An Overview of the Pathway Idea and Its Applications in Statistical and Physical Sciences

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 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.Axioms 2015, 4 531

Non-Debye impedance and relaxation models for dissipative electrochemical capacitors

“…24 belongs to a particular case of a type-1 beta family of functions and for q 1 > 1 (by writing 1−q 1 = −(q 1 −1)), Eq. 24 belongs to a particular case of a type-2 beta family of functions [38][39][40][41][42][43] . The half lifetime characteristic in this case is:…”

Multiple waves of COVID-19: A pathway model approach

“…It is worth noting that the right-hand side of (1), with q > 1, corresponds to the generalized type-2 beta density function, which is one of the limiting cases of the so-called pathway approach to describing certain complex systems Mathai (2005); Sebastian et al (2015). In the pathway approach the parameter q can be varied so as to produce a rather extensive family of probability distributions, ranging from the generalized type-1 beta functions (q < 1) to the generalized type-2 beta functions (q > 1), while also recovering in the limit q → 1 the generalized gamma distribution and other related distributions Sebastian et al (2015). In epidemic dynamics, which is our main focus here, the relevant range is q > 1.…”

“…The pathway approach in its probabilistic version has been applied to a great variety of physical phenomena, for instance in astrophysics and statistical mechanics Sebastian et al (2015). More recently, this approach was also used to model epidemic dynamics in the context of the COVID-19 pandemic Tsallis and Tirnakli (2020), where the main idea was to predict the day in which the peak of the curves of active cases and daily deaths would be achieved in various countries.…”

“…In this paper, we consider a different class of models -called the pathway approach -where the growth rate for the quantity of interest, say, the total number of cases or deaths, is given as a known, explicit function of time. The pathway approach was originally introduced to describe, in a unified manner, a large family of probability distributions Mathai (2005); Sebastian et al (2015), but recently it has been applied to one-wave COVID-19 curves of cases and deaths Tsallis and Tirnakli (2020). Here we extend the model to multiwave epidemic curves by assuming that the model parameters become time dependent, so as to reflect the changes in the underlying epidemiological conditions associated with the successive waves of infection.…”

Mittag-Leffler functions in superstatistics

“…f (β) = δ(β − β 0 ) we retrieve the Boltzmann-Gibbs statistic. In more complex situations the f (β) distribution may allow long tail which implies a number of new shapes for the generalised Boltzmann factor [19,28,42]. An important point to consider is how to define the probability distribution associated with generalised Boltzmann factor, introduced in the article [6,48] as follows…”