Special Issue "Special Functions: Fractional Calculus and the Pathway for Entropy Dedicated to Professor Dr. A.M. Mathai on the occasion of his 80th Birthday"

A special issue of Axioms (ISSN 2075-1680).

Deadline for manuscript submissions: closed (28 April 2017)

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Special Issue Editor

Guest Editor
Prof. Dr. Hans J. Haubold

1. Office for Outer Space Affairs, United Nations, Vienna International Centre, A-1400 Vienna, Austria
2. Centre for Mathematical and Statistical Sciences, Peechi Campus, KFRI, Peechi, Kerala 680653, India
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Interests: special functions; fractional calculus; entropic functional; mathematical physics; applied analysis; statistical distributions; geometrical probabilities; multivariate analysis

Special Issue Information

Dear Colleagues,

This Special Issue of Axioms is an attempt to capture the broad spectrum of scientific endeavors of Professor Dr. A.M. Mathai at the occasion of his anniversary.

A.M. Mathai was born on 28 April 1935 in Arakulam, near Palai, in the Idukki district of Kerala, India as the eldest son of Aley and Arakaparampil Mathai. After completing his high school education in 1953 from St. Thomas High School, Palai joined St. Thomas College, Palai with record marks and obtained his B.Sc. degree in Mathematics in 1957. In 1959, he completed his Master's Degree in Statistics from the University of Kerala, Thiruvananthapuram, Kerala, India; he was first in class, and achieved the first rank and a gold medal. Then, he joined St. Thomas College, Palai, University of Kerala, as a Lecturer in Statistics and served there until 1961. He obtained a Canadian Commonwealth scholarship in 1961 and went to the University of Toronto, Canada to complete his M.A. degree in Mathematics in 1962. He was awarded a Ph.D. from the University of Toronto, Canada in 1964. Then, he joined McGill University, Canada as an Assistant Professor until 1968. From 1968 to 1978, he was an Associate Professor there. He became a Full Professor of McGill in 1979 and served the Department of Mathematics and Statistics until 2000. He is also the founder of the Canadian Journal of Statistics and the Statistical Society of Canada. As of this date, A.M. Mathai is an Emeritus Professor of Mathematics and Statistics at McGill University, Canada, and Director of the Centre for Mathematical and Statistical Sciences, India. He has published over 300 research papers and more than 25 books on topics in mathematics, statistics, physics, astrophysics, chemistry, and biology. He is a Fellow of the Institute of Mathematical Statistics, National Academy of Sciences of India, President of the Mathematical Society of India, and a Member of the International Statistical Institute.

He was instrumental in the implementation of the United Nations Basic Space Science Initiative (1991-2012).

Prof. Dr. Hans J. Haubold
Guest Editor

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Keywords

  • special functions
  • fractional calculus
  • entropic functional
  • mathematical physics
  • applied analysis
  • statistical distributions
  • geometrical probabilities
  • multivariate analysis

Published Papers (18 papers)

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Editorial

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Open AccessEditorial Scientific Endeavors of A.M. Mathai: An Appraisal on the Occasion of his Eightieth Birthday, 28 April 2015
Axioms 2015, 4(3), 213-234; https://doi.org/10.3390/axioms4030213
Received: 6 May 2015 / Revised: 6 May 2015 / Accepted: 24 June 2015 / Published: 3 July 2015
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Abstract
A.M. Mathai is Emeritus Professor of Mathematics and Statistics at McGill University, Canada. He is currently the Director of the Centre for Mathematical and Statistical Sciences India. His research contributions cover a wide spectrum of topics in mathematics, statistics, physics, astronomy, and biology.
[...] Read more.
A.M. Mathai is Emeritus Professor of Mathematics and Statistics at McGill University, Canada. He is currently the Director of the Centre for Mathematical and Statistical Sciences India. His research contributions cover a wide spectrum of topics in mathematics, statistics, physics, astronomy, and biology. He is a Fellow of the Institute of Mathematical Statistics, National Academy of Sciences of India, and a member of the International Statistical Institute. He is a founder of the Canadian Journal of Statistics and the Statistical Society of Canada. He was instrumental in the implementation of the United Nations Basic Space Science Initiative (1991–2012). This paper highlights research results of A.M. Mathai in the period of time from 1962 to 2015. He published over 300 research papers and over 25 books. Full article

Research

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Open AccessArticle Fractional Integration and Differentiation of the Generalized Mathieu Series
Received: 27 April 2017 / Revised: 15 June 2017 / Accepted: 22 June 2017 / Published: 27 June 2017
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Abstract
We aim to present some formulas for the Saigo hypergeometric fractional integral and differential operators involving the generalized Mathieu series Sμ(r), which are expressed in terms of the Hadamard product of the generalized Mathieu series Sμ(
[...] Read more.
We aim to present some formulas for the Saigo hypergeometric fractional integral and differential operators involving the generalized Mathieu series S μ ( r ) , which are expressed in terms of the Hadamard product of the generalized Mathieu series S μ ( r ) and the Fox–Wright function p Ψ q ( z ) . Corresponding assertions for the classical Riemann–Liouville and Erdélyi–Kober fractional integral and differential operators are deduced. Further, it is emphasized that the results presented here, which are for a seemingly complicated series, can reveal their involved properties via the series of the two known functions. Full article
Open AccessArticle An Evaluation of ARFIMA (Autoregressive Fractional Integral Moving Average) Programs
Received: 13 March 2017 / Revised: 3 May 2017 / Accepted: 14 June 2017 / Published: 17 June 2017
Cited by 2 | PDF Full-text (569 KB) | HTML Full-text | XML Full-text | Supplementary Files
Abstract
Strong coupling between values at different times that exhibit properties of long range dependence, non-stationary, spiky signals cannot be processed by the conventional time series analysis. The autoregressive fractional integral moving average (ARFIMA) model, a fractional order signal processing technique, is the generalization
[...] Read more.
Strong coupling between values at different times that exhibit properties of long range dependence, non-stationary, spiky signals cannot be processed by the conventional time series analysis. The autoregressive fractional integral moving average (ARFIMA) model, a fractional order signal processing technique, is the generalization of the conventional integer order models—autoregressive integral moving average (ARIMA) and autoregressive moving average (ARMA) model. Therefore, it has much wider applications since it could capture both short-range dependence and long range dependence. For now, several software programs have been developed to deal with ARFIMA processes. However, it is unfortunate to see that using different numerical tools for time series analysis usually gives quite different and sometimes radically different results. Users are often puzzled about which tool is suitable for a specific application. We performed a comprehensive survey and evaluation of available ARFIMA tools in the literature in the hope of benefiting researchers with different academic backgrounds. In this paper, four aspects of ARFIMA programs concerning simulation, fractional order difference filter, estimation and forecast are compared and evaluated, respectively, in various software platforms. Our informative comments can serve as useful selection guidelines. Full article
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Open AccessArticle Multivariate Extended Gamma Distribution
Received: 7 June 2016 / Revised: 7 April 2017 / Accepted: 13 April 2017 / Published: 24 April 2017
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Abstract
In this paper, I consider multivariate analogues of the extended gamma density, which will provide multivariate extensions to Tsallis statistics and superstatistics. By making use of the pathway parameter β, multivariate generalized gamma density can be obtained from the model considered here.
[...] Read more.
In this paper, I consider multivariate analogues of the extended gamma density, which will provide multivariate extensions to Tsallis statistics and superstatistics. By making use of the pathway parameter β , multivariate generalized gamma density can be obtained from the model considered here. Some of its special cases and limiting cases are also mentioned. Conditional density, best predictor function, regression theory, etc., connected with this model are also introduced. Full article
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Open AccessArticle Operational Solution of Non-Integer Ordinary and Evolution-Type Partial Differential Equations
Received: 27 October 2016 / Revised: 28 November 2016 / Accepted: 6 December 2016 / Published: 13 December 2016
Cited by 4 | PDF Full-text (302 KB) | HTML Full-text | XML Full-text
Abstract
A method for the solution of linear differential equations (DE) of non-integer order and of partial differential equations (PDE) by means of inverse differential operators is proposed. The solutions of non-integer order ordinary differential equations are obtained with recourse to the integral transforms
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A method for the solution of linear differential equations (DE) of non-integer order and of partial differential equations (PDE) by means of inverse differential operators is proposed. The solutions of non-integer order ordinary differential equations are obtained with recourse to the integral transforms and the exponent operators. The generalized forms of Laguerre and Hermite orthogonal polynomials as members of more general Appèl polynomial family are used to find the solutions. Operational definitions of these polynomials are used in the context of the operational approach. Special functions are employed to write solutions of DE in convolution form. Some linear partial differential equations (PDE) are also explored by the operational method. The Schrödinger and the Black–Scholes-like evolution equations and solved with the help of the operational technique. Examples of the solution of DE of non-integer order and of PDE are considered with various initial functions, such as polynomial, exponential, and their combinations. Full article
Open AccessArticle Operational Approach and Solutions of Hyperbolic Heat Conduction Equations
Received: 27 October 2016 / Revised: 6 December 2016 / Accepted: 6 December 2016 / Published: 12 December 2016
Cited by 9 | PDF Full-text (5434 KB) | HTML Full-text | XML Full-text
Abstract
We studied physical problems related to heat transport and the corresponding differential equations, which describe a wider range of physical processes. The operational method was employed to construct particular solutions for them. Inverse differential operators and operational exponent as well as operational definitions
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We studied physical problems related to heat transport and the corresponding differential equations, which describe a wider range of physical processes. The operational method was employed to construct particular solutions for them. Inverse differential operators and operational exponent as well as operational definitions and operational rules for generalized orthogonal polynomials were used together with integral transforms and special functions. Examples of an electric charge in a constant electric field passing under a potential barrier and of heat diffusion were compared and explored in two dimensions. Non-Fourier heat propagation models were studied and compared with each other and with Fourier heat transfer. Exact analytical solutions for the hyperbolic heat equation and for its extensions were explored. The exact analytical solution for the Guyer-Krumhansl type heat equation was derived. Using the latter, the heat surge propagation and relaxation was studied for the Guyer-Krumhansl heat transport model, for the Cattaneo and for the Fourier models. The comparison between them was drawn. Space-time propagation of a power–exponential function and of a periodic signal, obeying the Fourier law, the hyperbolic heat equation and its extended Guyer-Krumhansl form were studied by the operational technique. The role of various terms in the equations was explored and their influence on the solutions demonstrated. The accordance of the solutions with maximum principle is discussed. The application of our theoretical study for heat propagation in thin films is considered. The examples of the relaxation of the initial laser flash, the wide heat spot, and the harmonic function are considered and solved analytically. Full article
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Open AccessArticle On the q-Laplace Transform and Related Special Functions
Received: 12 July 2016 / Revised: 24 August 2016 / Accepted: 30 August 2016 / Published: 6 September 2016
Cited by 1 | PDF Full-text (287 KB) | HTML Full-text | XML Full-text
Abstract
Motivated by statistical mechanics contexts, we study the properties of the q-Laplace transform, which is an extension of the well-known Laplace transform. In many circumstances, the kernel function to evaluate certain integral forms has been studied. In this article, we establish relationships
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Motivated by statistical mechanics contexts, we study the properties of the q-Laplace transform, which is an extension of the well-known Laplace transform. In many circumstances, the kernel function to evaluate certain integral forms has been studied. In this article, we establish relationships between q-exponential and other well-known functional forms, such as Mittag–Leffler functions, hypergeometric and H-function, by means of the kernel function of the integral. Traditionally, we have been applying the Laplace transform method to solve differential equations and boundary value problems. Here, we propose an alternative, the q-Laplace transform method, to solve differential equations, such as as the fractional space-time diffusion equation, the generalized kinetic equation and the time fractional heat equation. Full article
Open AccessArticle Applications of Skew Models Using Generalized Logistic Distribution
Received: 12 February 2016 / Revised: 6 April 2016 / Accepted: 7 April 2016 / Published: 15 April 2016
Cited by 1 | PDF Full-text (2033 KB) | HTML Full-text | XML Full-text
Abstract
We use the skew distribution generation procedure proposed by Azzalini [Scand. J. Stat., 1985, 12, 171–178] to create three new probability distribution functions. These models make use of normal, student-t and generalized logistic distribution, see Rathie and Swamee [Technical
[...] Read more.
We use the skew distribution generation procedure proposed by Azzalini [Scand. J. Stat., 1985, 12, 171–178] to create three new probability distribution functions. These models make use of normal, student-t and generalized logistic distribution, see Rathie and Swamee [Technical Research Report No. 07/2006. Department of Statistics, University of Brasilia: Brasilia, Brazil, 2006]. Expressions for the moments about origin are derived. Graphical illustrations are also provided. The distributions derived in this paper can be seen as generalizations of the distributions given by Nadarajah and Kotz [Acta Appl. Math., 2006, 91, 1–37]. Applications with unimodal and bimodal data are given to illustrate the applicability of the results derived in this paper. The applications include the analysis of the following data sets: (a) spending on public education in various countries in 2003; (b) total expenditure on health in 2009 in various countries and (c) waiting time between eruptions of the Old Faithful Geyser in the Yellow Stone National Park, Wyoming, USA. We compare the fit of the distributions introduced in this paper with the distributions given by Nadarajah and Kotz [Acta Appl. Math., 2006, 91, 1–37]. The results show that our distributions, in general, fit better the data sets. The general R codes for fitting the distributions introduced in this paper are given in Appendix A. Full article
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Open AccessArticle Entropy Production Rate of a One-Dimensional Alpha-Fractional Diffusion Process
Received: 30 December 2015 / Accepted: 2 February 2016 / Published: 5 February 2016
Cited by 6 | PDF Full-text (237 KB) | HTML Full-text | XML Full-text
Abstract
In this paper, the one-dimensional α-fractional diffusion equation is revisited. This equation is a particular case of the time- and space-fractional diffusion equation with the quotient of the orders of the time- and space-fractional derivatives equal to one-half. First, some integral representations
[...] Read more.
In this paper, the one-dimensional α-fractional diffusion equation is revisited. This equation is a particular case of the time- and space-fractional diffusion equation with the quotient of the orders of the time- and space-fractional derivatives equal to one-half. First, some integral representations of its fundamental solution including the Mellin-Barnes integral representation are derived. Then a series representation and asymptotics of the fundamental solution are discussed. The fundamental solution is interpreted as a probability density function and its entropy in the Shannon sense is calculated. The entropy production rate of the stochastic process governed by the α-fractional diffusion equation is shown to be equal to one of the conventional diffusion equation. Full article
Open AccessArticle On some Integral Representations of Certain G-Functions
Received: 3 November 2015 / Revised: 16 December 2015 / Accepted: 23 December 2015 / Published: 31 December 2015
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Abstract
This is a brief exposition of some statistical techniques utilized to obtain several useful integral equations involving G-functions. Full article
Open AccessArticle Some Aspects of Extended Kinetic Equation
Axioms 2015, 4(3), 412-422; https://doi.org/10.3390/axioms4030412
Received: 20 May 2015 / Revised: 27 August 2015 / Accepted: 31 August 2015 / Published: 18 September 2015
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Abstract
Motivated by the pathway model of Mathai introduced in 2005 [Linear Algebra and Its Applications, 396, 317–328] we extend the standard kinetic equations. Connection of the extended kinetic equation with fractional calculus operator is established. The solution of the general form of the
[...] Read more.
Motivated by the pathway model of Mathai introduced in 2005 [Linear Algebra and Its Applications, 396, 317–328] we extend the standard kinetic equations. Connection of the extended kinetic equation with fractional calculus operator is established. The solution of the general form of the fractional kinetic equation is obtained through Laplace transform. The results for the standard kinetic equation are obtained as the limiting case. Full article
Open AccessArticle Limiting Approach to Generalized Gamma Bessel Model via Fractional Calculus and Its Applications in Various Disciplines
Axioms 2015, 4(3), 385-399; https://doi.org/10.3390/axioms4030385
Received: 15 July 2015 / Revised: 10 August 2015 / Accepted: 10 August 2015 / Published: 26 August 2015
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Abstract
The essentials of fractional calculus according to different approaches that can be useful for our applications in the theory of probability and stochastic processes are established. In addition to this, from this fractional integral, one can list out almost all of the extended
[...] Read more.
The essentials of fractional calculus according to different approaches that can be useful for our applications in the theory of probability and stochastic processes are established. In addition to this, from this fractional integral, one can list out almost all of the extended densities for the pathway parameter q < 1 and q → 1. Here, we bring out the idea of thicker- or thinner-tailed models associated with a gamma-type distribution as a limiting case of the pathway operator. Applications of this extended gamma model in statistical mechanics, input-output models, solar spectral irradiance modeling, etc., are established. Full article
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Open AccessArticle An Overview of Generalized Gamma Mittag–Leffler Model and Its Applications
Axioms 2015, 4(3), 365-384; https://doi.org/10.3390/axioms4030365
Received: 6 May 2015 / Revised: 23 July 2015 / Accepted: 27 July 2015 / Published: 26 August 2015
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Abstract
Recently, probability models with thicker or thinner tails have gained more importance among statisticians and physicists because of their vast applications in random walks, Lévi flights, financial modeling, etc. In this connection, we introduce here a new family of generalized probability distributions associated
[...] Read more.
Recently, probability models with thicker or thinner tails have gained more importance among statisticians and physicists because of their vast applications in random walks, Lévi flights, financial modeling, etc. In this connection, we introduce here a new family of generalized probability distributions associated with the Mittag–Leffler function. This family gives an extension to the generalized gamma family, opens up a vast area of potential applications and establishes connections to the topics of fractional calculus, nonextensive statistical mechanics, Tsallis statistics, superstatistics, the Mittag–Leffler stochastic process, the Lévi process and time series. Apart from examining the properties, the matrix-variate analogue and the connection to fractional calculus are also explained. By using the pathway model of Mathai, the model is further generalized. Connections to Mittag–Leffler distributions and corresponding autoregressive processes are also discussed. Full article
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Open AccessArticle On the Fractional Poisson Process and the Discretized Stable Subordinator
Axioms 2015, 4(3), 321-344; https://doi.org/10.3390/axioms4030321
Received: 20 June 2015 / Revised: 27 July 2015 / Accepted: 28 July 2015 / Published: 4 August 2015
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Abstract
We consider the renewal counting number process N = N(t) as a forward march over the non-negative integers with independent identically distributed waiting times. We embed the values of the counting numbers N in a “pseudo-spatial” non-negative half-line x ≥ 0 and observe
[...] Read more.
We consider the renewal counting number process N = N(t) as a forward march over the non-negative integers with independent identically distributed waiting times. We embed the values of the counting numbers N in a “pseudo-spatial” non-negative half-line x ≥ 0 and observe that for physical time likewise we have t ≥ 0. Thus we apply the Laplace transform with respect to both variables x and t. Applying then a modification of the Montroll-Weiss-Cox formalism of continuous time random walk we obtain the essential characteristics of a renewal process in the transform domain and, if we are lucky, also in the physical domain. The process t = t(N) of accumulation of waiting times is inverse to the counting number process, in honour of the Danish mathematician and telecommunication engineer A.K. Erlang we call it the Erlang process. It yields the probability of exactly n renewal events in the interval (0; t]. We apply our Laplace-Laplace formalism to the fractional Poisson process whose waiting times are of Mittag-Leffler type and to a renewal process whose waiting times are of Wright type. The process of Mittag-Leffler type includes as a limiting case the classical Poisson process, the process of Wright type represents the discretized stable subordinator and a re-scaled version of it was used in our method of parametric subordination of time-space fractional diffusion processes. Properly rescaling the counting number process N(t) and the Erlang process t(N) yields as diffusion limits the inverse stable and the stable subordinator, respectively. Full article
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Open AccessArticle Closed-Form Representations of the Density Function and Integer Moments of the Sample Correlation Coefficient
Axioms 2015, 4(3), 268-274; https://doi.org/10.3390/axioms4030268
Received: 7 May 2015 / Revised: 8 June 2015 / Accepted: 18 June 2015 / Published: 20 July 2015
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Abstract
This paper provides a simplified representation of the exact density function of R, the sample correlation coefficient. The odd and even moments of R are also obtained in closed forms. Being expressed in terms of generalized hypergeometric functions, the resulting representations are
[...] Read more.
This paper provides a simplified representation of the exact density function of R, the sample correlation coefficient. The odd and even moments of R are also obtained in closed forms. Being expressed in terms of generalized hypergeometric functions, the resulting representations are readily computable. Some numerical examples corroborate the validity of the results derived herein. Full article
Open AccessArticle On Elliptic and Hyperbolic Modular Functions and the Corresponding Gudermann Peeta Functions
Axioms 2015, 4(3), 235-253; https://doi.org/10.3390/axioms4030235
Received: 18 May 2015 / Revised: 10 June 2015 / Accepted: 19 June 2015 / Published: 8 July 2015
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Abstract
In this article, we move back almost 200 years to Christoph Gudermann, the great expert on elliptic functions, who successfully put the twelve Jacobi functions in a didactic setting. We prove the second hyperbolic series expansions for elliptic functions again, and express generalizations
[...] Read more.
In this article, we move back almost 200 years to Christoph Gudermann, the great expert on elliptic functions, who successfully put the twelve Jacobi functions in a didactic setting. We prove the second hyperbolic series expansions for elliptic functions again, and express generalizations of many of Gudermann’s formulas in Carlson’s modern notation. The transformations between squares of elliptic functions can be expressed as general Möbius transformations, and a conjecture of twelve formulas, extending a Gudermannian formula, is presented. In the second part of the paper, we consider the corresponding formulas for hyperbolic modular functions, and show that these Möbius transformations can be used to prove integral formulas for the inverses of hyperbolic modular functions, which are in fact Schwarz-Christoffel transformations. Finally, we present the simplest formulas for the Gudermann Peeta functions, variations of the Jacobi theta functions. 2010 Mathematics Subject Classification: Primary 33E05; Secondary 33D15. Full article

Review

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Open AccessReview Approach of Complexity in Nature: Entropic Nonuniqueness
Received: 8 July 2016 / Revised: 8 August 2016 / Accepted: 8 August 2016 / Published: 12 August 2016
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Abstract
Boltzmann introduced in the 1870s a logarithmic measure for the connection between the thermodynamical entropy and the probabilities of the microscopic configurations of the system. His celebrated entropic functional for classical systems was then extended by Gibbs to the entire phase space of
[...] Read more.
Boltzmann introduced in the 1870s a logarithmic measure for the connection between the thermodynamical entropy and the probabilities of the microscopic configurations of the system. His celebrated entropic functional for classical systems was then extended by Gibbs to the entire phase space of a many-body system and by von Neumann in order to cover quantum systems, as well. Finally, it was used by Shannon within the theory of information. The simplest expression of this functional corresponds to a discrete set of W microscopic possibilities and is given by S B G = k i = 1 W p i ln p i (k is a positive universal constant; BG stands for Boltzmann–Gibbs). This relation enables the construction of BGstatistical mechanics, which, together with the Maxwell equations and classical, quantum and relativistic mechanics, constitutes one of the pillars of contemporary physics. The BG theory has provided uncountable important applications in physics, chemistry, computational sciences, economics, biology, networks and others. As argued in the textbooks, its application in physical systems is legitimate whenever the hypothesis of ergodicity is satisfied, i.e., when ensemble and time averages coincide. However, what can we do when ergodicity and similar simple hypotheses are violated, which indeed happens in very many natural, artificial and social complex systems. The possibility of generalizing BG statistical mechanics through a family of non-additive entropies was advanced in 1988, namely S q = k 1 i = 1 W p i q q 1 , which recovers the additive S B G entropy in the q→ 1 limit. The index q is to be determined from mechanical first principles, corresponding to complexity universality classes. Along three decades, this idea intensively evolved world-wide (see the Bibliography in http://tsallis.cat.cbpf.br/biblio.htm) and led to a plethora of predictions, verifications and applications in physical systems and elsewhere. As expected, whenever a paradigm shift is explored, some controversy naturally emerged, as well, in the community. The present status of the general picture is here described, starting from its dynamical and thermodynamical foundations and ending with its most recent physical applications. Full article
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Open AccessReview An Overview of the Pathway Idea and Its Applications in Statistical and Physical Sciences
Axioms 2015, 4(4), 530-553; https://doi.org/10.3390/axioms4040530
Received: 18 October 2015 / Revised: 4 December 2015 / Accepted: 7 December 2015 / Published: 19 December 2015
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Abstract
Pathway idea is a switching mechanism by which one can go from one functional form to another, and to yet another. It is shown that through a parameter α, called the pathway parameter, one can connect generalized type-1 beta family of densities, generalized
[...] Read more.
Pathway idea is a switching mechanism by which one can go from one functional form to another, and to yet another. It is shown that through a parameter α, called the pathway parameter, one can connect generalized type-1 beta family of densities, generalized type-2 beta family of densities, and generalized gamma family of densities, in the scalar as well as the matrix cases, also in the real and complex domains. It is shown that when the model is applied to physical situations then the current hot topics of Tsallis statistics and superstatistics in statistical mechanics become special cases of the pathway model, and the model is capable of capturing many stable situations as well as the unstable or chaotic neighborhoods of the stable situations and transitional stages. The pathway model is shown to be connected to generalized information measures or entropies, power law, likelihood ratio criterion or λ - criterion in multivariate statistical analysis, generalized Dirichlet densities, fractional calculus, Mittag-Leffler stochastic process, Krätzel integral in applied analysis, and many other topics in different disciplines. The pathway model enables one to extend the current results on quadratic and bilinear forms, when the samples come from Gaussian populations, to wider classes of populations. Full article
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