Transform information: A symmetry breaking measure

Vstovsky, G. V.

doi:10.1007/bf02551520

Cited by 38 publications

(23 citation statements)

References 44 publications

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“…Thus, the information measure of symmetry (IMS) is introduced. A connection between two fundamental concepts of information and symmetry breaking was addressed in reference [10]. Information content of spontaneous symmetry breaking was studied in reference [11].…”

Section: Introductionmentioning

confidence: 99%

Informational Measure of Symmetry vs. Voronoi Entropy and Continuous Measure of Entropy of the Penrose Tiling. Part II of the “Voronoi Entropy vs. Continuous Measure of Symmetry of the Penrose Tiling”

et al. 2021

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The notion of the informational measure of symmetry is introduced according to: Hsym(G)=−∑i=1kP(Gi)lnP(Gi), where P(Gi) is the probability of appearance of the symmetry operation Gi within the given 2D pattern. Hsym(G) is interpreted as an averaged uncertainty in the presence of symmetry elements from the group G in the given pattern. The informational measure of symmetry of the “ideal” pattern built of identical equilateral triangles is established as Hsym(D3)= 1.792. The informational measure of symmetry of the random, completely disordered pattern is zero, Hsym=0. The informational measure of symmetry is calculated for the patterns generated by the P3 Penrose tessellation. The informational measure of symmetry does not correlate with either the Voronoi entropy of the studied patterns nor with the continuous measure of symmetry of the patterns. Quantification of the “ordering” in 2D patterns performed solely with the Voronoi entropy is misleading and erroneous.

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Section: Introductionmentioning

confidence: 99%

Informational Measure of Symmetry vs. Voronoi Entropy and Continuous Measure of Entropy of the Penrose Tiling. Part II of the “Voronoi Entropy vs. Continuous Measure of Symmetry of the Penrose Tiling”

et al. 2021

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“…A connection between two fundamental concepts of information and symmetry breaking was addressed in ref. 10. Information content of spontaneous symmetry breaking was studied in ref.…”

Section: Introductionmentioning

confidence: 99%

Informational Measure of Symmetry vs. Voronoi Entropy and Continuous Measure of Entropy of the Penrose Tiling. Part II of the “Voronoi Entropy vs. Continuous Measure of Symmetry of the Penrose Tiling”.

Bormashenko¹,

Legchenkova²,

Frenkel³

et al. 2021

Preprint

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The notion of the informational measure of symmetry is introduced according to: HsymG=-i=1kPGilnPGi, where PGi is the probability of appearance of the symmetry operation Gi within the given 2D pattern. HsymG is interpreted as an averaged uncertainty in the presence of symmetry elements from the group G in the given pattern. The informational measure of symmetry of the “ideal” pattern built of identical equilateral triangles is established as HsymD3=1.792. The informational measure of symmetry of the random, completely disordered pattern is zero, Hsym=0. Informational measure of symmetry is calculated for the patterns generated by the P3 Penrose tessellation. Informational measure of symmetry does not correlate neither with the Voronoi entropy of the studied patterns nor with the continuous measure of symmetry of the patterns.

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“…Since for a physical system all the symmetric transformations form a symmetry group, a symmetry breaking perturbation gives rise to a reduction of the symmetry group into its corresponding subgroup. Based on the irreducible representation of the symmetry group and the Frobenius norm 22 , the SSD ranging from zero to unity is more capable to identify various nature of symmetries such as symmetry breaking, accidental degeneracy and spontaneous symmetry breaking 23 , in contrast to the previous explorations based on the abstract concepts of fuzz set 24 and transform information 25 . Recently, it was applied to the Frobenius norm-based measures for quantum coherence and asymmetry 26 .…”

Section: Introductionmentioning

confidence: 99%

Vector Form of Symmetry Degree

Dong

Zhang

Sun

et al. 2017

Sci Rep

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AbsractSymmetry degree is utilized to characterize the asymmetry of a physical system with respect to a symmetry group. The scalar form of symmetry degree (SSD) based on Frobenius-norm has been introduced recently to present a quantitative description of symmetry. Here we present the vector form of the symmetry degree (VSD) which possesses more advantages than the SSD. Mathematically, the dimension of VSD is defined as the conjugacy class number of the symmetry group, the square length of the VSD gives rise to the SSD and the direction of VSD is determined by the orders of the conjugacy classes. The merits of applying VSD both for finite and infinite symmetry groups include the additional information of broken symmetry operators with single symmetry breaking perturbation, and the capability of distinguishing distinct symmetry breaking perturbations which exactly give rise to degenerate SSD. Additionally, the VSD for physical systems under symmetry breaking perturbations can be regarded as a projection of the initial VSD without any symmetry breaking perturbations, which can be described by an evolution equation. There are the same advantages by applying VSD for the accidental degeneracy and spontaneous symmetry breaking.

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Transform information: A symmetry breaking measure

Cited by 38 publications

References 44 publications

Informational Measure of Symmetry vs. Voronoi Entropy and Continuous Measure of Entropy of the Penrose Tiling. Part II of the “Voronoi Entropy vs. Continuous Measure of Symmetry of the Penrose Tiling”

Informational Measure of Symmetry vs. Voronoi Entropy and Continuous Measure of Entropy of the Penrose Tiling. Part II of the “Voronoi Entropy vs. Continuous Measure of Symmetry of the Penrose Tiling”

Informational Measure of Symmetry vs. Voronoi Entropy and Continuous Measure of Entropy of the Penrose Tiling. Part II of the “Voronoi Entropy vs. Continuous Measure of Symmetry of the Penrose Tiling”.

Vector Form of Symmetry Degree

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