Revisiting some analytical and numerical interpretations of Cairns and Kappa–Cairns distribution functions

In this tutorial, three-dimensional (3D) Cairns and Kappa-Cairns distribution functions are re-examined both analytically and numerically. The difference between one-dimensional (1D) and (3D) Cairns distribution functions (CDF) has been explained by deriving reduced 1D Cairns distribution function. It is noted that expressions of 1D and reduced 1D distributions such as Maxwellian and Kappa distributions are similar to each other, while the plots of 1D and reduced 1D CDF are significantly different from each other. The effect of non-thermality parameter on the 3D CDF is also studied by plotting it as a function of magnitude of the 3D velocity. It shows that the peak of the distribution function shifts toward higher speeds with an increase in the value of non-thermality parameter. The longitudinal dielectric response function is derived by using the 3D CDF for studying kinetic behavior of electrostatic waves in an unmagnetized isotropic plasma. The dielectric function is written in terms of plasma dispersion function and is then used to derive the dispersion relations and Landau damping increments for electron plasma waves, ion acoustic waves, and dust acoustic waves in a Cairns distributed plasma. The expressions of the dispersion relation and Landau damping rate of Cairns distributed plasma change into the corresponding expressions of the Maxwellian distributed plasma when the nonthermality parameter is taken equal to zero. Mathematical manipulations have been done for 3D Kappa-Cairns distribution function (KCDF) to find the correct normalization factor. An appropriate and valid range of values of the spectral index κ is obtained by calculating second moment of the velocity by integrating KCDF over 3D velocity space. The effect of non-thermality parameter and the spectral index κ on the 3D KCDF is studied by plotting it as a function of magnitude of the 3D velocity. It is also seen that the velocities where the tails of KCDF exists are much higher compared to the velocities where the tail of Kappa distribution occurs. For both 3D CDF and KCDF, it has been observed that the difference in the distribution functions becomes negligible when the value of the non-thermality parameter becomes more than 0.5.