# Thermodynamic Efficiency of Interactions in Self-Organizing Systems

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## Abstract

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## 1. Introduction

A system is self-organizing if it acquires a spatial, temporal or functional structure without specific interference from the outside. By ‘specific’ we mean that the structure or functioning is not impressed on the system, but that the system is acted upon from the outside in a non-specific fashion. For instance, the fluid which forms hexagons is heated from below in an entirely uniform fashion, and it acquires its specific structure by self-organization.

Self-organization is a process in which pattern at the global level of a system emerges solely from numerous interactions among the lower-level components of the system. Moreover, the rules specifying interactions among the system’s components are executed using only local information, without reference to the global pattern.

## 2. Framework

## 3. Example: Curie–Weiss Model

#### 3.1. Varying External Field, B

#### 3.2. Varying Coupling Strength, J

## 4. Conclusions

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Conflicts of Interest

## References

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**Figure 1.**Derivatives of entropy and free energy as a function of temperature $\theta $ at zero magnetic field. The inset shows how the presence of a small magnetic field smooths out the singularity in $\partial S/\partial B$ at the critical point ${\theta}_{c}=J=1$.

**Figure 2.**Thermodynamic efficiency $\eta (\theta ,\delta B)$ as a function of $\theta $ at several small values of B. The critical point is at ${\theta}_{c}=1.0$ or, equivalently, at $t\equiv (\theta -{\theta}_{c})/{\theta}_{c}=0$. For $t>0$, $\eta $ is undefined at $B=0$. The solid lines $-1/2{t}^{-1}$ for $t<0$ and ${t}^{-1}$ for $t>0$ are analytic expressions for $\eta $ in the vicinity of the critical point.

**Figure 3.**Thermodynamic efficiency $\eta (J,\delta J)$ as a function of J at several small values of B at $\theta =1.0$. The critical point is at ${J}_{c}=1.0$ or, equivalently, at $\mathcal{J}\equiv (J-{J}_{c})/{J}_{c}=0$. For $\mathcal{J}<0$, $\eta $ is undefined at $B=0$. The solid lines $-2{\mathcal{J}}^{-1}$ for $\mathcal{J}<0$ and ${\mathcal{J}}^{-1}$ for $\mathcal{J}>0$ are analytic expressions for $\eta $ in the vicinity of the criticality.

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Nigmatullin, R.; Prokopenko, M.
Thermodynamic Efficiency of Interactions in Self-Organizing Systems. *Entropy* **2021**, *23*, 757.
https://doi.org/10.3390/e23060757

**AMA Style**

Nigmatullin R, Prokopenko M.
Thermodynamic Efficiency of Interactions in Self-Organizing Systems. *Entropy*. 2021; 23(6):757.
https://doi.org/10.3390/e23060757

**Chicago/Turabian Style**

Nigmatullin, Ramil, and Mikhail Prokopenko.
2021. "Thermodynamic Efficiency of Interactions in Self-Organizing Systems" *Entropy* 23, no. 6: 757.
https://doi.org/10.3390/e23060757