^{6}$, with density $\rho {\sigma}^{3}=0.864$, using red squares, blue triangles and green circles, respectively. The theoretical prediction presented in the text is plotted as the black dashed curve using ${\chi}_{T}^{\infty}=0.0295$.

**Figure 5.**Scaled fluctuations of the number of particles $\lambda {\chi}_{T}(L,{L}_{0})$, minus $c/{L}_{0}$, versus the ratio $\lambda =L/{L}_{0}$ for systems described by a TSLJ potential with ${r}_{c}/\sigma ={2}^{1/6}$. Results corresponding to sizes ${N}_{0}={10}^{4},\phantom{\rule{0.166667em}{0ex}}{10}^{5}$ and ${10}^{6}$, with density $\rho {\sigma}^{3}=0.864$, using red squares, blue triangles and green circles, respectively. The theoretical prediction Equation (7) presented in the text is plotted as the black solid curve using ${\chi}_{T}^{\infty}=0.0295$ and $c=0.415\sigma $.

**Figure 6.**Ratio ${\chi}_{T}^{\infty}={\kappa}_{T}/{\kappa}_{T}^{IG}$ at ${k}_{\mathrm{B}}T=1.2\u03f5$ as a function of the density for systems described by a TSLJ potential with ${r}_{c}/\sigma ={2}^{1/6}$, with ${\kappa}_{T}^{IG}={(\rho {k}_{\mathrm{B}}T)}^{-1}$ the isothermal compressibility of the ideal gas. The red curve is a guide to the eye.

**Figure 7.**Excess chemical potential ${\mu}^{ex}/\u03f5$ at ${k}_{\mathrm{B}}T=1.2\u03f5$ as a function of the density for systems described by a TSLJ potential with ${r}_{c}/\sigma ={2}^{1/6}$. Red squares indicate the data obtained with the spatially-resolved thermodynamic integration (SPARTIAN) method [36], and the blue triangles are the data points obtained with the method outlined in the text.

**Figure 8.**Reduced fluctuations as a function of $\lambda $ for systems described by a TSLJ potential with ${r}_{c}/\sigma =2.5$ with density $\rho {\sigma}^{3}=0.3$ at temperatures ${k}_{\mathrm{B}}T=2.00\u03f5$ and $1.15\u03f5$. For the latter case, it is apparent that the contribution proportional to ${\lambda}^{-1}$ is not negligible. The inset shows the full range $0<\lambda <1$. The black curves are the result of fitting the data to Equation (22).

**Figure 9.**Bulk isothermal compressibility ${\kappa}_{T}$ as a function of the density $\rho $ at ${k}_{\mathrm{B}}T=1.15\u03f5$ (red circles) and ${k}_{\mathrm{B}}T=2.00\u03f5$ (green squares) for systems described by a TSLJ potential with ${r}_{c}/\sigma =2.5$. The vertical black line indicates the location of the critical density $\rho {\sigma}^{3}=0.319$ [38].

**Figure 10.**Scaled finite-size Kirkwood–Buff integrals $\lambda {G}_{ij}(\lambda )$ as a function of $\lambda $ for different mole fractions: (

**a**) ${x}_{A}=0.20$; (

**b**) ${x}_{A}=0.30$; (

**c**) ${x}_{A}=0.50$ and (

**d**) ${x}_{A}=0.80$, for mixtures described by a TSLJ potential with ${r}_{c}/\sigma ={2}^{1/6}$. For clarity, only the cases ${G}_{AA}$ (red squares) and ${G}_{AB}$ (green circles) are plotted. In the asymptotic case $\lambda \to 1$, ${G}_{AB}\to 0$ and ${G}_{AA}\to 1/{\rho}_{A}$, as indicated by the horizontal green and red lines, respectively. The black curves correspond to Equation (26) with ${G}_{ij}^{\infty}$ and ${\alpha}_{ij}$ obtained from a simple regression analysis in the interval $\lambda <0.3$.

**Figure 11.**Isothermal compressibility at ${k}_{\mathrm{B}}T=1.20\u03f5$ and $P{\sigma}^{3}/\u03f5=9.8$ as a function of the mole fraction of type-A particles ${x}_{A}$ for mixtures described by a TSLJ potential with ${r}_{c}/\sigma ={2}^{1/6}$. The horizontal black lines indicate the compressibility for a pure system of type-A particles ${\kappa}_{T}^{A}\u03f5/{\sigma}^{3}=0.012(1)$ and for a pure system of type-B particles ${\kappa}_{T}^{B}\u03f5/{\sigma}^{3}=0.0281(8)$. The red line is a guide to the eye. The ideal case corresponds to ${\kappa}_{T}=(1-{x}_{A}){\kappa}_{T}^{B}+{x}_{A}{\kappa}_{T}^{A}$.

**Figure 12.**Excess chemical potential of type-A particles as a function of the mole fraction ${x}_{A}$ for mixtures described by a TSLJ potential with ${r}_{c}/\sigma ={2}^{1/6}$ at ${k}_{\mathrm{B}}T=1.2\u03f5$ and $P{\sigma}^{3}/\u03f5=9.8$. Data points obtained with the method in [36], in particular for ${x}_{A}=0.3$, are used as a reference for the data points obtained with Equations (30) and (31).

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**MDPI and ACS Style**

Heidari, M.; Kremer, K.; Potestio, R.; Cortes-Huerto, R.
Fluctuations, Finite-Size Effects and the Thermodynamic Limit in Computer Simulations: Revisiting the Spatial Block Analysis Method. *Entropy* **2018**, *20*, 222.
https://doi.org/10.3390/e20040222

**AMA Style**

Heidari M, Kremer K, Potestio R, Cortes-Huerto R.
Fluctuations, Finite-Size Effects and the Thermodynamic Limit in Computer Simulations: Revisiting the Spatial Block Analysis Method. *Entropy*. 2018; 20(4):222.
https://doi.org/10.3390/e20040222

**Chicago/Turabian Style**

Heidari, Maziar, Kurt Kremer, Raffaello Potestio, and Robinson Cortes-Huerto.
2018. "Fluctuations, Finite-Size Effects and the Thermodynamic Limit in Computer Simulations: Revisiting the Spatial Block Analysis Method" *Entropy* 20, no. 4: 222.
https://doi.org/10.3390/e20040222