# Full Statistics of Conjugated Thermodynamic Ensembles in Chains of Bistable Units

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## Abstract

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## 1. Introduction

## 2. Configurational Partition Functions and Force-Extension Relations in Gibbs and Helmholtz Ensembles

#### 2.1. The Gibbs Ensemble

#### 2.2. The Helmholtz Ensemble

## 3. Complete Probability Densities in the Gibbs and Helmholtz Ensembles

## 4. Probability Density of the Couple $({\dot{\mathit{x}}}_{\mathit{N}},{\mathit{x}}_{\mathit{N}})$ versus $\mathit{f}$ within the Gibbs Ensemble

## 5. Probability Density of the Couple $(\dot{\mathit{f}},\mathit{f})$ versus ${\mathit{x}}_{\mathit{N}}$ within the Helmholtz Ensemble

## 6. Discussion and Conclusions

## Author Contributions

## Funding

## Conflicts of Interest

## References

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**Figure 1.**Bistable symmetric potential energy of a single domain (blue dashed line) and its approximation by means of four parabolic profiles (red solid lines).

**Figure 2.**Average force-extension curves and average spin variables (plotted by means of dimensionless quantities) for the Gibbs ensemble with $N=5$ and $\frac{k{\ell}^{2}}{{k}_{B}T}$ = 10, 15, 30, 100.

**Figure 3.**Average force-extension curves and average spin variables (plotted by means of dimensionless quantities) for the Helmholtz ensemble with $N=5$ and $\frac{k{\ell}^{2}}{{k}_{B}T}$ = 10, 15, 30, 100.

**Figure 4.**Three-dimensional representation of the Gibbs density ${\varrho}_{G}({x}_{N};f)$ (see Equation (33)) obtained with $N=5$, $\ell =1$ (a.u.), $k=15$ (a.u.) and ${k}_{B}T$ = 0.7, 1.4, 2.1, 2.8 (a.u.).

**Figure 5.**Two-dimensional representation of the Gibbs density ${\varrho}_{G}({x}_{N};f)$ (see Equation (33)) obtained with $N=5$, $\ell =1$ (a.u.), $k=15$ (a.u.) and ${k}_{B}T$ = 0.7, 1.4, 2.1, 2.8 (a.u.).

**Figure 6.**Examples of multimodal curves obtained through the Gibbs density ${\varrho}_{G}({x}_{N};f)$ (see Equation (33)). On the left panel, the two-dimensional representation of the Gibbs density is shown with the cuts corresponding to the curves plotted on the right panel. We used $N=5$, $\ell =1$ (a.u.), $k=15$ (a.u.), ${k}_{B}T$ = 1 (a.u.) and different values of the applied force f, as indicated in the legend.

**Figure 7.**Variance of ${x}_{N}$ obtained by the Gibbs density ${\varrho}_{G}({x}_{N};f)$. As before, we used $N=5$, $\ell =1$ (a.u.), $k=15$ (a.u.) and ${k}_{B}T$ = 0.7, 1.4, 2.1, 2.8 (a.u.).

**Figure 8.**Three-dimensional representation of the Helmholtz density ${\varrho}_{H}(f;{x}_{N})$ (see Equation (54)) obtained with $N=5$, $\ell =1$ (a.u.), $k=15$ (a.u.) and ${k}_{B}T$ = 0.7, 1.4, 2.1, 2.8 (a.u.).

**Figure 9.**Two-dimensional representation of the Helmholtz density ${\varrho}_{H}(f;{x}_{N})$ (see Equation (54)) obtained with $N=5$, $\ell =1$ (a.u.), $k=15$ (a.u.) and ${k}_{B}T$ = 0.7, 1.4, 2.1, 2.8 (a.u.).

**Figure 10.**Examples of monomodal curves obtained through the Helmholtz density ${\varrho}_{H}(f;{x}_{N})$ (see Equation (54)). On the left panel, the two-dimensional representation of the Helmholtz density is shown with the cuts corresponding to the curves plotted on the right panel. We used $N=5$, $\ell =1$ (a.u.), $k=15$ (a.u.), ${k}_{B}T$ = 1 (a.u.) and different values of the prescribed position ${x}_{N}$, as indicated in the legend.

**Figure 11.**Variance of f obtained by the Helmholtz density ${\varrho}_{H}(f;{x}_{N})$. As before, we used $N=5$, $\ell =1$ (a.u.), $k=15$ (a.u.) and ${k}_{B}T$ = 0.7, 1.4, 2.1, 2.8 (a.u.).

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**MDPI and ACS Style**

Benedito, M.; Manca, F.; Giordano, S.
Full Statistics of Conjugated Thermodynamic Ensembles in Chains of Bistable Units. *Inventions* **2019**, *4*, 19.
https://doi.org/10.3390/inventions4010019

**AMA Style**

Benedito M, Manca F, Giordano S.
Full Statistics of Conjugated Thermodynamic Ensembles in Chains of Bistable Units. *Inventions*. 2019; 4(1):19.
https://doi.org/10.3390/inventions4010019

**Chicago/Turabian Style**

Benedito, Manon, Fabio Manca, and Stefano Giordano.
2019. "Full Statistics of Conjugated Thermodynamic Ensembles in Chains of Bistable Units" *Inventions* 4, no. 1: 19.
https://doi.org/10.3390/inventions4010019