Relativistic resistive dissipative magnetohydrodynamics from the relaxation time approximation

Here we derive the relativistic resistive dissipative second-order magnetohydrodynamic evolution equations using the Boltzmann equation, thus extending our work from the previous paper \href{https://link.springer.com/article/10.1007/JHEP03(2021)216}{JHEP 03 (2021) 216} where we considered the non-resistive limit. We solve the Boltzmann equation for a system of particles and antiparticles using the relaxation time approximation and the Chapman-Enskog like gradient expansion for the off-equilibrium distribution function, truncating beyond second-order. In the first order, the bulk and shear stress are independent of the electromagnetic field, however, the diffusion current, shows a dependence on the electric field. In the first order, the transport coefficients~(shear and bulk stress) are shown to be independent of the electromagnetic field. The diffusion current, however, shows a dependence on the electric field. In the second-order, the new transport coefficients that couple electromagnetic field with the dissipative quantities appear, which are different from those obtained in the 14-moment approximation~\cite{Denicol:2019iyh} in the presence of the electromagnetic field. Also we found out the various components of conductivity in this case.