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We derive an expression of the kinetic entropy current in the nonequilibrium$O(N)$ scalar theory from the Schwinger-Dyson (Kadanoff-Baym) equation with the1st order gradient expansion. We show that our kinetic entropy satisfies theH-theorem for the leading order of the gradient expansion with thenext-to-leading order self-energy of the $1/N$ expansion in the symmetricphase, and that entropy production occurs as the Green's function evolves withan nonzero collision term. Entropy production stops at local thermalequilibrium where the collision term contribution vanishes and the maximalentropy state is realized. Next we also compare our entropy density with thatin thermal equilibrium which is given from thermodynamic potential orequivalently 2 particle irreducible effective action. We find that our entropydensity corresponds to that in thermal equilibrium with the next-to-leadingorder skeletons of the $1/N$ expansion if skeletons with energy denominators inmomentum integral can be regularized appropriately. We have a possibility thatmemory correction terms remain in entropy current if not regularized.

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