Solving the Homogeneous Boltzmann Equation with Arbitrary Scattering Kernel

TL;DR

This paper introduces a generalized method for solving the nonlinear space homogeneous Boltzmann equation with arbitrary scattering kernels, applicable to both relativistic and classical cases, and capable of handling complex particle interactions.

Contribution

It extends existing methods by expanding the scattering matrix element in terms of cosine functions, enabling a unified approach for diverse particle scattering laws and quantum effects.

Findings

Method recovers previous results in Fermi-approximation

Applicable to reactive particle mixtures with quantum effects

Suitable for large networks of Boltzmann equations

Abstract

With applications in astroparticle physics in mind, we generalize a method for the solution of the nonlinear, space homogeneous Boltzmann equation with isotropic distribution function to arbitrary matrix elements. The method is based on the expansion of the matrix element in terms of two cosines of the "scattering angles". The scattering functions used by previous authors in particle physics for matrix elements in Fermi-approximation are retrieved as lowest order results in this expansion. The method is designed for the unified treatment of reactive mixtures of particles obeying different scattering laws, including the quantum statistical terms for blocking or stimulated emission, in possibly large networks of Boltzmann equations. Although our notation is the relativistic one, as it is used in astroparticle physics, the results can also be applied in the classical case.