Special functions : : Fractional Calculus and the Pathway for Entropy / / Hans J. Haubold, editor.

• TeilnehmendeR: Haubold, Hans J.,
• Place / Publishing House: Basel : : MDPI - Multidisciplinary Digital Publishing Institute,, [2018]; ©2018
• Year of Publication: 2018
• Language: English
• Physical Description: 1 online resource (304 pages)

Historically, the notion of entropy emerged in conceptually very distinct contexts. This book deals with the connection between entropy, probability, and fractional dynamics as they appeared, for example, in solar neutrino astrophysics since the 1970's (Mathai and Rathie 1975, Mathai and Pederzoli 1977, Mathai and Saxena 1978, Mathai, Saxena, and Haubold 2010). The original solar neutrino problem, experimentally and theoretically, was resolved through the discovery of neutrino oscillations and was recently enriched by neutrino entanglement entropy. To reconsider possible new physics of solar neutrinos, diffusion entropy analysis, utilizing Boltzmann entropy, and standard deviation analysis was undertaken with Super-Kamiokande solar neutrino data. This analysis revealed a non-Gaussian signal with harmonic content. The Hurst exponent is different from the scaling exponent of the probability density function and both Hurst exponent and scaling exponent of the Super-Kamiokande data deviate considerably from the value of 1/2, which indicates that the statistics of the underlying phenomenon is anomalous. Here experiment may provide guidance about the generalization of theory of Boltzmann statistical mechanics. Arguments in the so-called Boltzmann-Planck-Einstein discussion related to Planck's discovery of the black-body radiation law are recapitulated mathematically and statistically and emphasize from this discussion is pursued that a meaningful implementation of the complex 'entropy-probability-dynamics' may offer two ways for explaining the results of diffusion entropy analysis and standard deviation analysis. One way is to consider an anomalous diffusion process that needs to use the fractional space-time diffusion equation (Gorenflo and Mainardi) and the other way is to consider a generalized Boltzmann entropy by assuming a power law probability density function. Here new mathematical framework, invented by sheer thought, may provide guidance for the generalization of Boltzmann statistical mechanics. In this book Boltzmann entropy, generalized by Tsallis and Mathai, is considered. The second one contains a varying parameter that is used to construct an entropic pathway covering generalized type-1 beta, type-2 beta, and gamma families of densities. Similarly, pathways for respective distributions and differential equations can be developed. Mathai's entropy is optimized under various conditions reproducing the well-known Boltzmann distribution, Raleigh distribution, and other distributions used in physics. Properties of the entropy measure for the generalized entropy are examined. In this process the role of special functions of mathematical physics, particularly the H-function, is highlighted.

Table of Contents

About the Special Issue Editor -- Preface to "Special Functions: Fractional Calculus and the Pathway for Entropy" ix Constantino Tsallis -- Approach of Complexity in Nature: Entropic Nonuniqueness -- Reprinted from: Axioms 2016, 5(3), 20; doi: 10.3390/axioms5030020 1 -- Rudolf Gorenflo and Francesco Mainardi On the Fractional Poisson Process and the Discretized Stable Subordinator -- Reprinted from: Axioms 2015, 4(3), 321-344; doi: 10.3390/axioms4030321 15 -- Nicy Sebastian, Seema S. Nair and Dhannya P. Joseph -- An Overview of the Pathway Idea and Its Applications in Statistical and Physical Sciences -- Reprinted from: Axioms 2015, 4(4), 530-553; doi: 10.3390/axioms4040530 36 -- Yuri Luchko Entropy Production Rate of a One-Dimensional Alpha-Fractional Diffusion Process Reprinted from: Axioms 2016, 5(1), 6; doi: 10.3390/axioms5010006 58 -- Shanoja R. Naik and Hans J. Haubold On the q-Laplace Transform and Related Special Functions -- Reprinted from: Axioms 2016, 5(3), 24; doi: 10.3390/axioms5030024 69 -- Konstantin V. Zhukovsky and Hari M. Srivastava Operational Solution of Non-Integer Ordinary and Evolution-Type Partial Differential Equation -- Reprinted from: Axioms 2016, 5(4), 29; doi: 10.3390/axioms5040029 85 -- Konstantin Zhukovsky Operational Approach and Solutions of Hyperbolic Heat Conduction Equations -- Reprinted from: Axioms 2016, 5(4), 28; doi: 10.3390/axioms5040028 106 -- Ram K. Saxena and Rakesh K. Parmar Fractional Integration and Differentiation of the Generalized Mathieu Series -- Reprinted from: Axioms 2017, 6(3), 18; doi: 10.3390/axioms6030018 132 -- Kai Liu, YangQuan Chen and Xi Zhang An Evaluation of ARFIMA (Autoregressive Fractional Integral Moving Average) Programs -- Reprinted from: Axioms 2017, 6(2), 16; doi: 10.3390/axioms6020016 143 -- Pushpa Narayan Rathie, Paulo Silva and Gabriela Olinto Applications of Skew Models Using Generalized Logistic Distribution -- Reprinted from: Axioms 2016, 5(2), 10; doi: 10.3390/axioms5020010 159 -- Serge B. Provost Closed-Form Representations of the Density Function and Integer Moments of the Sample Correlation Coefficient -- Reprinted from: Axioms 2015, 4(3), 268-274; doi:10.3390/axioms4030268 185 -- Seemon Thomas On some Integral Representations of Certain G-Functions Reprinted from: Axioms 2016, 5(1), 1; doi: 10.3390/axioms5010001 191 -- Thomas Ernst On Elliptic and Hyperbolic Modular Functions and the Corresponding Gudermann Peeta Functions -- Reprinted from: Axioms 2015, 4(3), 235-253; doi: 10.3390/axioms4030235 196 -- Dilip Kumar Some Aspects of Extended Kinetic Equation -- Reprinted from: Axioms 2015, 4(3), 412-422; doi: 10.3390/axioms4030412 213 -- Seema S. Nair An Overview of Generalized Gamma Mittag-Leffler Model and Its Applications Reprinted from: Axioms 2015, 4(3), 365-384; doi: 10.3390/axioms4030365 222 -- Nicy Sebastian Limiting Approach to Generalized Gamma Bessel Model via Fractional Calculus and Its Applications in Various Disciplines Reprinted from: Axioms 2015, 4(3), 385-399; doi: 10.3390/axioms4030385 239 -- Dhannya P. Joseph Multivariate Extended Gamma Distribution -- Reprinted from: Axioms 2017, 6(2), 11; doi: 10.3390/axioms6020011 252 Hans J. Haubold and Arak M. Mathai Scientific Endeavors of A.M. Mathai: An Appraisal on the Occasion of his Eightieth Birthday, 28 April 2015 -- Reprinted from: Axioms 2015, 4(3), 213-234; doi: 10.3390/axioms4030213 264.