  We study non-equilibrium quantum dynamics of the single-component scalar
field theory in 1+1 space-time dimensions on the basis of the Kadanoff-Baym
equation including the next-to-leading-order (NLO) skeleton diagrams. As an
extension of the non-relativistic case, we derive relativistic kinetic entropy
at the first order in the gradient expansion of the Kadanoff-Baym equations.
The derived entropy satisfies the H theorem. Next we perform numerical
simulations in spatially homogeneous configurations to investigate
thermalization properties of the system by evaluating the system entropy. We
find that at later times the kinetic entropy increases approaching the
equilibrium value, although the limited time interval in the early stage
invalidates the use of it.
