# Thermodynamic Efficiency of Interactions in Self-Organizing Systems

^{1}

^{2}

^{*}

## Abstract

**:**

## 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

- Haken, H. Information and Self-Organization: A Macroscopic Approach to Complex Systems; Springer: Berlin/Heidelberg, Germany, 1988. [Google Scholar]
- Haken, H. Synergetics, an Introduction: Nonequilibrium Phase Transitions and Self-Organization in Physics, Chemistry, and Biology, 3rd ed.; Springer: New York, NY, USA, 1983. [Google Scholar]
- Camazine, S.; Deneubourg, J.L.; Franks, N.R.; Sneyd, J.; Theraulaz, G.; Bonabeau, E. Self-Organization in Biological Systems; Princeton University Press: Princeton, NJ, USA, 2001. [Google Scholar]
- Bonabeau, E.; Theraulaz, G.; Deneubourg, J.L.; Camazine, S. Self-organisation in social insects. Trends Ecol. Evol.
**1997**, 12, 188–193. [Google Scholar] [CrossRef][Green Version] - Polani, D. Measuring self-organization via observers. In Advances in Artificial Life, Proceedings of the 7th European Conference on Artificial Life (ECAL), Dortmund, Germany, 14–17 September 2003; Banzhaf, W., Christaller, T., Dittrich, P., Kim, J.T., Ziegler, J., Eds.; Springer: Berlin/Heidelberg, Germany, 2003; pp. 667–675. [Google Scholar]
- Kauffman, S.A. Investigations; Oxford University Press: Oxford, UK, 2000. [Google Scholar]
- Carteret, H.; Rose, K.; Kauffman, S. Maximum Power Efficiency and Criticality in Random Boolean Networks. Phys. Rev. Lett.
**2008**, 101, 218702. [Google Scholar] [CrossRef][Green Version] - Barato, A.; Hartich, D.; Seifert, U. Efficiency of cellular information processing. New J. Phys.
**2014**, 16, 103024. [Google Scholar] [CrossRef][Green Version] - Kempes, C.; Wolpert, D.; Cohen, Z.; Pérez-Mercader, J. The thermodynamic efficiency of computations made in cells across the range of life. Philos. Trans. R. Soc. Math. Phys. Eng. Sci.
**2017**, 375. [Google Scholar] [CrossRef] [PubMed] - Crosato, E.; Spinney, R.E.; Nigmatullin, R.; Lizier, J.T.; Prokopenko, M. Thermodynamics and computation during collective motion near criticality. Phys. Rev. E
**2018**, 97, 012120. [Google Scholar] [CrossRef] [PubMed][Green Version] - Brody, D.; Rivier, N. Geometrical aspects of statistical mechanics. Phys. Rev. E
**1995**, 51, 1006–1011. [Google Scholar] [CrossRef] [PubMed] - Brody, D.; Ritz, A. Information geometry of finite Ising models. J. Geom. Phys.
**2003**, 47, 207–220. [Google Scholar] [CrossRef] - Janke, W.; Johnston, D.; Kenna, R. Information geometry and phase transitions. Phys. Stat. Mech. Appl.
**2004**, 336, 181–186. [Google Scholar] [CrossRef][Green Version] - Wang, X.; Lizier, J.; Prokopenko, M. Fisher Information at the Edge of Chaos in Random Boolean Networks. Artif. Life
**2011**, 17, 315–329. [Google Scholar] [CrossRef] [PubMed] - Prokopenko, M.; Lizier, J.; Obst, O.; Wang, X. Relating Fisher information to order parameters. Phys. Rev. E
**2011**, 84, 041116. [Google Scholar] [CrossRef][Green Version] - Machta, B.B.; Chachra, R.; Transtrum, M.K.; Sethna, J.P. Parameter Space Compression Underlies Emergent Theories and Predictive Models. Science
**2013**, 342, 604–607. [Google Scholar] [CrossRef][Green Version] - Kubo, R.; Toda, M.; Hashitsume, N. Statistical Physics II: Nonequilibrium Statistical Mechanics, 2nd ed.; Springer: Berlin/Heidelberg, Germany, 1991. [Google Scholar]
- Crooks, G. Measuring Thermodynamic Length. Phys. Rev. Lett.
**2007**, 99, 100602. [Google Scholar] [CrossRef][Green Version] - Crosato, E.; Nigmatullin, R.; Prokopenko, M. On critical dynamics and thermodynamic efficiency of urban transformations. R. Soc. Open Sci.
**2018**, 5, 180863. [Google Scholar] [CrossRef] [PubMed][Green Version] - Harding, N.; Nigmatullin, R.; Prokopenko, M. Thermodynamic efficiency of contagions: A statistical mechanical analysis of the SIS epidemic model. Interface Focus
**2018**, 8, 20180036. [Google Scholar] [CrossRef] [PubMed] - Wei, B.B. Insights into phase transitions and entanglement from density functional theory. New J. Phys.
**2016**, 18, 113035. [Google Scholar] [CrossRef][Green Version] - Kochmanski, M.; Paszkiewicz, T.; Wolski, S. Curie-Weiss magnet—A simple model of phase transition. Eur. J. Phys.
**2013**, 34, 1555. [Google Scholar] [CrossRef][Green Version] - Seifert, U. Stochastic thermodynamics, fluctuation theorems, and molecular machines. Rep. Prog. Phys.
**2012**, 75, 126001. [Google Scholar] [CrossRef] [PubMed][Green Version] - Bérut, A.; Petrosyan, A.; Ciliberto, S. Information and thermodynamics: Experimental verification of Landauer’s Erasure principle. J. Stat. Mech. Theory Exp.
**2015**, 2015, P06015. [Google Scholar] [CrossRef] - Nisoli, C. Write it as you like it. Nat. Nanotechnol.
**2018**, 13, 5–6. [Google Scholar] [CrossRef] - Lao, Y.; Caravelli, F.; Sheikh, M.; Sklenar, J.; Gardeazabal, D.; Watts, J.D.; Albrecht, A.M.; Scholl, A.; Dahmen, K.; Nisoli, C.; et al. Classical topological order in the kinetics of artificial spin ice. Nat. Phys.
**2018**, 14, 723–727. [Google Scholar] [CrossRef] - Wolpert, D.H. The stochastic thermodynamics of computation. J. Phys. A
**2019**, 52, 193001. [Google Scholar] [CrossRef][Green Version] - Grégoire, G.; Chaté, H. Onset of Collective and Cohesive Motion. Phys. Rev. Lett.
**2004**, 92, 025702. [Google Scholar] [CrossRef] [PubMed][Green Version] - Buhl, J.; Sumpter, D.; Couzin, I.; Hale, J.; Despland, E.; Miller, E.; Simpson, S. From Disorder to Order in Marching Locusts. Science
**2006**, 312, 1402–1406. [Google Scholar] [CrossRef][Green Version] - Vicsek, T.; Czirók, A.; Ben-Jacob, E.; Cohen, I.; Sochet, O. Novel Type of Phase Transition in a System of Self-Driven Particles. Phys. Rev. Lett.
**2006**, 75, 1226. [Google Scholar] [CrossRef][Green Version] - Szabo, B.; Szollosi, G.; Gönci, B.; Jurányi, Z.; Selmeczi, D.; Vicsek, T. Phase transition in the collective migration of tissue cells: Experiment and model. Phys. Rev. Stat. Nonlinear Soft Matter Phys.
**2007**, 74, 061908. [Google Scholar] [CrossRef][Green Version] - Mora, T.; Bialek, W. Are biological systems poised at criticality? J. Stat. Phys.
**2011**, 144, 268–302. [Google Scholar] [CrossRef][Green Version] - Bialek, W.; Cavagna, A.; Giardina, I.; Mora, T.; Silvestri, E.; Viale, M.; Walczak, A. Statistical mechanics for natural flocks of birds. Proc. Natl. Acad. Sci. USA
**2012**, 109, 4786–4791. [Google Scholar] [CrossRef] [PubMed][Green Version] - György, S.; István, B. Evolutionary potential games on lattices. Phys. Rep.
**2016**, 624, 1–60. [Google Scholar] [CrossRef][Green Version] - Yoshiki, K.; Ikuko, N. Statistical macrodynamics of large dynamical systems. Case of a phase transition in oscillator communities. J. Stat. Phys.
**1987**, 49, 569–605. [Google Scholar] - Miritello, G.; Pluchino, A.; Rapisarda, A. Central Limit Behavior in the Kuramoto model at the ‘Edge of Chaos’. Phys. Stat. Mech. Appl.
**2009**, 388, 4818–4826. [Google Scholar] [CrossRef][Green Version] - Newman, M.; Watts, D. Scaling and percolation in the small-world network model. Phys. Rev. E
**1999**, 60, 7332–7342. [Google Scholar] [CrossRef][Green Version] - Sander, L.; Warren, C.; Sokolov, I.; Simon, C.; Koopman, J. Percolation on heterogeneous networks as a model for epidemics. Math. Biosci.
**2002**, 180, 293–305. [Google Scholar] [CrossRef] - Wang, W.; Liu, Q.H.; Zhong, L.F.; Tang, M.; Gao, H.; Stanley, H. Predicting the epidemic threshold of the Susceptible–Infected–Recovered model. Sci. Rep.
**2015**, 6, 24676. [Google Scholar] [CrossRef][Green Version] - Wilson, A.; Dearden, J. Phase Transitions and Path Dependence in Urban Evolution. J. Geogr. Syst.
**2011**, 13, 1–16. [Google Scholar] [CrossRef] - Slavko, B.; Glavatskiy, K.; Prokopenko, M. Dynamic resettlement as a mechanism of phase transitions in urban configurations. Phys. Rev. E
**2019**, 99, 042143. [Google Scholar] [CrossRef] [PubMed] - Bak, P.; Tang, C.; Wiesenfeld, K. Self-organized criticality: An explanation of the 1/f noise. Phys. Rev. Lett.
**1987**, 59, 381–384. [Google Scholar] [CrossRef] [PubMed] - Newman, M.E.J. The structure and function of complex networks. SIAM Rev.
**2003**, 45, 167–256. [Google Scholar] [CrossRef][Green Version] - Liu, Y.Y.; Barabási, A.L. Control principles of complex systems. Rev. Mod. Phys.
**2016**, 88, 035006. [Google Scholar] [CrossRef][Green Version] - Daniels, B.C.; Krakauer, D.C.; Flack, J.C. Control of finite critical behaviour in a small-scale social system. Nat. Commun.
**2017**, 8, 14301. [Google Scholar] [CrossRef] [PubMed]

**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.

Publisher’s Note: MDPI stays neutral with regard to jurisdictional claims in published maps and institutional affiliations. |

© 2021 by the authors. Licensee MDPI, Basel, Switzerland. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC BY) license (https://creativecommons.org/licenses/by/4.0/).

## Share and Cite

**MDPI and ACS Style**

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