### The Distributions in Nature and Entropy Principle

#### Abstract

**Abstract. **The derivation of the maximum entropy distribution of particles in boxes yields two kinds of distributions: a "bell-like" distribution and a long-tail distribution. The first one is obtained when the ratio between particles and boxes is low, and the second one - when the ratio is high. The obtained long tail distribution yields correctly the empirical Zipf law, Pareto's 20:80 rule and Benford's law. Therefore, it is concluded that the long tail and the "bell-like" distributions are outcomes of the tendency of statistical systems to maximize entropy.

**Keywords.** Maximum Entropy, Long tail distribution, 20:80 Pareto's rule, Zipf Law, Benford's law, Bell-like distribution.

**JEL. **C62**.**

#### Keywords

#### References

Bak, P. (1996). How Nature Works: The Science of Self-Organized Criticality, Springer-Verlag, New York.

Benford, F. (1938). The law of anomalous numbers, Proceedings of the American Mathematical Society, 78(4), 551-572.

Cohen, D., & Talbot K. (1984). Prime numbers and the first digit phenomenon, Journal of Number Theory, 18(3), 261-268. doi. 10.1016/0022-314X(84)90061-1

Jurgan, J.M. (1951). Quality Control Handbook, McGraw-Hill Book Company.

Kafri, O. (2007). The second law as a cause for the evolution, Unpublished Paper, [Retrieved from].

Kafri, O. (2009). Entropy principle in direct derivation of Benford's law, Unpublished Paper, [Retrieved from].

Kafri, O. (2016). Sociological inequality and the second law, Journal of Economic and Social Thought, 3(4), xxx-xxx.

Kossovsky, A.E. (2012). Towards a better understanding of the leading digits phenomena, Unpublished Paper, [Retrieved from].

Miller, G.A. & Newman E.B. (1958). Tests of a statistical explanation of the rank-frequency relation for words in written English, American Journal of Psychology, 71(1), 209-218. doi. 10.2307/1419208

Newcomb, S. (1881). Note on the frequency of use of the different digits in natural numbers, American Journal of Mathematics, 4, 39-40.

Newman, M.E. (2006). Power-law, Pareto Distribution and Zipf's law, Unpublished Paper, [Retrieved from].

Nigrini, M. (1996). A taxpayer compliance application of Benford's law, Journal of the American Taxation Association, 18(1), 72-91.

Pareto, V. (1897). Coursd' Economie Politique. Droz, Geneva Switzerland, 1896, Rouge, Lausanne et Paris.

Planck, M. (1901). On the law of distribution of energy in the normal spectrum, Annalen der Physik, 4, 553-562.

Zipf, G.K. (1949). Human Behavior and the Principle of Least-Effort, Addison-Wesley.

DOI: http://dx.doi.org/10.1453/ter.v3i4.1122

### Refbacks

- There are currently no refbacks.

.......................................................................................................................................................................................................................................................................................................................................

Turkish Economic Review - Turk. Econ. Rev. - TER - **www.kspjournals.org**

ISSN: 2149-0414

Editor: ** ter@ksplibrary.org** Secretarial:

**Istanbul - Turkey.**

**secretarial@ksplibrary.org**Copyright © KSP Library