DIPLODOCUS Framework

DIPLODOCUS

Distribution In PLateaux methODOlogy for the compUtation of transport equationS.

Diplodocus is a novel framework for evolving the a set of particle distribution functions through the seven dimensions of phase space: one time, three space and three momentum. Evolution includes advection of particles, continuous forcing and discrete interactions (collisions) between particles.

More details of the framework and its numerical implementation can be found in the DIPLODOCUS series of papers, the first two are currently available Everett and Cotter (2026) and Everett et al. (2026)

Transport Equations for Particles

Transport equations refers to a set of equations that dictate the evolution of particles through phase space. In Diplodocus, that takes the form of an integrated Boltzmann equation:

$${\int }_{\partial Q}f(\mathit{x},\mathit{p})\mathit{\omega }={\int }_Q\mathit{C}(\mathit{x},\mathit{p}),$$

$Q$ is a volume in phase space, $f(\mathit{x},\mathit{p})$ is the particle distribution function, $\mathit{\omega }$ is the differential element for hypersurfaces $\partial Q$ bounding $Q$, such that $f(\mathit{x},\mathit{p})\mathit{\omega }$ measures the fluxes of particles through those boundaries. The integration of $\mathit{C}(\mathit{x},\mathit{p})$ over the volume $Q$ then measures the change of particles caused by collisions within $Q$.

Distribution-In-Plateaux

To solve the evolution described by the transport equations computationally, a method called "Distribution-In-Plateaux" is used to discretise particle distribution functions. In brief, a continuous distribution over a surface in phase space is divided into a number of plateaux over sub-areas of the surface. The "height" of this plateaux is then taken to be the average value of the continuous distribution over that sub-area.

This discretisation allows interactions between particles to be pre-computed while simultaneously providing a flexible and conservative numerical scheme for the transport of distribution function through phase space.

Pre-computation of collision terms is implemented in the DiplodocusCollisions.jl package.

Transport of the distribution functions is handled by the DiplodocusTransport.jl package.

Plotting of results is split into a separate package DiplodocusPlots.jl.

Reference

• Everett, C. N. and Cotter, G. (2026). DIPLODOCUS I: Framework for the evaluation of relativistic transport equations with continuous forcing and discrete particle interactions. The Open Journal of Astrophysics 9, arXiv:2508.13296 [astro-ph.HE].

• Everett, C. N.; Klinger-Plaisier, M. and Cotter, G. (2026). DIPLODOCUS II: Implementation of transport equations and test cases relevant to micro-scale physics of jetted astrophysical sources. The Open Journal of Astrophysics 9, arXiv:2510.12505 [astro-ph.HE].