DIPLODOCUS II: Implementation of transport equations and test cases relevant to micro-scale physics of jetted astrophysical sources

TL;DR

DIPLODOCUS provides a comprehensive framework for transporting particle distribution functions through phase space, incorporating collisions, emissive processes, and external forces relevant to micro-scale jet physics. The Julia-based Diplodocus.jl package implements Monte-Carlo pre-computation of collision matrices and a robust upwind transport scheme, enabling both isotropic comparisons with single-zone codes like AM3 and anisotropic investigations such as helical magnetic-field configurations. The paper validates the approach across elastic collisions, gyration/drift in ExB fields, radiation reaction, and synchrotron/SSC emissions, highlighting both agreements with established models and the new insights arising from full angular and non-linear coupling. The results demonstrate Diplodocus.jl’s potential to refine jet modeling and parameter estimation by capturing anisotropic micro-physics with a scalable, open-source framework, setting the stage for macro-scale tests in the next paper.

Abstract

DIPLODOCUS (Distribution-In-PLateaux methODOlogy for the CompUtation of transport equationS) is a framework being developed for the general transport of particle distribution functions through the seven dimensions of phase space, including forcing terms and interactions between particles. Following Paper I, which details the background analytic framework, this second paper provides an overview of the numerical implementation in the form of the code package Diplodocus$.$jl, written in Julia, including the description of a novel Monte-Carlo sampling technique for the pre-computation of anisotropic collision integrals. In addition to the discussion of numerical implementation, a selection of test cases are presented to examine the package's capabilities. These test cases focus on micro-scale physical effects: binary collisions, emissive interactions and external forces that are relevant to the modelling of jetted astrophysical sources, such as Active Galactic Nuclei and X-Ray Binaries.

Figures (24)

• Figure 1: A generic function $F(x,y)$ (Eq. (\ref{['eqn: MC test func']})), with magnitude given by the function $H(x,y)$ and $x$ is a dependent variable constrained by $x=G(y)$. Accurate Monte-Carlo integration of this function over sub-domains $x$ (e.g. the sub-domain $\Delta x$) is difficult as $x$ cannot be directly sampled and a large contribution to the integral estimate can arise from a small domain $y$ values.
• Figure 2: Isotropic emission spectrum for inverse-Compton scattering of photons with momenta $p=\qty{e-8}\,[m_ec]$ by a electrons with momenta $p=\qty{e6}\,[m_ec]$, generated using the methods discussed in Sections \ref{['subsec: importance sampling']} and \ref{['subsec:weighting']}, with the differential cross section given in Appendix \ref{['app: diff x sections']}. The scale factor $s$ adjust which angles of the outgoing photon are sampled and therefore affect the area of the dependent photon momentum that is sampled. The total integral estimate (black dots) is then calculated using a weighted average (Eq. (\ref{['eqn: weighted average']})) and closely matches the approximate spectrum given by sarkarDissectingComptonScattering2019 (white line).
• Figure 3: Locations of the values $h^\pm$ and the fluxes $\mathcal{B}^\pm$ with respect to boundaries of phase-space sub-domains (grid cells) in the $\beta$ coordinate direction.
• Figure 4: Time evolution of the hard sphere distribution in momentum space undergoing elastic collisions. The distribution is presented in polar coordinates, with $\theta=\arccos u$, $r=\log_{10}p$, with the colour scale indicating the magnitude of the distribution.
• Figure 5: Time evolution of the angle averaged momentum distribution of hard spheres undergoing elastic collisions. Additionally plotted (black line) is the expected Maxwell-Jüttner (thermal) distribution.