Electric Conductivity from the solution of the Relativistic Boltzmann Equation

TL;DR

This paper numerically investigates the electric conductivity of a relativistic fluid using the Boltzmann equation, comparing numerical methods with analytic approximations, and finds that the Relaxation Time Approximation significantly underestimates conductivity.

Contribution

The study provides a detailed numerical analysis of electric conductivity in relativistic fluids, highlighting the limitations of the Relaxation Time Approximation for different scattering scenarios.

Findings

Numerical results agree well between the external field and Green-Kubo methods.

RTA underestimates conductivity by up to a factor of 2 in certain conditions.

RTA underestimates conductivity by 60-80% in realistic quark-gluon plasma models.

Abstract

We present numerical results of electric conductivity $\sigma_{el}$ of a fluid obtained solving the Relativistic Transport Boltzmann equation in a box with periodic boundary conditions. We compute $\sigma_{el}$ using two methods: the definition itself, i.e. applying an external electric field, and the evaluation of the Green-Kubo relation based on the time evolution of the current-current correlator. We find a very good agreement between the two methods. We also compare numerical results with analytic formulas in Relaxation Time Approximation (RTA) where the relaxation time for $\sigma_{el}$ is determined by the transport cross section $\sigma_{tr}$, i.e. the differential cross section weighted with the collisional momentum transfer. We investigate the electric conductivity dependence on the microscopic details of the 2-body scatterings: isotropic and anisotropic cross-section, and massless and massive particles. We find that the RTA underestimates considerably $\sigma_{el}$; for example at screening masses $m_D \sim \,T$ such underestimation can be as large as a factor of 2. Furthermore, we study a more realistic case for a quark-gluon system (QGP) considering both a quasi-particle model, tuned to lQCD thermodynamics, as well as the case of a pQCD gas with running coupling. Also for these cases more directly related to the description of the QGP system, we find that RTA significantly underestimate the $\sigma_{el}$ by about a $60-80\%$.