An approximate Kappa generator for particle simulations

TL;DR

This paper addresses the need for a GPU-friendly random number generator for the Kappa velocity distribution, a key model in space-plasma simulations, by replacing rejection-based sampling with an inverse-transform strategy. It introduces an approximate CDF $G(x)$ built from a $q$-exponential form and derives the corresponding sampling procedure with three uniform variates, avoiding control-flow divergence inherent to rejection methods. The authors provide explicit parameterizations for $a$, $b$, $c$, and $q^*$, along with a closed-form surrogate for $c$, and demonstrate that the approximation is highly accurate for $κ < 4$, with energy-density errors below $10^{-3}$ except around $κ ≈ 4.1$. Benchmarking shows substantial speedups on GPUs compared to standard gamma-based or Pareto rejection methods, highlighting the method’s practical value for large-scale, SIMT-based kinetic simulations of Kappa-distributed plasmas.

Abstract

A random number generator for the Kappa velocity distribution in particle simulations is proposed. Approximating the cumulative distribution function with the q-exponential function, an inverse transform procedure is constructed. The proposed method provides practically accurate results, in particular for k<4. It runs fast on graphics processing units (GPUs). The derivation, numerical validation, and relevance to GPU execution models are discussed.

Figures (8)

• Figure 1: The two parameters, $a$ (Eq. \ref{['eq:a']}) and $b/c$ (Eq. \ref{['eq:b']}), are displayed as a function of $\kappa$.
• Figure 2: The relative entropy $\mathcal{D}$ (colormap) is shown as a function of $(\kappa,c)$. The optimum line (black) and the approximate line (Eq. \ref{['eq:c']}; red) are overplotted.
• Figure 3: Comparison of the Kappa distribution (Eq. \ref{['eq:kappa']}) and the proposed approximation (Eq. \ref{['eq:approx']}). (a) Phase space density as a function of $v$. The thin black lines indicate the Kappa distribution, and the color thick lines indicate the proposed approximation. (a) Phase space density of the proposed approximation, normalized by the exact Kappa distribution.
• Figure 4: (a) The relative entropy between the approximate distribution (Eq. \ref{['eq:approx_x']}) and the exact distribution (Eq. \ref{['eq:kappax']}) as a function of $\kappa$. (b) The normalized difference in energy density, $\pm \Delta \mathcal{E} / \mathcal{E}_{\kappa} = \pm (\mathcal{E}_{\rm approx}-\mathcal{E}_{\kappa}) / \mathcal{E}_{\kappa}$ as a function of $\kappa$.
• Figure 5: Monte Carlo generation of the approximated Kappa distribution (Eq. \ref{['eq:approx']}; histograms) and the exact solutions (Eq. \ref{['eq:kappa']}) of the Kappa distribution (solid curves) (a) on the logarithmic scale and (b) on the linear scale.