Convergence problem of gradient expansion in the relaxation-time approximation

We obtain a formal integral solution to the $3+1\text{ }\text{ }\mathrm{D}$ Boltzmann equation in relaxation-time approximation. The gradient series obtained from this integral solution contains exponentially decaying nonhydrodynamic terms. It is shown that this gradient expansion can have a finite radius of convergence under certain assumptions of analyticity. We then argue that, in the relaxation-time model, proximity to local thermal equilibrium is not necessary for the system to be described by hydrodynamic equations.