Nonthermal and suprathermal distributions as a consequence of superstatistics

We propose to put the well-known nonthermal and suprathermal empirical distributions, used in plasma physics, onto a more rigorous foundation. Their use is frequently criticized because of a lack of formal derivation and physical explanation. A connection between these non-Maxwellian distributions and the Beck-Cohen superstatistics is suggested. They are perceived as a consequence of typical temperature fluctuations. We show that the suprathermal distribution is generated by the $\Gamma$ distribution of the inverse temperature, in the same way as the Tsallis $q$ statistics. The nonthermal distribution also follows from the ${\chi }^2$ distribution, with a small variance. Our contribution provides a possible physical meaning for these ad hoc distributions.

Figure 1

Variation of the $\Gamma$ distribution with the inverse temperature $\beta$ for different values of the parameters $b$ and $c$ ($b=1,c=2$, solid line; $b=2,c=2$, dashed line; $b=1,c=1$, dotted line).

Figure 2

Plot of the suprathermal distribution for different values of $c=3$ ($\kappa =2$, solid line), $c=3.5$ ($\kappa =2.5$, dashed line), and $c=4$ ($\kappa =3$, dotted line), with $b=1$.

Figure 3

Plot of the nonthermal distribution for $c=1/2$ and $b=1$.