Bivariate superstatistics: an application to statistical plasma physics

Abstract

From a bivariate superstatistics, we show that it is possible to obtain a generalized Kappa distribution, currently known for their great performance describing many anisotropic high-energy tail plasmas. Some particular cases obtained through this procedure are shown in this paper, and, on the other hand, the bimodality effect of marginal on the stationary superstatistical distribution is explored.

Graphic abstract

Appendix

Let the marginal case

$$\begin{aligned} B(v)= & {} \displaystyle \frac{\alpha }{\lambda \pi ^{3/2}Z(\alpha , \lambda , \delta )} \left( 1+\frac{v^2}{\lambda }\right) ^{-(\alpha +1)}\\&\quad \displaystyle \times \left[ 2 - 2\frac{\delta (\alpha +1)}{\lambda +v^2} + \frac{\delta ^2 (\alpha +1)(\alpha +2)}{(\lambda +v^2)^2}\right] . \end{aligned}$$

By using the change of variables $x=v^2$ and the fact that

$$\begin{aligned} \displaystyle \int _0^{\infty } \displaystyle \frac{\alpha }{\lambda } \left( 1+\frac{x}{\lambda }\right) ^{-(\alpha +1)} \mathrm{{d}}x = 1, \end{aligned}$$

we obtain the n-moment of the velocity, given by $\mathbb {E}(v^n) =0$ for n odd, and

$$\begin{aligned} \begin{array}{lll} \mathbb {E}(v^n) &{} = &{} \displaystyle \frac{\lambda ^{\frac{n-1}{2}}}{\pi ^{3/2}} \frac{\Gamma (\alpha -\frac{n-1}{2})\Gamma (\frac{n-1}{2}+1)}{\Gamma (\alpha )} \frac{1}{Z(\alpha , \lambda , \delta )} \\ &{}&{} \times \displaystyle \left[ 2 {-} 2\frac{\delta (\alpha {-}\frac{n{-}1}{2})}{\lambda } {+} \frac{\delta ^2 (\alpha -\frac{n-1}{2}) (\alpha +1 -\frac{n-1}{2})}{\lambda ^2}\right] , \end{array}\nonumber \\ \end{aligned}$$

(17)

if n is even and $\alpha > \frac{n-1}{2}$. Finally,

$$\begin{aligned} \begin{array}{lll} \mathrm{{Kurt}}(v) = \displaystyle \frac{3\lambda ^{\frac{1}{2}}\pi }{(\alpha -\frac{3}{2})} \frac{\Gamma (\alpha )}{\Gamma (\alpha - \frac{1}{2} )} \frac{ \displaystyle \left( 2 {-} 2\frac{\delta \alpha }{\lambda } {+} \frac{\delta ^2 \alpha (\alpha {+}1)}{\lambda ^2}\right) \left( 2 {-} 2\frac{\delta (\alpha {-}\frac{3}{2})}{\lambda } {+} \frac{\delta ^2 (\alpha {-}\frac{3}{2}) (\alpha {-} \frac{1}{2})}{\lambda ^2}\right) }{\displaystyle \left( 2 {-} 2\frac{\delta (\alpha {-}\frac{1}{2})}{\lambda } {+} \frac{\delta ^2 (\alpha {-}\frac{1}{2}) (\alpha {+} \frac{1}{2})}{\lambda ^2}\right) ^2}, \end{array} \end{aligned}$$

(18)

for $\alpha >\frac{3}{2}$.