Submitted 26. 7. 95 to the journal Symmetry: Culture and Science,

Editor G. Darvas, Hungary. Without any reply.

Entropy as a Measure of Symmetry

Milan Kunz

Jurkovičova 13, 63800 Brno, The Czech Republic

In sciences, it is not allowed to speculate, what would pass if, but the late development of the symmetry theory led to a great confusion in modern science. Its consequences are spread from mathematics, over physics, biology, social sciences to philosophy.

One word to change, such a small correction could be enough, and J. Von Neumann could not say to Shannon¹: "You should call it entropy, for two reasons. In the first place, your uncertainty function has been used in statistical mechanics under that name so, it already has a name. In the second place, and more important, no one knows what entropy really is, so in a debate you will always have the advantage."

The word, which could change this uncertainty was the word symmetry. It should be used by Boltzmann², when he explained his idea, what the entropy of the ideal gas is. He had another problem, too. His proof needed a quantum hypothesis, 30 years ahead Planck. He had no faith in his thoughts and abandoned them for conventional mathematics. Instead the term "symmetry", he used the nothing saying word "probability". Since this probability is determined by the symmetry, he was correct, but the basic idea of his proof of the H theorem was not recognized and remained obscure (Kac³ "a demonstration").

Orbits in multidimensional spaces

S. Weinberg⁴ mentioned in his lecture about importance of mathematics the case of the Ramanudjan-Hardy equation, determining the number of partitions of the number n into n parts⁵. Hardy thought, that it never would have physical applications. This part of the number theory became important in the theory of elementary particles. It remained completely forgotten, that even before Hardy, the partitions had a more practical use, they were the base of the Boltzmann´s proof.

Boltzmann gave an example for 7 particles partitioning 7 quantas of energy. All possible cases can be written in a table, where upper indexes count particles with the same energy

70^6
	610^5
	520^5	51^20^4
	430^5	4210^4	41^30^3
		3310^4	321^20^3	31^40^2
		32^20^4
			2^310^3	2^21^30^2	21^50
						1^7

It is a diagram of the seven dimensional plane orthgonal to the diagonal unit vector I, its crossection. The columns represent consecutively vertices, points on lines, two dimensional bodies (surfaces in three dimensions), and so on. In rows, the partitions are arranged according to the size of the largest part. Each partition of the number seven represents one orbit (Boltzmann used the term "complexion"), and so they count them. All points lying on such an orbit have same length, they are obtained by cyclic permutations of the partition vector and therefore the orbits are spherical. At the beginning, all energy is concentrated in one particle. After a big bang, the particle, all energy of the system is concentrated in, collides at its flight with other particles and the energy dissipates. The system goes spontaneously on the orbit with the largest symmetry, determined by the group of cyclic permutations. The number of points on each orbit is determined by the polynomial coefficient for n permutations, and the function H_n is just the logarithm of this number

H_n = ln(n!/P n_k!) (1)

There is no doubt about it, the factorials of large numbers are approximated quite satisfactorily using the Stirling formula. A problem is, that the system must be very large, it must contain a huge number of particles for obtaining sufficient many simultaneous collisions, to keep it on one orbit⁶. A question is, if the function H_n corresponds really to the entropy. We will see later, that additional terms must be added at real gases, but the function H_n itself is defined without any uncertainty as a measure of the symmetry.

Information entropy

Shannon⁷ build his theory of communication using axioms. They need not to be explained. Nevertheless, he used binary logarithm and then the Hm function has a quite certain interpretation⁸.

To index a set of m objects by a regular code (symbols 0 and 1), we need mlog₂m digits, for example 000, 001, 010, 011, 100, 101, 110, 111 for 8 items. If these objects are classified into n groups with the index j, say aaaabbcd, we need only ....m_jlog₂m_j digits, in our example 10 digits: a00, a01, a10, a11, b0,, b1, c, d. The difference (24- 10) divided by n = 8 is a measure of information, we have about the set. The fractions mj/m, obtained after manipulations with the formula, are again interpreted as a probabilities pj.

There remains a question, if both functions H are related and how. This question was answered differently, but always wrongly.

We have shown, that Boltzmann connected function H_n with cyclic permutations of the partition vector. At messages, there are possible two kinds of permutations, either permutations of the order of symbols in a string, e. g. when aaaabbcd permutes to babacada, or substitutions, when e. g. ddddccba is obtained.

This can be done by a formal mathematical operation, if we write a string of symbols as a naive matrix^9,10 N

a	b	c	d
0	1	0	0
1	0	0	0
0	1	0	0
1	0	0	0
0	0	1	0
1	0	0	0
0	0	0	1
1	0	0	0

Both symmetry operations are then performed by multiplying the matrix N by the unit permutation matrices P from the left and from the right. The number of strings leading to each point on the partition orbit is determined by the polynomial coefficient for m permutations

m!/P mj! = m!/P n_km_k!

The function Hm could be again just the logarithm of this number, since natural and binary logarithms differ only by a factor. The function Hm measures the number of messages which can be formed from a given set of symbols. But at each string, the Boltzmann function H_n is defined too, therefore the total number of strings of length m going to a n dimensional plane is

(n!/P n_k!)(m!/P n_km_k!) = n^m

The logarithm of the product is a sum, therefore both measures H are additive. In our example, the vector of frequencies mk is (4, 2, 1, 1) and n(0) = 0, n(1) = 2, n(2) = 1 n(3) = 0, n(4) = 1, etc. Since 0! =1, the zero frequencies can be neglected and

H_n = log(4!/2!1!²).

The existence of two measures of symmetry can explain the observed redundancy of natural languages. It is true, that if all symbols are used equally, it is possible to formulate the greatest number of messages. But many of them were alike. It is better to explain this difficulty speaking about words instead of symbols. In a message without redundancy, there were no key words, and we could not recognize, what is spoken about.

Symmetry of graphs

At ideal gases, all energy is concentrated in a point. At real gases the energy is dispersed, but only within a molecule, its quanta can not be loose through the whole matrix as symbols are in a naive matrix. The ideal gas can be formally considered as a quadratic form P^TN^TNP where P^T is the transposed permutation matrix for n permutations and N^TN is the diagonal vector. Its molecules are represented as points on vector axes. A real molecule forms a blot in the matrix representing the system.

Molecules are isomorphic with graphs, mapping their structures. A graph is described by an incidence matrix. This matrix is either a difference of two naive matrices (oriented graph) or their sum (unoriented graph). The symmetry of graphs is determined similarly as at naive matrices, by multiplying the incidence matrix by the unit permutation matrices from the left and from the right. This leads to the wreath products of the cyclic groups¹¹ and to rather complicated formulas transforming the group of cyclic permutations Sn into graph groups.

But here, the relation of the entropy with symmetry was stated by many authors many times before. Unfortunately, in specialized journals, only^12,13.

Moreover, there exists the entropy of mixing. Consider now, that the string aaaabbcd represents 8 molecules of 4 different kinds (another embodiment of this kind of entropy is sorting hot and cool molecules inside the system). The entropy of such a mixture depends on mixing of molecules inside the system. If the original arrangement permutes to babacada, its entropy must change. Both H_n and Hm do not measure this effect^14,15.

Discussion

If we interpret the spontaneous growing of entropy as the spontaneous growing of symmetry in the Universe, then we do not need the term negentropy for living organisms¹⁶. They have a greater number of symmetry elements of higher order, a greater complexity than non-living things, only. The increase of symmetry is a spontaneous process. Elementary particles form atoms, atoms molecules, molecules structures as crystals or living cells. Living cells assemble into organisms, organisms into societies. In each step, new symmetry elements appear to the old ones^17,18.

On the molecular level, an integrating factor exists, the temperature. The physical entropy is a logarithmic measure of the amount of energy needed to increase the temperature.

Outside physics, we can calculate functions H on many levels. But we do not know, if an integrating factor exists here. The apparent disorder is only unrecognized symmetry.

References

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2. L. Boltzmann, Über die Beziehung zwischen dem zweiten Hauptsatze der mechanishen Wärmetheorie und die Wahrscheinlichkeitsrechnung, Wiener Berichte 76, (1877), 373.

3. M. KAC in J. MEHRA, Ed. The Physicist's Conception of Nature, Reidel, Dordrecht, 1073, p.560.

4. S. Weinberg, Mathematics, the unifying thread in Science, Notices AMS, 33 (1986), 716.

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6. M. Kunz: How to distinguish distinguishability: Physical and combinatorial definitions, Physics Letters A 135 (1989) 421-424.

7. C. E. Shannon, The Mathematical Theory of Communication, Bell System Technical Journal, 27 (1948), 379, 623.

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9. M. Kunz, Information processing in linear vector space, Information Processing and Management, 20 (1984) 519-524.

10. M. Kunz, About metrics of bibliometrics, J. Chem. Inform. Comput. Sci., 33, 193-196.

11. F. Harary, E. M. Palmer, Graphical Enumeration, Academic Press, New York, 1973.

12. M. Gordon, W. B. Temple, Chemical Combinatorics. Part I. Chemical Kinetics, Graph Theory and Combinatorial Entropy, J. Chem. Soc (A), Inorg. Phys. Theor., 1970, 729.

13. R. M. Dannenfelser, N. Surendran, S. H. Yalkowsky, SAR, QSAR Environ. Res., 1 (1993), 273.

14. M. Kunz, Time Spectra of Patent Information, Scientometrics, 11 (1987) 163.

15. M. Kunz: A note about the negentropy principle, MATCH, 23 (1988) 3.

16. E. Schroedinger, What Is Life?, Cambridge University Press, Cambridge, 1944.

17. J. Tonnelat, Conformity of Evolution towards Complexity from Thermodynamic Conclusions, Arch. Int. Physiol. Biochim. 94 (1986) C11.

18. M. W. Evans, Three Principles of Group Theoretical Mechanics, Phys. Lett. A, 134 (1989) 409.