Kappa distribution
= Kappa distribution
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A <Kappa distribution> is a heavy-tailed distribution used to model particle velocities with a suprathermal population. One isotropic three-dimensional convention is $f_\kappa(\mathbf v)=A_\kappa[1+v^2/(\kappa\theta^2)]^{-(\kappa+1)}$, where $A_\kappa$ normalizes the <velocity> density and $\theta$ is a speed parameter.
For fixed $\theta$, the limit $\kappa\to\infty$ is proportional to $e^{-v^2/\theta^2}$. Finite $\kappa$ gives a power-law tail instead. The kinetic second moment requires $\kappa>3/2$: multiplying the large-speed density by $v^4$ gives a radial <integral> proportional to $\int v^{2-2\kappa}dv$. A fitted speed parameter must not be identified with <temperature> without checking the convention.<Electron kappa distribution and quasi-thermal noise by Peter H. Yoon>[Introduction and velocity-distribution conventions]{ref}