Nonlinear Boltzmann equation for the homogeneous isotropic case: Minimal deterministic Matlab program

TL;DR

This paper presents a minimal, open-source Matlab program for solving the homogeneous isotropic Boltzmann equation deterministically, improving accuracy and conservation properties for applications in thermodynamics, econophysics, and sociodynamics.

Contribution

It introduces a simplified, accurate Matlab implementation of a deterministic solution method for the HIBE, incorporating conservation corrections and reformulation in particle kinetic energy.

Findings

Enhanced accuracy in relaxation rate computation

Exact conservation of particle number and energy during collisions

Open-source, minimal Matlab code for broad use

Abstract

The homogeneous isotropic Boltzmann equation (HIBE) is a fundamental dynamic model for many applications in thermodynamics, econophysics and sociodynamics. Despite recent hardware improvements, the solution of the Boltzmann equation remains extremely challenging from the computational point of view, in particular by deterministic methods (free of stochastic noise). This work aims to improve a deterministic direct method recently proposed [V.V. Aristov, Kluwer Academic Publishers, 2001] for solving the HIBE with a generic collisional kernel and, in particular, for taking care of the late dynamics of the relaxation towards the equilibrium. Essentially (a) the original problem is reformulated in terms of particle kinetic energy (exact particle number and energy conservation during microscopic collisions) and (b) the computation of the relaxation rates is improved by the DVM-like correction, where DVM stands for Discrete Velocity Model (ensuring that the macroscopic conservation laws are exactly satisfied). Both these corrections make possible to derive very accurate reference solutions for this test case. Moreover this work aims to distribute an open-source program (called HOMISBOLTZ), which can be redistributed and/or modified for dealing with different applications, under the terms of the GNU General Public License. The program has been purposely designed in order to be minimal, not only with regards to the reduced number of lines (less than 1,000), but also with regards to the coding style (as simple as possible).