# Angular mode expansion of the Boltzmann equation in the small-angle approximation

## TL;DR

This paper introduces an angular mode expansion method to solve the Boltzmann equation for a gluon plasma, demonstrating rapid convergence and improved computational efficiency, and investigates thermalization dynamics during longitudinal expansion.

## Contribution

The paper presents a novel angular mode expansion approach for the Boltzmann equation, showing its effectiveness and faster convergence compared to traditional methods in expanding gluon plasma scenarios.

## Key findings

- Mode expansion converges rapidly for practical cases.
- Thermalization involves pressure equilibration lag due to expansion.
- First angular mode relaxation is hindered by expansion effects.

## Abstract

We use an expansion in angular mode functions in order to solve the Boltzmann equation for a gluon plasma undergoing longitudinal expansion. By comparing with the exact solution obtained numerically by other means we show that the expansion in mode functions converges rapidly for all cases of practical interest, and represents a substantial gain in numerical effort as compared to more standard methods. We contrast the cases of a non expanding plasma and of longitudinally expanding plasmas, and follow in both cases the evolutions towards thermalization. In the latter case, we observe that, although the spherical mode function appears to be well reproduced after some time by a local equilibrium distribution function depending on slowly varying temperature and chemical potential, thereby suggesting thermalization of the system, the longitudinal and transverse pressures take more time to equilibrate. This is because the expansion hinders the relaxation of the first angular mode function. This feature was also observed in a simpler context where the Boltzmann equation is solved in terms of special moments within the relaxation time approximation, and attributed there to the particular coupling between the first two moments of the distribution function. The present analysis confirms this observation in a more realistic setting.