  We show that the second accelerating expansion of the universe appears
smoothly from the decelerating universe remarkably after the initial inflation
in the two-dimensional soluble semi-classical dilaton gravity along with the
modified Poisson brackets of noncommutativity between the relevant fields.
However, the ordinary solution coming from the equations of motion following
the conventional Poisson algebra describes permanent accelerating universe
without any phase change. In this modified model, it turns out that the phase
transition is related to the noncommutative Poisson algebra.
