On Extended d-D Kappa Distribution

Abstract

The thermal Doppler broadening of spectral profiles for particle populations in the absence or presence of potential fields can be described by kappa distributions. The kappa distribution provides a replacement for the Maxwell–Boltzmann distribution, which can be considered as a generalization for describing systems characterized by local correlations among their particles, as found in space and astrophysical plasmas. This paper presents all special cases of kappa distributions as members of a general pathway family of densities introduced by Mathai. The aim of the present paper is to bring to attention the application of various forms of the kappa distribution, its various special cases and its generalizations, which, in scalar-variable and multivariate situations, belong to a general family of distributions known as Mathai’s pathway models, comprising three different families of functions, namely the generalized type-1 beta, type-2 beta and gamma families. Through one parameter, known as the pathway parameter, one will be able to reach all the three families of functions and the stages of transitioning from one family to another. After pointing out the connection of multivariate (vector-variate) kappa distributions to the multivariate pathway model, the multivariate kappa distribution is extended to the real matrix-variate case by working out the various forms and by evaluating the normalizing constants of the various forms of the matrix-variate case explicitly. It is also pointed out that the pathway models are available for the scalar, vector and rectangular matrix-variate cases in the real domain as well as in the complex domain.

1. Introduction

Kappa distributions are used across the physics of astrophysical plasma processes, describing the velocities and energies of particles from solar wind and planetary magnetospheres to the heliosheath, and beyond to interstellar and intergalactic plasmas [1,2,3]. From the point of view of physics, the connection of kappa distributions with statistical mechanics and thermodynamics is important. The statistical origin of kappa distributions came from the maximization of Tsallis q-entropy [4]. From the point of view of astrophysics, kappa distributions were introduced in the 1960s by Binsack [5], Olbert [6], and Vasyliunas [7].

A particle is in thermal equilibrium when the exchange of heat or entropy in the system stops. The kappa index controls the exchange of entropy and it is also a measure of departure from the equilibrium state. The particle velocity distribution can be given in terms of a kappa distribution. In the case of a collisional plasma, where no local correlations among particles exist, the system is stabilized into a Maxwell–Boltzmann distribution. The kappa index is inversely proportional to the correlation between the energies of two particles. The d-D kappa distribution describes the particle velocity with dimensionality d. Livadiotis [8] and Livadiotis and McComas [9] take this particle velocity density as the following:

$$f(X;\theta ,k_d)=c_d{[1+{\displaystyle \frac{1}{k_d-\frac{d}{2}}}{\displaystyle \frac{{(X-\mu )}^{\prime }(X-\mu )}{{\theta }^2}}]}^{-(k_d+1)}$$

(1)

where X is a $d\times 1$ velocity vector, a prime denotes the transpose and $\mu =E\left[X\right]=<X>$ is the expected value of X, which is also a location parameter vector, where $E\left[\right(·\left)\right]$ denotes the expected value with respect to the density $f(X;\theta ,k_d)$, and the normalizing constant $c_d$ is the following:

$$c_d={\displaystyle \frac{\mathrm{\Gamma }(k_d+1)}{\mathrm{\Gamma }(k_d-\frac{d}{2}+1){\left[\pi (k_d-\frac{d}{2}){\theta }^2\right]}^{\frac{d}{2}}}}$$

(2)

and $\theta =\sqrt{k_{B}T/m}$ is the thermal speed of the particle with mass m and temperature T expressed in speed units. If $k_d-{\displaystyle \frac{d}{2}}$ is invariant over the dimensionality d then let $k_o$ be this constant value. Then, $k_d=k_o+{\displaystyle \frac{d}{2}}$. Then, the normalizing constant becomes

$$c_d={\displaystyle \frac{\mathrm{\Gamma }(k_o+1+\frac{d}{2})}{\mathrm{\Gamma }(k_o+1){\left[\pi k_o{\theta }^2\right]}^{\frac{d}{2}}}}$$

(3)

and the functional part is

$${[1+{\displaystyle \frac{1}{k_d-\frac{d}{2}}}{\displaystyle \frac{{(X-\mu )}^{\prime }(X-\mu )}{{\theta }^2}}]}^{-(k_d+1)}={[1+{\displaystyle \frac{1}{k_o}}{\displaystyle \frac{{(X-\mu )}^{\prime }(X-\mu )}{{\theta }^2}}]}^{-(k_o+1+{\displaystyle \frac{d}{2}})}$$

(4)

Since $k_d-{\displaystyle \frac{d}{2}}=k_o$ is invariant $k_3-{\displaystyle \frac{3}{2}}=k_d-{\displaystyle \frac{d}{2}}$, $k_3=k_d-{\displaystyle \frac{(d-3)}{2}}$. If $\theta$ depends on the kappa index, then $\theta$ is of the form ${\theta }_k$. Two forms of ${\theta }_k$ are usually taken. One is ${\theta }_k^2={\displaystyle \frac{(k_3-\frac{3}{2})}{k_3}}{\displaystyle \frac{(d-1)}{2}}{\theta }^2$ and the other is ${\theta }_{k,d}^2={\displaystyle \frac{k_o}{k_d}}{\theta }^2={\displaystyle \frac{(k_d-\frac{d}{2})}{k_d}}{\theta }^2$. If ${\theta }_k$ is used then (4) becomes

$${[1+{\displaystyle \frac{1}{k_d}}{\displaystyle \frac{{(X-\mu )}^{\prime }(X-\mu )}{{\theta }_k^2}}]}^{-(k_d+{\displaystyle \frac{1}{2}}(d-1))}.$$

(5)

On the other hand, if ${\theta }_{k,d}$ is used then (4) becomes

$${[1+{\displaystyle \frac{1}{k_d}}{\displaystyle \frac{{(X-\mu )}^{\prime }(X-\mu )}{{\theta }_{k,d}^2}}]}^{-(k_d+1)}.$$

(6)

Thus, one can write the kappa density in (1) in a number of different ways. One set of representations will be for $d=1,2,3$; another set will be with $\theta$ replaced by ${\theta }_k$ or ${\theta }_{k,d}$. When $k_o\to \infty$, (1) will approach the density

$$f_1(X;\theta ,k_d)=c_{\ast }{\mathrm{e}}^{-{\displaystyle \frac{{(X-\mu )}^{\prime }(X-\mu )}{{\theta }^2}}}$$

(7)

where $c_{\ast }={\left(\pi {\theta }^2\right)}^{-{\displaystyle \frac{d}{2}}}$. This density (6) is a multivariate real Gaussian and also becomes a $d-D$ Maxwellian distribution. Now, we look at the connection of this distribution with a very general family of models.

This paper is organized as follows: Section 2 deals with the multivariate pathway model and establishes the connection of multivariate kappa distribution to the multivariate pathway model. Section 3 derives Livadiotis’ d-D density through an optimization of Mathai’s entropy under two moment-type constraints. Section 4 provides the derivation of a matrix-variate generalization of Livadiotis’ multivariate density. Section 5 gives some concluding remarks and provides a glimpse to possible extensions of the present work.

2. A Pathway Family of Densities

All the forms of the kappa distribution considered in Section 1 are special cases of a general pathway family of densities introduced in Mathai [10]. Let X be a $p\times 1$ real-vector random variable with the covariance matrix $\Sigma =\mathrm{Cov}\left(X\right)=E\left[(X-E\left(X\right)){(X-E\left(X\right))}^{\prime }\right]$, where a prime denotes the transpose and $E(·)$ is the expected value of $(·)=<(·)>$ with respect to the density of X. The p components of X may be correlated among themselves. One can remove the effect of correlations by taking $Y={\Sigma }^{-{\displaystyle \frac{1}{2}}}(X-\mu )$, where ${\Sigma }^{{\displaystyle \frac{1}{2}}}$ is the real positive definite square root of the positive definite matrix $\Sigma >O$. Then, the covariance matrix in Y is the identity matrix, free of all correlations. The Euclidean distance of Y from the origin is also ${(X-\mu )}^{\prime }{\Sigma }^{-1}(X-\mu )$, $\mu =E\left(X\right)$. Also, ${(X-\mu )}^{\prime }{\Sigma }^{-1}(X-\mu )=\mathrm{a} \mathrm{positive} \mathrm{constant}$ is an offset ellipsoid and it is also known in statistical literature as the ellipsoid of concentration. If the components of X are to be simply weighted, rather than removing the effect of correlations, then one can replace ${\Sigma }^{-1}$ in the quadratic form by a real positive definite matrix $A>O$ and then we may consider ${(X-\mu )}^{\prime }A(X-\mu )$. Consider the following density:

$$g\left(X\right)\mathrm{d}X=c{\left[{(X-\mu )}^{\prime }{\Sigma }^{-1}(X-\mu )\right]}^{\gamma }{[1+a{\left({(X-\mu )}^{\prime }{\Sigma }^{-1}(X-\mu )\right)}^{\delta }]}^{-\eta }\mathrm{d}X$$

(8)

where $a>0,\delta >0,\eta >0,$ and $\gamma >0$ are all real scalar parameters, and c is the normalizing constant, which can be evaluated as

$$c={\displaystyle \frac{\delta \mathrm{\Gamma }\left(\frac{p}{\delta }\right)\mathrm{\Gamma }\left(\eta \right)a^{\frac{\gamma }{\delta }+\frac{p}{2\delta }}}{{|\Sigma |}^{\frac{1}{2}}{\pi }^{\frac{p}{2}}\mathrm{\Gamma }(\frac{\gamma }{\delta }+\frac{p}{2\delta })\mathrm{\Gamma }(\eta -\frac{\gamma }{\delta }-\frac{p}{2\delta })}},\eta -{\displaystyle \frac{\gamma }{\delta }}-{\displaystyle \frac{p}{2\delta }}>0$$

(9)

(see the derivation of c in Appendix A). Consider $\delta =1,\Sigma =I$ (identity matrix), $a={\displaystyle \frac{1}{{\theta }^2(k_d-\frac{d}{2})}},\eta =k_d+1,\gamma =0$. Then, (8) reduces to (1), which is the density given by Livadiotis [8]. The general representation in (8) and (9) has many advantages. This (8) is a member of the pathway family of distributions defined in Mathai [10]. Let $a={\displaystyle \frac{b}{\alpha -a_o}},b>0$, $\alpha >a_o$ and $\eta =\rho (\alpha -a_o),\rho >0$. Then, (8) becomes

$$g_1\left(X\right)\mathrm{d}X=c_1{\left[{(X-\mu )}^{\prime }{\Sigma }^{-1}(X-\mu )\right]}^{\gamma }{[1+{\displaystyle \frac{b}{\alpha -a_o}}{\left({(X-\mu )}^{\prime }{\Sigma }^{-1}(X-\mu )\right)}^{\delta }]}^{-\rho (\alpha -a_0)}\mathrm{d}X.$$

(10)

Note that when $\alpha \to a_o$ from the right, $g_1\left(X\right)$ approaches the density

$$g_2\left(X\right)\mathrm{d}X=c_2{\left[{(X-\mu )}^{\prime }{\Sigma }^{-1}(X-\mu )\right]}^{\gamma }{\mathrm{e}}^{-b\rho {\left[{(X-\mu )}^{\prime }{\Sigma }^{-1}(X-\mu )\right]}^{\delta }}\mathrm{d}X$$

(11)

where

$$c_2={\displaystyle \frac{\delta \mathrm{\Gamma }\left(\frac{p}{2}\right){\left(b\rho \right)}^{\frac{\gamma }{\delta }+\frac{p}{2\delta }}}{{\pi }^{\frac{p}{2}}{|\Sigma |}^{\frac{1}{2}}\mathrm{\Gamma }(\frac{\gamma }{\delta }+\frac{p}{2\delta })}},\delta >0,b>0,\rho >0.$$

(12)

The density in (11) can be taken as a power-transformed real multivariate Maxwell–Boltzmann density. For $\delta =1$, (11) is a form of multivariate Maxwell–Boltzmann density. Hence, if (11) is the stable density in a physical system, then the unstable neighborhood and transitional stages are determined by (10) for various values of the pathway parameter $\alpha$. We may also note that for $\alpha =k_d,a_o={\displaystyle \frac{d}{2}},b={\displaystyle \frac{1}{{\theta }^2}},\delta =1,\gamma =0,\rho ={\displaystyle \frac{k_d+1}{k_d-\frac{d}{2}}}$, one also has (1). This density in (10) has another advantage. For $\alpha <a_o$, we can write $\alpha -a_o=-(a_o-\alpha ),a_o-\alpha >0$ so that the density in (10) switches into a type-1 beta form of the density, given by

$$g_3\left(X\right)\mathrm{d}X=c_3{\left[{(X-\mu )}^{\prime }{\Sigma }^{-1}(X-\mu )\right]}^{\gamma }{[1-{\displaystyle \frac{b}{a_o-\alpha }}{\left({(X-\mu )}^{\prime }{\Sigma }^{-1}\left(X_{\mu }\right)\right)}^{\delta }]}^{\rho (a_o-\alpha )}\mathrm{d}X,\alpha <a_o$$

(13)

where $[1-{\displaystyle \frac{b}{a_o-\alpha }}{\left({(X-\mu )}^{\prime }{\Sigma }^{-1}(X-\mu )\right)}^{\delta }]>0$ so that the density is defined within the ellipsoid

$${(X-\mu )}^{\prime }{\Sigma }^{-1}(X-\mu )={({\displaystyle \frac{a_o-\alpha }{b}})}^{{\displaystyle \frac{1}{\delta }}} \mathrm{for} \alpha <a_o,b>0,\delta >0$$

and

$$c_3={\displaystyle \frac{\delta \mathrm{\Gamma }\left(\frac{p}{\delta }\right)\mathrm{\Gamma }(1+\rho (a_o-\alpha )+\frac{\gamma }{\delta }+\frac{p}{2\delta }){\left(\frac{b}{a_o-\alpha }\right)}^{\frac{\gamma }{\delta }+\frac{p}{2\delta }}}{{|\Sigma |}^{\frac{1}{2}}{\pi }^{\frac{p}{2}}\mathrm{\Gamma }(\frac{\gamma }{\delta }+\frac{p}{2\delta })\mathrm{\Gamma }(1+\rho (a_o-\alpha ))}},\alpha <a_o.$$

(14)

This normalizing constant is evaluated via the procedure in Appendix A.1 and the final integral is evaluated by using a type-1 beta integral. Thus, $g_1,g_2,$ and $g_3$ belong to the pathway family of densities. If the power-transformed Maxwell–Boltzmann density in (11) is the stable density in a physical system, then the unstable neighborhoods and the transitional stages are given by $g_1$ in (10) and $g_3$ in (13). One can switch among a generalized type-1 beta family, a type-2 beta family and a gamma family or Maxwell–Boltzmann family of distributions through the pathway parameter $\alpha$. In the model in (10), one can identify $\alpha$ with $k_d$ and $a_o$ with ${\displaystyle \frac{d}{2}}$ if convenient.

3. Livadiotis’ $\mathit{d}-\mathit{D}$ Density Through an Entropy Optimization

Let X be a $p\times 1$ real-vector random variable. Let $f\left(X\right)$ be a density function, that is, $f\left(X\right)\ge 0$ for all X and ${\int }_{X}f\left(X\right)\mathrm{d}X=1$, where $f\left(X\right)$ is a real-valued scalar function of X. Consider Mathai’s entropy for the vector random variable, namely

$$M_{\alpha }\left(f\right)={\displaystyle \frac{{\int }_X{\left[f\left(X\right)\right]}^{1+\frac{a_o-\alpha }{\eta }}\mathrm{d}X-1}{\alpha -a_o}}$$

(15)

where $a_o$ is a fixed quantity or anchoring point, $\alpha$ is the pathway parameter and the deviation of $\alpha$ from $a_o$ is measured in $\eta >0$ units. Then, when $\alpha \to a_o$, we can see that (15) reduces to Shannon’s entropy $S\left(f\right)=-K{\int }_{X}f\left(X\right)lnf\left(X\right)\mathrm{d}X$, where K is a constant. Shannon’s entropy is for the scalar variable case, and the corresponding real vector-variate form is denoted here as $S\left(f\right)$. Let $\Sigma >O$ be the covariance matrix of X. Consider the ellipsoid of concentration ${(X-\mu )}^{\prime }{\Sigma }^{-1}\left(X_{\mu }\right)=$ a positive constant, where $\mu$ is a $p\times 1$ location parameter vector. We will set moment-type constraints on the ellipsoid of concentration and then optimize (15). Consider the following constraints:

$$E{\left[{(X-\mu )}^{\prime }{\Sigma }^{-1}(X-\mu )\right]}^{\gamma ({\displaystyle \frac{a_o-\alpha }{\eta }})}=\mathrm{fixed}, E{\left[{(X-\mu )}^{\prime }{\Sigma }^{-1}(X-\mu )\right]}^{\gamma ({\displaystyle \frac{a_o-\alpha }{\eta }})+\delta }=\mathrm{fixed}$$

(16)

for some parameters $\gamma >0,\eta >0,\delta >0$. If we use a calculus of variation for the optimization, then the Euler equation becomes the following, where ${\lambda }_1$ and ${\lambda }_2$ are Lagrangian multipliers:

$${\displaystyle \frac{\partial }{\partial f}}[f^{1+{\displaystyle \frac{a_o-\alpha }{\eta }}}-{\lambda }_1{\left[{(X-\mu )}^{\prime }{\Sigma }^{-1}(X-\mu )\right]}^{\gamma ({\displaystyle \frac{a_o-\alpha }{\eta }})}f+{\lambda }_2{\left[{(X-\mu )}^{\prime }{\Sigma }^{-1}(X-\mu )\right]}^{\gamma ({\displaystyle \frac{a_o-\alpha }{\eta }})+\delta }f]=0.$$

This gives the solution for f as the following:

$$f={\nu }_1{\left[{(X-\mu )}^{\prime }{\Sigma }^{-1}(X-\mu )\right]}^{\gamma }{[1-{\displaystyle \frac{{\lambda }_2}{{\lambda }_1}}{\left[{(X-\mu )}^{\prime }{\Sigma }^{-1}(X-\mu )\right]}^{\delta }]}^{{\displaystyle \frac{\eta }{a_o-\alpha }}}$$

(17)

where ${\nu }_1,{\lambda }_1,{\lambda }_2$ are constants. Let ${\displaystyle \frac{{\lambda }_2}{{\lambda }_1}}=b(a_o-\alpha ),b>0$ and let ${\nu }_1$ be the normalizing constant to make (17) a density. Then, for $u={(X-\mu )}^{\prime }{\Sigma }^{-1}(X-\mu )$, we have the following densities from (17):

$$\begin{array}{cc}\hfill f_1\left(X\right)& ={\nu }_1{u}^{\gamma }{[1-b(a_o-\alpha )u^{\delta }]}^{{\displaystyle \frac{\eta }{a_0-\alpha }}},\alpha <a_o\hfill \end{array}$$

(18)

$$\begin{array}{cc}\hfill f_2\left(X\right)& ={\nu }_2{u}^{\gamma }{[1+b(\alpha -a_o)u^{\delta }]}^{-{\displaystyle \frac{\eta }{\alpha -a_o}}},\alpha >a_o\hfill \end{array}$$

(19)

$$\begin{array}{cc}\hfill f_3\left(X\right)& ={\nu }_3{u}^{\gamma }{\mathrm{e}}^{-b\eta u^{\delta }},\alpha \to a_o\hfill \end{array}$$

(20)

where we need an additional condition $1-b(a_o-\alpha )u^{\delta }>0$ in (18) in order to make it a density. Note that in the limiting form, we have the following properties:

$$\begin{array}{cc}\hfill {\mathrm{e}}^{-b\eta u^{\delta }}& =\underset{\alpha \to a_o}{lim}{[1-b(a_o-\alpha )u^{\delta }]}^{{\displaystyle \frac{\eta }{a_o-\alpha }}}=\underset{\alpha \to a_o}{lim}{[1+b(\alpha -a_o)u^{\delta }]}^{-{\displaystyle \frac{\eta }{\alpha -a_o}}}\hfill \\ \hfill & =\underset{\alpha \to a_o}{lim}{[1-{\displaystyle \frac{b}{a_o-\alpha }}{u}^{\delta }]}^{\eta (a_o-\alpha )}=\underset{\alpha \to a_o}{lim}{[1+{\displaystyle \frac{b}{\alpha -a_o}}{u}^{\delta }]}^{-\eta (\alpha -a_o)}\hfill \end{array}$$

(21)

Hence, one can take any one of the formats in (21) in the limiting case. Livadiotis’ density in (1) is available from (19) by taking $b(\alpha -a_o)={\displaystyle \frac{1}{\theta (k_d-\frac{d}{2})}}$ and ${\displaystyle \frac{\eta }{\alpha -a_o}}=k_d+1$.

For $\gamma =0,\delta =1,\eta =1,a_o=1,$ and $b=1$, Equations (18)–(20) give a real multivariate version of Tsallis statistics of non-extensive statistical mechanics [4]. For $\delta =1,\eta =1,a_o=1$, $b=1,$ and $\delta =1$, (19) and (20) can be taken as a multivariate extension of superstatistics of Beck and Cohen [11]. Matrix-variate versions of Tsallis statistics [4] and Beck and Cohen superstatistics [11] can also be defined.

4. A Matrix-Variate Generalization of Livadiotis’ Density

Let $Y=\left(y_{ij}\right)$ be a $p\times q,p\le q$ and of rank p matrix with distinct real scalar variables, $y_{ij}$, as elements. Let $g\left(Y\right)$ be a real-valued scalar function of Y such that $g\left(Y\right)\ge 0$ for all Y and ${\int }_{Y}g\left(Y\right)\mathrm{d}Y=1$, so that $g\left(Y\right)$ is a density function, where $\mathrm{d}Y={\wedge }_{i=1}^p{\wedge }_{j=1}^q\mathrm{d}{y}_{ij}=$ is the wedge product of all distinct differentials in Y. Let M be a $p\times q,p\le q$ parameter matrix. Let $A>O$ be $p\times p$ and $B>O$ be $q\times q$ positive definite constant matrices. Let $A^{{\displaystyle \frac{1}{2}}}$ and $B^{{\displaystyle \frac{1}{2}}}$ be the positive definite square roots of A and B, respectively. Let

$$U=A^{{\displaystyle \frac{1}{2}}}(Y-M)B^{{\displaystyle \frac{1}{2}}},V=U{U}^{\prime }=A^{{\displaystyle \frac{1}{2}}}(Y-M)B{(Y-M)}^{\prime }{A}^{{\displaystyle \frac{1}{2}}}.$$

(22)

Then, the determinant of V, namely $\left|V\right|$, can be interpreted in different ways. $\left|V\right|$ is the product of the eigenvalues of the matrix V. Let $U_1,\dots ,U_p$ be the p linearly independent rows of U, the $p\times q,p\le q$ and of rank p matrix U. Then, $U_j,j=1,\dots ,p$ can be taken as p linearly independent points in a q-dimensional Euclidean space with $p\le q$. Then, ${\left|V\right|}^{{\displaystyle \frac{1}{2}}}={\left|U{U}^{\prime }\right|}^{{\displaystyle \frac{1}{2}}}=$, which corresponds to the volume of the p-parallelotope generated in the convex hull of the p linearly independent points, taken in the given order. Thus, $\left|V\right|$ is also the square of the volume content of this parallelotope. Consider Mathis’s entropy in (15) with $f\left(X\right)$ replaced by $g\left(Y\right)$ where Y is now $p\times q,p\le q$ matrix of rank p. Consider the optimization of (15) with the real-valued scalar function $g\left(Y\right)$ under the following constraints:

$${E[|V|}^{\gamma ({\displaystyle \frac{a_o-\alpha }{\eta }})}]={\mathrm{fixed}, E\left[\right|V|}^{\gamma ({\displaystyle \frac{a_o-\alpha }{\eta }})}{\left[\mathrm{tr}\left(V\right)\right]}^{\delta }]=\mathrm{fixed}$$

(23)

Then, proceeding as in Section 3, we can end up with the following pathway densities:

$$\begin{array}{cc}\hfill g_1\left(Y\right)& =C_1{\left|V\right|}^{\gamma }{[1-b(a_o-\alpha )\left(\mathrm{tr}\left(V\right)\right)]}^{{\displaystyle \frac{\eta }{a_o-\alpha }}},\alpha <a_o\hfill \end{array}$$

(24)

$$\begin{array}{cc}\hfill g_2\left(Y\right)& =C_2{\left|V\right|}^{\gamma }{[1+b(\alpha -a_o)\left(\mathrm{tr}\left(V\right)\right)]}^{-{\displaystyle \frac{\eta }{\alpha -a_o}}},\alpha >a_o\hfill \end{array}$$

(25)

$$\begin{array}{cc}\hfill g_3\left(Y\right)& ={\left|V\right|}^{\gamma }{\mathrm{e}}^{-b\eta \mathrm{tr}\left(V\right)}\hfill \end{array}$$

(26)

where $C_1,C_2,$ and $C_3$ are the normalizing constants, and the additional condition needed in (24) is $1-b(a_o-\alpha )\mathrm{tr}\left(V\right)>0$ to make it a density, where

$$V=A^{{\displaystyle \frac{1}{2}}}(Y-M)B{(Y-M)}^{\prime }{A}^{{\displaystyle \frac{1}{2}}}.$$

(27)

The normalizing constants are evaluated in Appendix A.2 by using the following steps:

$$U=A^{{\displaystyle \frac{1}{2}}}(Y-M)B^{{\displaystyle \frac{1}{2}}}\Rightarrow \mathrm{d}U={\left|A\right|}^{{\displaystyle \frac{q}{2}}}{\left|B\right|}^{{\displaystyle \frac{p}{2}}}\mathrm{d}Y$$

and

$$V=U{U}^{\prime }\Rightarrow \mathrm{d}U={\displaystyle \frac{{\pi }^{\frac{pq}{2}}}{{\mathrm{\Gamma }}_p\left(\frac{q}{2}\right)}}{\left|V\right|}^{{\displaystyle \frac{q}{2}}-{\displaystyle \frac{p+1}{2}}}\mathrm{d}V$$

(28)

where ${\mathrm{\Gamma }}_p\left(\alpha \right)$ is a real matrix-variate gamma function, associated with a real matrix-variate gamma integral, as follows:

$$\begin{array}{cc}\hfill {\Gamma }_p\left(\alpha \right)& ={\int }_{S>O}{\left|S\right|}^{\alpha -{\displaystyle \frac{p+1}{2}}}{\mathrm{e}}^{-\mathrm{tr}\left(S\right)}\mathrm{d}S,\Re \left(\alpha \right)>{\displaystyle \frac{p-1}{2}}\hfill \end{array}$$

(29)

$$\begin{array}{cc}\hfill & ={\pi }^{{\displaystyle \frac{p(p-1)}{4}}}\Gamma \left(\alpha \right)\Gamma (\alpha -{\displaystyle \frac{1}{2}})...\Gamma (\alpha -{\displaystyle \frac{p-1}{2}}),\Re \left(\alpha \right)>{\displaystyle \frac{p-1}{2}}\hfill \end{array}$$

(30)

where $\Re (·)$ denotes the real part of $(·)$. Note that (26) is a real rectangular matrix-variate Maxwell–Boltzmann density.

• Note. Results parallel to all the results in Section 3 and Section 4 are also available in the complex domain. Corresponding physics can also be dealt with in the complex domain. A real rectangular matrix-variate version of Tsallis statistics [4] is available from Equations (24)–(26) for $\gamma +{\displaystyle \frac{q}{2}}={\displaystyle \frac{p+1}{2}},b=1,\alpha =1,a_o=1,\eta =1$. Corresponding results in the complex domain can also be worked out. For $b=1,\alpha =1,$ and $\eta =1$, (25) and (26) give a real rectangular matrix-variate version of superstatistics [11]. Corresponding versions in the complex domain can also be worked out. Such rectangular or square matrix-variate versions may not be available in the literature.

5. Conclusions

The manuscript introduces a single pathway density that brings together all univariate, multivariate and matrix-variate kappa distributions, including Livadiotis’s plasma models, Tsallis statistics, and their Maxwell–Boltzmann limit, by varying one parameter alpha, called the pathway parameter. It is shown how each case follows from an entropy maximization principle. This unified approach based on Mathai’s pathway covers generalized type-2 beta (kappa), type-1 beta and gamma/Maxwell–Boltzmann/Gaussian cases. All normalizing constants come from detailed multivariate integrals as shown in Appendix A.

For those who would like to estimate the parameters of the models introduced in this paper, the method of moments is the most appropriate procedure because a pathway model essentially consists of generalized type-1 beta, type-2 beta and gamma families of functions. For these families, the method of moments is the best procedure. Since the pathway model consists of three families of functions, there will be an infinite amount of possible shapes for the curves if one plans to plot the density for various parameter values in the scalar variable cases. Since the authors do not have any specific parameter points in mind, no attempt to plot the curves in the scalar case, for some parameter points, is made here. Plotting will essentially be equivalent to plotting generalized type-1 beta or type-2 beta or gamma densities. Since the authors do not have any dataset of interest at hand to fit these models, no fitting is attempted here.

For studying specific properties of each model, the steps will essentially be the steps used in the derivations of the normalizing constants. These steps are given in detail in Appendix A. Hence, no specific property is derived or discussed in the present paper. For example, if one wishes to study the properties of the positive definite quadratic form ${(X-\mu )}^{\prime }{\Sigma }^{-1}(X-\mu )$ in Appendix A.1, compute the total integral (detailed steps are given there) with the parameter $\gamma$ replaced by $\gamma +h$ for an arbitrary h, including complex values. Then, divide this total integral with the same quantity at $h=0$. The resulting quantity is $E{\left[{(X-\mu )}^{\prime }{\Sigma }^{-1}(X-\mu )\right]}^h$ or the h-th moment for an arbitrary h. One can study all sorts of properties by using this h-th moment for arbitrary h; one can take the inverse Mellin transform (for $h=s-1$, where a complex s is the Mellin transform of the quadratic form, whenever it exists) to obtain the exact density of the quadratic form.

If physical interpretations can be given, then one can look into the following possible generalizations: Let X be a $p\times q,p\le q$ matrix of rank p with distinct $pq$ real scalar variables as elements. Let $Y=A^{{\displaystyle \frac{1}{2}}}(X-M)B^{{\displaystyle \frac{1}{2}}}$, a doubly scaled $(X-M)$, where M is a a $p\times q$ parameter matrix, $A^{{\displaystyle \frac{1}{2}}}>O$ and $B^{{\displaystyle \frac{1}{2}}}>O$ are the positive definite square roots of the positive definite constant matrices $A>O,B>O$, respectively. We can show that $\mathrm{d}X={\left|A\right|}^{-{\displaystyle \frac{q}{2}}}{\left|B\right|}^{-{\displaystyle \frac{p}{2}}}\mathrm{d}X$ (see [12]),

where $\left|\right(·\left)\right|$ means the determinant of $(·)$:

$$\begin{array}{cc}\hfill f_1\left(Y\right)\mathrm{d}Y& =c_1{\left[\mathrm{tr}\left(Y{Y}^{\prime }\right)\right]}^{\eta }{[1-a_1(a_0-\alpha ){\left(\mathrm{tr}\left(Y{Y}^{\prime }\right)\right)}^{\rho }]}^{{\displaystyle \frac{\tau }{a_0-\alpha }}}{\mathrm{e}}^{-a{\left[\mathrm{tr}\left(Y{Y}^{\prime }\right)\right]}^{\delta }}\mathrm{d}Y\hfill \\ \hfill f_2\left(Y\right)\mathrm{d}Y& =c_2{\left|Y{Y}^{\prime }\right|}^{\gamma }{\left[\mathrm{tr}\left(Y{Y}^{\prime }\right)\right]}^{\eta }{[1-a_1(a_0-\alpha ){\left(\mathrm{tr}\left(Y{Y}^{\prime }\right)\right)}^{\rho }]}^{{\displaystyle \frac{\tau }{a_0-\alpha }}}{\mathrm{e}}^{-a{\left[\mathrm{tr}\left(Y{Y}^{\prime }\right)\right]}^{\delta }}\mathrm{d}Y\hfill \\ \hfill f_3\left(Y\right)\mathrm{d}Y& =c_3{\left|Y{Y}^{\prime }\right|}^{\gamma }{|I{\mathrm{bYY}}^{\prime }|}^{\eta }{\mathrm{e}}^{-a{\left[\mathrm{tr}\left(Y{Y}^{\prime }\right)\right]}^{\delta }}\mathrm{d}Y\hfill \end{array}$$

where $c_j,j=1,2,3$ are the normalizing constants, $a_1>0,\delta >0,\tau \ge 0,\gamma >0,b>0$, $I\pm bY{Y}^{\prime }>O$ (positive definite), $a\ge 0$. $f_1\left(Y\right)$ for $a=0$ is a standard scalar variable pathway model and all the cases $\alpha <a_0,\alpha >a_0$ and $\alpha \to a_0$ are available in the literature. $f_1\left(Y\right)$ for $a\ne 0,\tau \ne 0$ has recently been solved by the first author for the two cases $\rho >0$ and $\rho <0$. $f_2\left(Y\right)$ for $\tau =0$ is the famous Wishart–Kotz model. This is available in the literature. $f_2\left(Y\right)$ for $\tau \ne 0,a\ne 0$ has recently been solved by the first author for the two cases $\rho <0$ and $\rho >0$. $f_3\left(Y\right)$ for $a=0$ are the standard matrix-variate type-1 and type-2 beta densities and are available in the literature. But the case $a\ne 0$ is still a problem. Physical interpretations of all the three models for the general or specific parameters are still open problems. For more information on the mathematical aspects of univariate, multivariate and matrix-variate distributions in the real and complex domains, and for the various scalar and matrix scaling models and texture models, one may refer to Mathai et al. [13].