# Attraction Controls the Entropy of Fluctuations in Isosceles Triangular Networks

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## Abstract

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## 1. Introduction

## 2. Isosceles Triangular Network Model

## 3. Conclusions

## Acknowledgments

## Author Contributions

## Conflicts of Interest

## References

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**Figure 1.**(

**a**) Straight and (

**b**) maximally zigzagging or bent configuration. Shells, which order is indicated with ${n}_{s}$, are denoted by increasingly darker colors for increasing ${n}_{s}$. Thicker, gray lines correspond to springs of longer (for $\alpha >\pi /3$) or shorter (for $\alpha <\pi /3$) length at rest; (

**c**,

**d**) are the unit cell for the straight- and bent-stripes confgirations, respectively. Numbers associated to particles correspond to the particle positions in tables in Section 2; (

**e**) Parameters associated to every plaquette: a, b, c and $\alpha $.

**Figure 2.**(

**a**) Entropy difference per particle $\Delta s$, between straight- and bent-stripe configurations vs. deformation angle $\alpha $ for increasing numbers of shells ${n}_{s}$ of fluctuating particles; (

**b**) $\Delta s$ vs. $\alpha $ for $n=1,2,3$ particles free to move for harmonic (open symbols) and repulsive harmonic (filled symbols) interactions.

**Figure 3.**Example of straight (

**a**) and bent (

**b**) configuration for $n=1$ particle free to move. The deviation of the central particle from its equilibrium position is described by polar coordinates ($\rho ,\gamma $).

**Figure 4.**Color diagram of $1/f$, as defined in the text, as a function of the angles $\alpha $ and $\gamma $, for harmonic and repulsive harmonic interactions and for straight- and bent-stripes configurations.

**Table 1.**Distances between the neighboring particles and the central particle 0 in the unit cell of the straight-stripe configuration. Particle positions are graphically shown in Figure 1c.

Particles | $\mathit{dx}$ | $\mathit{dy}$ | ${\mathit{dr}}_{\mathbf{0}}$ |
---|---|---|---|

1, 0 | $2a$ | 0 | $2a$ |

2, 0 | a | b | c |

3, 0 | $-a$ | b | c |

4, 0 | $-2a$ | 0 | $2a$ |

5, 0 | $-a$ | $-b$ | c |

6, 0 | a | $-b$ | c |

**Table 2.**Distances between the neighboring particles and 0 and 1 particles in the unit cell of the maximally zigzagging-stripe configuration. Particle positions are graphically shown in Figure 1d.

Particles | $\mathit{dx}$ | $\mathit{dy}$ | ${\mathit{dr}}_{\mathbf{0}}$ |
---|---|---|---|

1, 0 | $2a$ | 0 | $2a$ |

4, 0 | a | b | c |

5, 0 | $2a\left(\right)open="("\; close=")">1-8{\displaystyle \frac{{a}^{2}}{{c}^{2}}}$ | $4b\left(\right)open="("\; close=")">1-{\displaystyle \frac{{b}^{2}}{{c}^{2}}}$ | $2a$ |

6, 0 | $a\left(\right)open="("\; close=")">1-4{\displaystyle \frac{{b}^{2}}{{c}^{2}}}$ | $b\left(\right)open="("\; close=")">3-4{\displaystyle \frac{{b}^{2}}{{c}^{2}}}$ | c |

7, 0 | $-a$ | $-b$ | c |

8, 0 | a | $-b$ | c |

2, 1 | $-a\left(\right)open="("\; close=")">1-4{\displaystyle \frac{{b}^{2}}{{c}^{2}}}$ | $-b\left(\right)open="("\; close=")">3-4{\displaystyle \frac{{b}^{2}}{{c}^{2}}}$ | c |

3, 1 | a | b | c |

4, 1 | $-a$ | b | c |

8, 1 | $-a$ | $-b$ | c |

9, 1 | $-2a\left(\right)open="("\; close=")">1-2{\displaystyle \frac{{b}^{2}}{{c}^{2}}}$ | $-4b\left(\right)open="("\; close=")">1-{\displaystyle \frac{{b}^{2}}{{c}^{2}}}$ | $2a$ |

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Leoni, F.; Shokef, Y.
Attraction Controls the Entropy of Fluctuations in Isosceles Triangular Networks. *Entropy* **2018**, *20*, 122.
https://doi.org/10.3390/e20020122

**AMA Style**

Leoni F, Shokef Y.
Attraction Controls the Entropy of Fluctuations in Isosceles Triangular Networks. *Entropy*. 2018; 20(2):122.
https://doi.org/10.3390/e20020122

**Chicago/Turabian Style**

Leoni, Fabio, and Yair Shokef.
2018. "Attraction Controls the Entropy of Fluctuations in Isosceles Triangular Networks" *Entropy* 20, no. 2: 122.
https://doi.org/10.3390/e20020122