  We derive an expression of the kinetic entropy current in the nonequilibrium
$O(N)$ scalar theory from the Schwinger-Dyson (Kadanoff-Baym) equation with the
1st order gradient expansion. We show that our kinetic entropy satisfies the
H-theorem for the leading order of the gradient expansion with the
next-to-leading order self-energy of the $1/N$ expansion in the symmetric
phase, and that entropy production occurs as the Green's function evolves with
an nonzero collision term. Entropy production stops at local thermal
equilibrium where the collision term contribution vanishes and the maximal
entropy state is realized. Next we also compare our entropy density with that
in thermal equilibrium which is given from thermodynamic potential or
equivalently 2 particle irreducible effective action. We find that our entropy
density corresponds to that in thermal equilibrium with the next-to-leading
order skeletons of the $1/N$ expansion if skeletons with energy denominators in
momentum integral can be regularized appropriately. We have a possibility that
memory correction terms remain in entropy current if not regularized.
