Entropy current for the relativistic Kadanoff-Baym equation and H-theorem in $O(N)$ theory with NLO self-energy of $1/N$ expansion

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 Greens 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.