# Revisit to electrical and thermal conductivities, Lorenz number and Knudsen number in thermal QCD in a strong magnetic field

## TL;DR

This paper investigates how electrical and thermal conductivities in a thermal QCD medium are affected by magnetic fields and expansion anisotropies, revealing their temperature dependence and implications for magnetic field decay and equilibrium.

## Contribution

It provides a detailed analysis of conductivities in a magnetic field and expansion anisotropy using relativistic Boltzmann transport, highlighting their contrasting effects on transport properties.

## Key findings

- Conductivities increase with magnetic field-driven anisotropy.
- Electrical conductivity decreases with temperature, opposite to expansion-driven anisotropy.
- Knudsen number indicates the system remains near equilibrium under studied conditions.

## Abstract

We have explored how the electrical ($\sigma_{\rm el}$) and thermal ($\kappa$) conductivities in a thermal QCD medium get affected in weak-momentum anisotropies arising either due to a strong magnetic field or due to asymptotic expansion. This study facilitates to understand the longevity of strong magnetic field through $\sigma_{el}$, Lorenz number in Wiedemann-Franz law, and the validity of equilibrium by the Knudsen number. We calculate the conductivities by solving relativistic Boltzmann transport equation in relaxation-time approximation within quasiparticle model at finite T and strong B. We have found that $\sigma_{el}$ and $\kappa$ get enhanced in a magnetic field-driven anisotropy, but $\sigma_{el}$ decreases with temperature, opposite to its faster increase in expansion-driven anisotropy. Whereas $\kappa$ increases slowly with temperature, contrary to its rapid increase in expansion-driven anisotropy. The above findings are broadly attributed to three factors: the stretching and squeezing of distribution function in anisotropies generated by the magnetic field and asymptotic expansion, respectively, the dispersion relation and resulting phase-space factor, the relaxation-time in absence and presence of strong magnetic field. So $\sigma_{\rm el}$ extracts the time-dependence of magnetic field, which decays slower than in vacuum but expansion-driven anisotropy makes the decay faster. The variation in $\kappa$ transpires that Knudsen number decreases with T but expansion-driven anisotropy reduces its value and magnetic field-driven anisotropy raises its value but to less than one, thus the system can still be in equilibrium. The ratio, $\kappa/\sigma_{el}$ in magnetic field-driven anisotropy increases linearly with temperature but with a value smaller than in expansion-driven anisotropy. Thus the Lorenz number can make the distinction between different anisotropies.