  We investigate the influence of chiral symmetry which varies along the
space-time evolution of the system by considering the chiral phase transition
in an non-equilibrium expanding quark-antiquark system. The chiral symmetry is
described by the mean field order parameter, whose values is the solution of a
self-consistent equation, and affects the space-time evolution of the system
through the force term in the Vlasov equation. The Vlasov equation and the gap
equation are solved concurrently and continuously for a longitudinal
boost-invariant and transversely rotation-invariant system. This numerical
framework enables us to carefully investigate how the phase transition and
collision affect the evolution of the system. It is observed that the chiral
phase transition gives rise to a kink in the flow velocity, which is caused by
the force term in the Vlasov equation. The kink is enhanced by larger
susceptibility and tends to be smoothed out by non-equilibrium effect. The
spatial phase boundary appears as a "wall" for the quarks, as the quarks with
low momentum are bounced back, while those with high momentum go through the
wall but are slowed down.
