A hydrodynamic approach to the study of anisotropic instabilities in dissipative relativistic plasmas

We develop a purely hydrodynamic formalism to describe collisional, anisotropic instabilities in a relativistic plasma, that are usually described with kinetic theory tools. Our main motivation is the fact that coarse-grained models of high particle number systems give more clear and comprehensive physical descriptions of those systems than purely kinetic approaches, and can be more easily tested experimentally as well as numerically. In particular, we aim at developing a theory that describes both a background non-equilibrium fluid configurations and its perturbations, to be able to account for the backreaction of the latter on the former. Our system of equations includes the usual conservation laws for the energy-momentum tensor and for the electric current, and the equations for two new tensors that encode the information about dissipation. To make contact with kinetic theory, we write the different tensors as the moments of a non-equilibrium one-particle distribution function (1pdf) which, for illustrative purposes, we take in the form of a Grad-like ansatz. Although this choice limits the applicability of the formalism to states not far from equilibrium, it retains the main features of the underlying kinetic theory. We assume the validity of the Vlasov-Boltzmann equation, with a collision integral given by the Anderson-Witting prescription, which is more suitable for highly relativistic systems than Marle's (or Bhatnagar, Gross and Krook) form, and derive the conservation laws by taking its corresponding moments. We apply our developments to study the emergence of instabilities in an anisotropic, but axially symmetric background. For small departures of isotropy we find the dispersion relation for normal modes, which admit unstable solutions for a wide range of values of the parameter space.