# Polyadic Entropy, Synergy and Redundancy among Statistically Independent Processes in Nonlinear Statistical Physics with Microphysical Codependence

## Abstract

**:**

## 1. Introduction

#### 1.1. General Motivation

#### 1.2. Statistical and Information Entropy

#### 1.3. Information Redundancy

#### 1.4. Information Synergy

## 2. Polyadic “Non-Extensive” Entropies

#### 2.1. The Fundamentals

#### 2.2. Dyadic Systems

#### 2.3. Triadic Systems

## 3. Polyadic Synergy and Redundancy

#### 3.1. Synergy and Redundancy Emerging among Statistically Independent Variables

#### 3.2. Dyadic Form

#### 3.3. Triadic Form

## 4. Concluding Remarks

- Factorable probabilities do not necessarily lead to additive entropies.
- Microscale codependence does not necessarily lead to macroscale codependence.
- Macroscale independendence does not necessarily imply microscale independence.

## Acknowledgments

## Conflicts of Interest

## References

- Shannon, C.E. The mathematical theory of communication. Bell Syst. Tech. J.
**1948**, 27, 379–423. [Google Scholar] [CrossRef] - Cover, T.M.; Thomas, J.A. Elements of Information Theory; John Wiley and Sons: Hoboken, NJ, USA, 1991. [Google Scholar]
- Schneidman, E.; Still, S.; Berry, M.J.; Bialek, W. Network information and connected correlations. Phys. Rev. Lett.
**2003**, 91, 238701. [Google Scholar] [CrossRef] [PubMed] - McGill, W.J. Multivariate information transmission. Psychometrika
**1954**, 19, 97–116. [Google Scholar] [CrossRef] - Pires, C.A.L.; Perdigão, R.A.P. Non-Gaussian interaction information: Estimation, optimization and diagnostic application of triadic wave resonance. Nonlinear Process. Geophys.
**2015**, 22, 87–108. [Google Scholar] [CrossRef] - Lage, E.J.S. Física Estatística; Fundação Calouste Gulbenkian: Lisbon, Portugal, 1995. (In Portuguese) [Google Scholar]
- Callen, H.B. Thermodynamics and an Introduction to Thermostatistics, 2nd ed.; John Wiley and Sons: Hoboken, NJ, USA, 2001. [Google Scholar]
- Kaniadakis, G.; Lissia, M.; Scarfone, A.M. Two-parameter deformations of logarithm, exponential and entropy: A consistent framework for generalized statistical mechanics. Phys. Rev. E
**2005**, 71, 046128. [Google Scholar] [CrossRef] [PubMed] - Pires, C.A.L.; Perdigão, R.A.P. Minimum Mutual Information and Non-Gaussianity Through the Maximum Entropy Method: Estimation from finite samples. Entropy
**2012**, 14, 1103–1126. [Google Scholar] [CrossRef] - Pires, C.A.L.; Perdigão, R.A.P. Minimum Mutual Information and Non-Gaussianity Through the Maximum Entropy Method: Theory and Properties. Entropy
**2013**, 15, 721–752. [Google Scholar] [CrossRef] - Jakulin, A.; Bratko, I. Quantifying and Visualizing Attribute Interactions. arXiv. 2004. arXiv:cs/0308002. Available online: https://arxiv.org/abs/cs/0308002v3 (accessed on 3 January 2018).
- Borges, E.P.; Roditi, I. A family of non-extensive entropies. Phys. Lett. A
**1998**, 246, 399–402. [Google Scholar] [CrossRef] - Curado, E.M.F.; Nobre, F.D. Derivation of nonlinear Fokker-Planck equations by means of approximations to the master equation. Phys. Rev. E
**2003**, 67, 021107. [Google Scholar] [CrossRef] [PubMed] - Schwämmle, V.; Curado, E.M.F.; Nobre, F.D. A general nonlinear Fokker-Planck equation and its associated entropy. Eur. Phys. J. B
**2007**, 58, 159–165. [Google Scholar] [CrossRef] - Perdigão, R.A.P.; Blöschl, G. Spatiotemporal flood sensitivity to annual precipitation: Evidence for landscape-climate coevolution. Water Resour. Res.
**2014**, 50, 5492–5509. [Google Scholar] [CrossRef] - Perdigão, R.A.P.; Pires, C.A.L.; Hall, J. Synergistic Dynamic Theory of Complex Coevolutionary Systems: Disentangling Nonlinear Spatiotemporal Controls on Precipitation. arXiv. 2016. arXiv:1611.03403. Available online: https://arxiv.org/abs/1611.03403 (accessed on 3 January 2018).
- Tsallis, C. Nonextensive physics: A possible connection between generalized statistical mechanics and quantum groups. Phys. Lett. A
**1994**, 195, 329–334. [Google Scholar] [CrossRef] - Perdigão, R.A.P. Mathematical Physics and Predictability of Non-Periodic Emergence and Extremes in Complex Coevolutionary Systems; APMG: Lisbon, Portugal, 2017; ISBN 9789899566002. [Google Scholar]

© 2018 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**

Perdigão, R.A.P.
Polyadic Entropy, Synergy and Redundancy among Statistically Independent Processes in Nonlinear Statistical Physics with Microphysical Codependence. *Entropy* **2018**, *20*, 26.
https://doi.org/10.3390/e20010026

**AMA Style**

Perdigão RAP.
Polyadic Entropy, Synergy and Redundancy among Statistically Independent Processes in Nonlinear Statistical Physics with Microphysical Codependence. *Entropy*. 2018; 20(1):26.
https://doi.org/10.3390/e20010026

**Chicago/Turabian Style**

Perdigão, Rui A. P.
2018. "Polyadic Entropy, Synergy and Redundancy among Statistically Independent Processes in Nonlinear Statistical Physics with Microphysical Codependence" *Entropy* 20, no. 1: 26.
https://doi.org/10.3390/e20010026