# Entropy, Information, and Symmetry: Ordered is Symmetrical

## Abstract

**:**

## 1. Introduction

## 2. Symmetry and Entropy of Binary Systems

## 3. Conclusions

## Funding

## Acknowledgments

## Conflicts of Interest

## References

- Clausius, R. On the Application of the Mechanical theory of Heat to the Steam-Engine. In The Mechanical Theory of Heat—with Its Applications to the Steam Engine and to Physical Properties of Bodies; John van Voorst: London, UK, 1856; 1 Paternoster Row, MDCCCLXVII; pp. 136–207. [Google Scholar]
- Müller, I. A History of Thermodynamics; The Doctrine of Energy and Entropy; Springer: Berlin, Germany, 2007. [Google Scholar]
- Falk, G. Entropy, a resurrection of caloric-a look at the history of thermodynamics. Eur. J. Phys.
**1985**, 6, 108–115. [Google Scholar] [CrossRef][Green Version] - Martin, J.S.; Smith, N.A.; Francis, C.D. Removing the entropy from the definition of entropy: Clarifying the relationship between evolution, entropy, and the second law of thermodynamics. Evol. Educ. Outreach
**2013**, 6, 30. [Google Scholar] [CrossRef] - Clausius, R. On Several Convenient Forms of the Fundamental Equations of the Mechanical Theory of Heat. In The Mechanical Theory of Heat with Its Applications to the Steam-Engine and to the Physical Properties of Bodies; Tyndall, J., Translator; John Van Voorst: London, UK, 1867; pp. 327–365. [Google Scholar]
- Gaudenzi, R. Entropy? Exercices de Style. Entropy
**2019**, 21, 742. [Google Scholar] [CrossRef][Green Version] - Baierlein, R. Thermal Physics; Cambridge University Press: Cambridge, UK, 1999. [Google Scholar]
- Landau, L.D.; Lifshitz, E.M. Statistical Physics, 3rd ed.; Elsevier: Oxford, UK, 2011; Course of Theoretical Physics; Volume 5. [Google Scholar]
- Kittel, C. Thermal Physics; J. Wiley & Sons: New York, NY, USA, 1969. [Google Scholar]
- Wright, P.G. Entropy and disorder. Contemp. Phys.
**1970**, 11, 581–588. [Google Scholar] [CrossRef] - Styer, D. Entropy as disorder: History of misconception. Phys. Teach.
**2019**, 57, 454–458. [Google Scholar] [CrossRef][Green Version] - Darrigol, O. Number and measure: Hermann von Helmholtz at the crossroads of mathematics, physics, and psychology. Stud. Hist. Philos. Sci. A
**2003**, 34, 515–573. [Google Scholar] [CrossRef] - Myszkowski, N.; Storme, M.; Zenasni, F. Order in complexity: How Hans Eysenck brought differential psychology and aesthetics together. Personal. Individ. Differ.
**2016**, 103, 156–162. [Google Scholar] [CrossRef] - Shaki, S.; Fischer, M.H.; Petrusic, W.M. Reading habits for both words and numbers contribute to the SNARC effect. Psychon. Bull. Rev.
**2009**, 16, 328–331. [Google Scholar] [CrossRef] [PubMed][Green Version] - Yodogawa, E. Symmetropy, an entropy-like measure of visual symmetry. Percept. Psychophys.
**1982**, 32, 230–240. [Google Scholar] [CrossRef] [PubMed][Green Version] - Zabrodsky, H.; Avnir, D.; Peleg, S. Symmetry as a continuous feature. IEEE Trans. Pattern Anal. Mach. Intell.
**1995**, 17, 1154–1166. [Google Scholar] [CrossRef][Green Version] - Petitjean, M. Chirality and symmetry measures: A transdisciplinary review. Entropy
**2003**, 5, 271–312. [Google Scholar] [CrossRef] - Weyl, H. Symmetry; Princeton University Press: Princeton, NJ, USA, 1989. [Google Scholar]
- Van Fraassen, B.C. Laws and Symmetry; Oxford University Press: Oxford, UK, 1989. [Google Scholar]
- Rosen, J. Symmetry in Science: An Introduction to the General Theory; Springer: Berlin, Germany, 1995. [Google Scholar]
- Hall, B.C. Quantum Theory for Mathematicians; Graduate Texts in Mathematics; Springer: Berlin, Germany, 2013. [Google Scholar]
- Chatterjee, S.K. Crystallography and the World of Symmetry; Springer: Berlin, Germany, 2008. [Google Scholar]
- El-Batanouny, M.; Wooten, F. Symmetry and Condensed Matter Physics—A Computational Approach; Cambridge University Press: Cambridge, UK, 2010. [Google Scholar]
- Landauer, R. Dissipation and heat generation in the computing process. IBM J. Res. Dev.
**1961**, 5, 183. [Google Scholar] [CrossRef] - Landauer, R. Information is physical. Phys. Today
**1991**, 44, 23–29. [Google Scholar] [CrossRef] - Landauer, R. Minimal energy requirements in communication. Science
**1996**, 272, 1914–1918. [Google Scholar] [CrossRef] [PubMed] - Barthélemy, M. Spatial networks. Phys. Rep.
**2011**, 499, 1–101. [Google Scholar] [CrossRef][Green Version] - Bormashenko, E.; Frenkel, M.; Vilk, A.; Legchenkova, I.; Fedorets, A.; Aktaev, N.; Dombrovsky, L.; Nosonovsky, M. Characterization of self-assembled 2D patterns with Voronoi Entropy. Entropy
**2018**, 20, 956. [Google Scholar] [CrossRef][Green Version] - Fedorets, A.A.; Frenkel, M.; Bormashenko, E.; Nosonovsky, M. Small levitating ordered droplet clusters: Stability, symmetry, and Voronoi Entropy. J. Phys. Chem. Lett.
**2017**, 8, 5599–5602. [Google Scholar] [CrossRef] [PubMed] - Bormashenko, E.; Legchenkova, I.; Frenkel, M. Symmetry and Shannon Measure of ordering: Paradoxes of Voronoi tessellation. Entropy
**2019**, 21, 452. [Google Scholar] [CrossRef][Green Version]

**Figure 1.**(

**A**) The binary one-dimensional (1D) system of N non-interacting elementary magnets (the external magnetic field equals zero ($\overrightarrow{H}=0$). All of the up/down arrangements of the magnets are available. (

**B**) The axis of symmetry shown with the dashed line restricts the number of available configurations of magnets. (

**C**) Two-dimensional (2D) binary system of elementary magnets. Axes of symmetry shown with dashed lines restrict the available arrangements of the magnets.

**Figure 2.**(

**A**) Particles of two kinds (blue and red) are located within the chamber divided equally by the permeable partition. The arrangements at which two particles are simultaneously located at one side of the partition are permitted. (

**B**) Particles of two kinds (blue and red) are located within the chamber divided equally by the permeable partition. Only the arrangements symmetric relatively to axis $O{O}^{\prime}$ shown with the red dashed line are permitted.

© 2019 by the author. 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 (http://creativecommons.org/licenses/by/4.0/).

## Share and Cite

**MDPI and ACS Style**

Bormashenko, E. Entropy, Information, and Symmetry: Ordered is Symmetrical. *Entropy* **2020**, *22*, 11.
https://doi.org/10.3390/e22010011

**AMA Style**

Bormashenko E. Entropy, Information, and Symmetry: Ordered is Symmetrical. *Entropy*. 2020; 22(1):11.
https://doi.org/10.3390/e22010011

**Chicago/Turabian Style**

Bormashenko, Edward. 2020. "Entropy, Information, and Symmetry: Ordered is Symmetrical" *Entropy* 22, no. 1: 11.
https://doi.org/10.3390/e22010011