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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.

Abstract:

Particle interactions are key elements of many dynamical systems. In the context of systems described by a Boltzmann equation, such interactions may be described by a collision integral, a multidimensional integral over the momentum-phase space of the interaction. This integral is often simplified by assuming isotropic particle distributions; however, such an assumption places constraints on the dynamics of the system. This paper presents computational forms of the collision integral for relativistic, binary interactions at three levels of anisotropy, including a novel form in the isotropic case. All these forms are split into two parts, an absorption and an emission spectrum, which may be pre-calculated via numerical integration for simulation purposes. We demonstrate the use of these forms by comparison with the analytically integrated, isotropic emission spectrum of electron–positron annihilation, which are shown to agree to numerical precision. The emission spectrum is then further extended to axisymmetric particle distributions, where two-dimensional spectral maps can be generated to provide new insight.