# Exact solution of the (0+1)-dimensional Boltzmann equation for a massive gas

## TL;DR

This paper presents an exact integral solution to the (0+1)-dimensional Boltzmann equation for a massive relativistic gas, enabling precise numerical analysis of transport coefficients and comparison with viscous hydrodynamics.

## Contribution

It provides the first exact integral solution for the massive system's kinetic equation in this context, allowing detailed comparison with hydrodynamic models.

## Key findings

- Exact integral equation solved numerically with high precision.
- Predictions for shear and bulk viscosities match known results.
- Time evolution of bulk pressure aligns with second-order viscous hydrodynamics.

## Abstract

We solve the one-dimensional boost-invariant kinetic equation for a relativistic massive system with the collision term treated in the relaxation time approximation. The result is an exact integral equation which can be solved numerically by the method of iteration to arbitrary precision. We compare predictions for the shear and bulk viscosities of a massive system with those obtained from the exact solution. Finally, we compare the time evolution of the bulk pressure obtained from our exact solution with results obtained from the dynamical equations of second-order viscous hydrodynamics.