Loading kappa-type distributions in particle simulations

The kappa distribution is one of the most important velocity distributions in space plasmas. It has both a quasi-Maxwellian core and a suprathermal tail, and it has been considered throughout the heliosphere and magnetospheres. Despite a strong demand for numerical studies in a kappa-distributed plasma, it is not clear how to generate kappa distributions in particle velocities in particle-in-cell (PIC) and Monte Carlo simulations. In this contribution, we present numerical procedures to generate three kappa-type velocity distributions in particle simulations. First, we review an algorithm for the nonrelativistic kappa distribution. Mathematically, the kappa distribution is equivalent to the multivariate t-distribution [1]. A random variate following the multivariate t-distribution can be generated from the normal and chi-squared distributions. Second, we propose algorithms to generate a kappa loss-cone (KLC) distribution [2], which is often considered in the planetary magnetosphere. We have constructed two procedures. Using the mathematical properties of the beta prime distribution, we can straightforwardly generate the KLC distribution in particle simulations. Third, we propose a procedure for initializing a relativistic kappa distribution. Although the Lorentz factor makes this problem difficult, we have successfully developed a rejection-based algorithm [3]. The rejection part extends an earlier method for a relativistic Maxwell distribution [4], and it accepts particles at the rate of 95% or higher. Our method also use the beta prime distribution. As a result, we can successfully generate a power-law tail of the relativistic kappa distribution. References:[1] R. F. Abdul & R. L. Mace, Phys. Plasmas 22, 102107 (2015)[2] D. Summers & R. M. Thorne, J. Plasma Phys. 53, 293 (1995)[3] S. Zenitani & S. Nakano, Phys. Plasmas 29, 113904 (2022)[4] E. Canfield, W. M. Howard, & E. P. Liang, Astrophys. J 323, 565 (1987)