Perturbative approaches in relativistic kinetic theory and the emergence of first-order hydrodynamics

Hydrodynamics can be formulated in terms of a perturbative series derivatives the temperature, chemical potential, and flow velocity around an equilibrium state. Different formulations for this have been proposed over years, which consequently led to development various hydrodynamic theories. In work, we discuss relativistic generalizations expansions put forward by Chapman Enskog, Hilbert, using general matching conditions kinetic theory. This allows us describe, comprehensive way, how different out-of-equilibrium definitions fields affect series. We provide method systematically deriving formulation recently Bemfica, Disconzi, Noronha, Kovtun (BDNK) from The transport coefficients that appear BDNK (at first-order) are explicitly computed new relaxation time approximation Boltzmann equation. Assuming Bjorken flow, also determine attractors theory compare overall evolution obtained with generated Israel-Stewart equations motion