Generalized Measure of Entropy, Mathai's Distributional Pathway Model, and Tsallis Statistics

The pathway model of Mathai (2005) mainly deals with the rectangular matrix-variate case. In this paper the scalar version is shown to be associated with a large number of probability models used in physics. Different families of densities are listed here, which are all connected through the pathway parameter 'alpha', generating a distributional pathway. The idea is to switch from one functional form to another through this parameter and it is shown that basically one can proceed from the generalized type-1 beta family to generalized type-2 beta family to generalized gamma family when the real variable is positive and a wider set of families when the variable can take negative values also. For simplicity, only the real scalar case is discussed here but corresponding families are available when the variable is in the complex domain. A large number of densities used in physics are shown to be special cases of or associated with the pathway model. It is also shown that the pathway model is available by maximizing a generalized measure of entropy, leading to an entropic pathway. Particular cases of the pathway model are shown to cover Tsallis statistics (Tsallis, 1988) and the superstatistics introduced by Beck and Cohen (2003).Comment: LaTeX, 13 pages, title changed, introduction, conclusions, and references update

Pathway Model, Superstatistics, Tsallis Statistics, and a Generalized Measure of Entropy

The pathway model of Mathai (2005) is shown to be inferable from the maximization of a certain generalized entropy measure. This entropy is a variant of the generalized entropy of order 'alpha', considered in Mathai and Rathie (1975), and it is also associated with Shannon, Boltzmann-Gibbs, Renyi, Tsallis, and Havrda-Charvat entropies. The generalized entropy measure introduced here is also shown to haveinteresting statistical properties and it can be given probabilistic interpretations in terms of inaccuracy measure, expected value, and information content in a scheme. Particular cases of the pathway model are shown to be Tsallis statistics (Tsallis, 1988) and superstatistics introduced by Beck and Cohen (2003). The pathway model's connection to fractional calculus is illustrated by considering a fractional reaction equation.Comment: LaTeX, 22 page

On extended thermonuclear functions through pathway model

The major problem in the cosmological nucleosynthesis is the evaluation of the reaction rate. The present scenario is that the standard thermonuclear function in the Maxwell-Boltzmann form is evaluated by using various techniques. The Maxwell-Boltzmannian approach to nuclear reaction rate theory is extended to cover Tsallis statistics (Tsallis, 1988) and more general cases of distribution functions. The main purpose of this paper is to investigate in some more detail the extended reaction probability integral in the equilibrium thermodynamic argument and in the cut-off case. The extended reaction probability integrals will be evaluated in closed form for all convenient values of the parameter by means of residue calculus. A comparison of the standard reaction probability integrals with the extended reaction probability integrals is also done.Comment: 21 pages, LaTe

On positivity of the Kadison constant and noncommutative Bloch theory

In an earlier paper, we established a natural connection between the Baum-Connes conjecture and noncommutative Bloch theory, viz. the spectral theory of projectively periodic elliptic operators on covering spaces. We elaborate on this connection here and provide significant evidence for a fundamental conjecture in noncommutative Bloch theory on the non-existence of Cantor set type spectrum. This is accomplished by establishing an explicit lower bound for the Kadison constant of twisted group C*-algebras in a large number of cases, whenever the multiplier is rational.Comment: Latex2e, 16 pages, final version, to appear in a special issue of Tohoku Math. J. (in press

Fusion yield: Guderley model and Tsallis statistics

The reaction rate probability integral is extended from Maxwell-Boltzmann approach to a more general approach by using the pathway model introduced by Mathai [Mathai A.M.:2005, A pathway to matrix-variate gamma and normal densities, Linear Algebra and Its Applications}, 396, 317-328]. The extended thermonuclear reaction rate is obtained in closed form via a Meijer's G-function and the so obtained G-function is represented as a solution of a homogeneous linear differential equation. A physical model for the hydrodynamical process in a fusion plasma compressed and laser-driven spherical shock wave is used for evaluating the fusion energy integral by integrating the extended thermonuclear reaction rate integral over the temperature. The result obtained is compared with the standard fusion yield obtained by Haubold and John in 1981.[Haubold, H.J. and John, R.W.:1981, Analytical representation of the thermonuclear reaction rate and fusion energy production in a spherical plasma shock wave, Plasma Physics, 23, 399-411]. An interpretation for the pathway parameter is also given.Comment: 17 pages, LaTe

Extended Reaction Rate Integral as Solutions of Some General Differential Equations

Here an extended form of the reaction rate probability integral, in the case of nonresonant thermonuclear reactions with the depleted tail and the right tail cut off, is considered. The reaction rate integral then can be looked upon as the inverse of the convolution of the Mellin transforms of Tsallis type statistics of nonextensive statistical mechanics and stretched exponential as well as that of superstatistics and stretched exponentials. The differential equations satisfied by the extended probability integrals are derived. The idea used is a novel one of evaluating the extended integrals in terms of some special functions and then by invoking the differential equations satisfied by these special functions. Some special cases of limiting situations are also discussed.Comment: 9 pages, LaTe

Explicit Evaluations of Matrix-variate Gamma and Beta Integrals in the Real and Complex Cases

Matrix transformations in terms of triangular matrices is the easiest method of evaluating matrix-variate gamma and beta integrals in the real and complex cases. Here we give several procedures of explicit evaluation of gamma and beta integrals in the general real and complex situations. The procedure also reveals the structure of these matrix-variate integrals. Apart from the evaluation of matrix-variate gamma and beta integrals, the procedure can also be applied to evaluate such integrals explicitly in similar situations. Various methods described here will be useful to those who are working on integrals involving real-valued scalar functions of matrix argument in general and gamma and beta integrals in particular.Comment: 17 pages, LaTe