Crossover from generalized to conventional hydrodynamics in nearly integrable systems under relaxation time approximation

Upon breaking the integrability, the equations of generalized hydrodynamics (GHD) are supplemented by a Boltzmann collision term. Such terms are typically complicated and stem from a perturbative treatment of integrability-breaking terms in the hamiltonian. In our work, we study a simplified version of the collision operator in a form of relaxation time approximation familiar from kinetic theory. We explicitly compute transport coefficients which characterize the Navier-Stokes (NS) hydrodynamic regime emerging at large space-time scales. We also thoroughly study the crossover between GHD and NS hydrodynamic descriptions, identifying relevant characteristic space-time scales for the transition. In particular, we show how the emergence of NS hydrodynamics is visible in dynamics of conserved and non-conserved charge densities, and in hydrodynamic two-point functions.