Kappa distribution

The Kappa distribution or the generalized Lorentzian distribution is a one-parameter generalization of the Maxwellian (Gaussian) distribution. It is particularly useful in modeling suprathermal populations in space and astrophysical plasmas. The Kappa distribution reads as

$f^{\kappa }(v)=\frac{1}{2\pi {\left(\kappa w_{\kappa }^2\right)}^{3/2}}\frac{\mathrm{\Gamma }(\kappa +1)}{\mathrm{\Gamma }(\kappa -1/2)\mathrm{\Gamma }(3/2)}{(1+\frac{v^2}{\kappa w_{\mathrm{\kappa }}^2})}^{-(\kappa +1)}$

where $w_{\kappa }$ is the thermal velocity and $\mathrm{\Gamma }(x)$ is the Gamma function. The Kappa distribution interpolates between the Maxwellian distribution ($\kappa \to \mathrm{\infty }$) and the Lorentzian distribution ($\kappa =2$).

The use of the Kappa distribution in plasma physics dates back to the work of Vasyliunas in 1968. An alternative distribution that differs slightly from the distribution introduced by Vasyliunas was introduced by Leubner to make a bridge with nonextensive statistics. It reads as

$f^{\kappa }(v)=\frac{1}{v_{th}}\frac{1}{\sqrt{\kappa }}\frac{\mathrm{\Gamma }[\kappa ]}{\mathrm{\Gamma }[\kappa -1/2]}{[1+\frac{1}{\kappa }\frac{v^2}{v_{th}^2}]}^{-\kappa }$

Although their microscopic origin is not clear to date, some mechanisms are known to produce these distributions or very similar ones. Among these mechanisms, the interaction with external radiation or the presence of temperature fluctuations

References

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