DIPLODOCUS I: Framework for the evaluation of relativistic transport equations with continuous forcing and discrete particle interactions

TL;DR

DIPLODOCUS introduces a covariant, mesoscopic framework for relativistic transport of particle distributions in seven-dimensional phase space, incorporating both continuous conservative/non-conservative forces and discrete collisions. It first formulates a coordinate-free Boltzmann equation on a manifold, then extends to coordinate-dependent dynamics with a 7D mass-shell space $\mathcal{M}_m$ defined by $p^0=\sqrt{m^2+(p^i)^2}$ and a stationary observer formalism to define moments and fluxes. The key advancement is Distribution-In-Plateaux (DIP), a discretization that partitions phase space into plateaux with constant $f_{\alpha\beta\gamma\delta ijk}$, enabling pre-computed collision gain/loss arrays $G$ and flux coefficients $\mathcal{A},\mathcal{B},...$, yielding a conservative, scalable transport scheme. This framework decouples the numerical transport from spacetime geometry, paving the way for robust kinetic modelling of relativistic astrophysical systems such as AGN jets, with Paper II detailing the numerical implementation and Paper III exploring macroscopic tests.

Abstract

DIPLODOCUS (Distribution-In-PLateaux methODOlogy for the CompUtation of transport equationS) is a novel framework being developed for the mesoscopic modelling of astrophysical systems via the transport of particle distribution functions through the seven dimensions of phase space, including continuous forces and discrete interactions between particles. This first paper in a series provides an overview of the analytical framework behind the model, consisting of an integral formulation of the relativistic transport equations (Boltzmann equations) and a discretisation procedure for the particle distribution function (Distribution-In-Plateaux). The latter allows for the evaluation of anisotropic interactions, and generates a conservative numerical scheme for a distribution function's transport through phase space.

Figures (7)

• Figure 1: The worldline of a single particle through phase space ($\mathcal{M}$) between two points $A$ and $B$, defined by a curve $\gamma(\lambda)$, with $\lambda$ being some affine parameter of that curve, and tangent to the vector $\bm{v}$.
• Figure 2: A congruence of worldlines $\gamma_{q}$ through phase space $\mathcal{M}$ defined by the vector field $\bm{v}$ at every point $q\in\mathcal{M}$.
• Figure 3: Introduction of the distribution function $f$, a scalar field on the manifold $\mathcal{M}$ along with an integration domain $Q$ with volume element $\bm{\Omega}$ and boundary $\partial Q$ with hypersurface element $\bm{\omega}$. The integral $\int_{\partial Q}f\bm{\omega}$ then measures the number of worldlines (particles) passing through the hypersurface $\partial Q$.
• Figure 4: Collisions (discrete transfers of momentum) are described by the termination (crosses) and starting (bar) of particle worldlines within phase space. Here a particle $\mathfrak{a}$ at a position $q_\mathfrak{a}$ collides with a particle $\mathfrak{b}$ at a position $q_\mathfrak{b}$, terminating each worldline. The locations $q_\mathfrak{a}$ and $q_\mathfrak{b}$ need not be coincident as the particles may have different momenta (a coordinate in phase space). The collision may cause a discrete exchange of momentum, whereby the particles then re-appear at points $q'_\mathfrak{a}$ and $q'_\mathfrak{b}$, starting a new pair of worldlines. The integral $\int_Q\bm{C}$ then counts the number of collisions (worldlines that are started and/or terminated) within the volume $Q$.
• Figure 5: Timelike $\Sigma$ and spacelike $\Lambda$ hypersurfaces in phase space $\mathcal{M}_m$ defined by a set of stationary observers with trajectory tangent to $\bm{n}$. The spacelike hypersurfaces $\Sigma$ define a set of simultaneous events as measured by the observer.