Investigating the domain of validity of the Gubser solution to the Boltzmann equation

TL;DR

This paper examines the exact relativistic Boltzmann equation solution in a conformal system, revealing conditions under which the distribution function remains positive and identifying the solution's domain of validity.

Contribution

It provides a detailed analysis of the Gubser solution's validity, highlighting how boundary conditions affect the positivity and physical applicability of the distribution function.

Findings

Distribution function can become negative in certain phase space regions.

Positivity constraints restrict the solution's domain of validity.

Physical boundary conditions ensure a positive definite distribution function.

Abstract

We study the evolution of the one particle distribution function that solves exactly the relativistic Boltzmann equation within the relaxation time approximation for a conformal system undergoing simultaneously azimuthally symmetric transverse and boost-invariant longitudinal expansion. We show, for arbitrary values of the shear viscosity to entropy density ratio, that the distribution function can become negative in certain kinematic regions of the available phase space depending on the boundary conditions. For thermal equilibrium initial conditions, we determine numerically the physical boundary in phase space where the distribution function is always positive definite. The requirement of positivity of this particular exact solution restricts its domain of validity, and it imposes physical constraints on its applicability.