Electric conductivity from the solution of the relativistic Boltzmann equation

We present numerical results of electric conductivity σₑₗ of a fluid obtained solving the relativistic transport Boltzmann equation in a box with periodic boundary conditions. We compute σₑₗ 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 σₑₗ is determined by the transport cross section σₜᵣ, i.e., the differential cross section weighted with the collisional momentum transfer. We investigate the electric conductivity dependence on the microscopic details of the two-body scatterings: isotropic and anisotropic cross section as well as massless and massive particles. We find that the RTA underestimates considerably σₑₗ; for example, at screening masses mD~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 quasiparticle model tuned to lattice QCD (lQCD) thermodynamics, as well as the case of a perturbative QCD (pQCD) gas with running coupling. Also, for these cases more directly related to the description of the QGP system, we find that the RTA significantly underestimates the σₑₗ by about 60%--80%.