  By employing a Chapman-Enskog like iterative solution of the Boltzmann
equation in relaxation-time approximation, we derive a new expression for the
entropy four-current up to third order in gradient expansion. We show that
unlike second-order and third-order entropy four-current obtained using Grad's
method, there is a non-vanishing entropy flux in the present third-order
expression. We further quantify the effect of the higher-order entropy density
in the case of boost-invariant one-dimensional longitudinal expansion of a
system. We demonstrate that the results obtained using third-order evolution
equation for shear stress tensor, derived by employing the method of
Chapman-Enskog expansion, show better agreement with the exact solution of the
Boltzmann equation as well as with the parton cascade BAMPS, as compared to
those obtained using the third-order equations from the method of Grad's
14-moment approximation.
