Loading non-Maxwellian Velocity Distributions in Particle Simulations

TL;DR

The paper addresses the challenge of generating non-Maxwellian velocity distribution functions (VDFs) for particle-in-cell simulations in heliophysics. It develops dedicated, algorithmically explicit sampling strategies for nine VDFs, including the $(r,q)$ distribution, regularized and subtracted kappa distributions, ring and shell components (and their Maxwellian variants), as well as super-Gaussian and filled-shell forms, all using only three elemental variates: uniform, normal, and gamma. Key contributions are the beta-prime and piecewise rejection sampling methods, explicit efficiency analyses, and validated Monte Carlo tests demonstrating faithful reproduction of target distributions. The practical impact lies in enabling accurate, efficient kinetic modeling of diverse, scientifically relevant VDFs in the solar wind and planetary magnetospheres, with ready-to-implement recipes and clear guidance on method selection.

Abstract

Numerical procedures for generating non-Maxwellian velocity distributions in particle simulations are presented. First, Monte Carlo methods for an $(r,q)$ distribution that generalizes flattop and kappa distributions are discussed. Then, two rejection methods for the regularized kappa distribution are presented, followed by a comparison in the $κ$ space. A simple recipe is proposed for the subtracted kappa distribution. Properties and numerical recipes for the ring and shell distributions with a finite Gaussian width are discussed. The ring and shell Maxwellians are further introduced as alternatives to the ring and shell distributions. Finally, the super-Gaussian and the filled-shell distributions are presented.

Figures (5)

• Figure 1: (a) Phase-space density of the $(r,q)$ distribution with $(r,q,\theta_\parallel, \theta_\perp)=(2,2,\theta,\theta)$. The distribution function (black line; Eq. \ref{['eq:rq']}) and normalized Monte Carlo results (blue histogram) of $10^6$ particles. The solid and dashed orange lines show the left and right envelope functions (Eq. \ref{['eq:rq_env']}). (b) Acceptance efficiency for the $(r,q)$ distribution (Eq. \ref{['eq:rq_eff']}). The vertical red line corresponds to the kappa distribution and the red dashed line corresponds to the flattop distribution. Below the blue dashed line, the beta-prime method requires a gamma distribution with a shape parameter less than unity in the denominator.
• Figure 2: (a) Radial velocity distribution of the regularized kappa distribution. The distribution function with $(\kappa, \theta, \alpha)=(1,1,0.05)$ (red line; Eq. \ref{['eq:RKD']}) is compared with numerical results of $10^6$ particles (blue histogram). For reference, the standard kappa distribution $(\kappa, \theta, \alpha)=(1,1,0)$ (black line; Eq. \ref{['eq:kappa']}) and normalized numerical results of $10^6$ particles (gray histogram) are presented. (b) Acceptance efficiency. The theory (solid lines; Eq. \ref{['eq:RKD_eff']}) and numerical results (squares) of the post-rejection method, and the theory (dashed lines; Eq. \ref{['eq:RKD_eff2']}) and numerical results (circles) of the piecewise rejection method are presented. The dash-dotted and dotted purple lines indicate upper-bounds (Eqs. \ref{['eq:RKD_eff3']} and \ref{['eq:RKD_eff4']}).
• Figure 3: Monte Carlo sampling of the subtracted kappa distribution ($\theta=1.0$, $\kappa=3.5$, $\beta=0.5$) with $10^6$ particles. Phase-space density is shown in the $v_{\perp}$--$v_{\parallel}$ plane. The filling parameter is (a) $\Delta=0.0$ and (b) $\Delta=0.2$, respectively.
• Figure 4: Left) Perpendicular velocity distributions of the ring distributions, $2\pi v_\perp \times g_\textrm{ring}(v_\perp)$. Right) Radial velocity distributions of the shell distributions, $4\pi v^2 \times f(v)$. (a) The ring distribution (Eq. \ref{['eq:ring0']}), (b) the shell distribution (Eq. \ref{['eq:shell0']}), (c) the ring Maxwellian (Eq. \ref{['eq:ringM']}), and (d) the shell Maxwellian (Eq. \ref{['eq:shellM']}) are presented by the black lines. The blue histograms are numerical results with $10^6$ particles. (e) The ratios of the ring distribution to the ring Maxwellian for $V=5\theta_\perp$ (dashed lines) and for $V=\theta_\perp$ (dotted lines). (f) The ratios of the shell distribution to the shell Maxwellian for $V=5\theta$ (dashed lines) and for $V=\theta$ (dotted lines). The orange dashed lines in Panels (a,b) are envelope functions for the piecewise rejection method, described in Appendix A.
• Figure 5: The phase-space densities (black lines) and Monte Carlo results of $10^6$ particles (blue histograms) for (a) the super Gaussian distribution ($p=3$, $\theta=1$) and (b) the filled-shell distribution ($p=-3/2$, $V=2$) are presented.