  We study the dynamics of the chiral phase transition expected during the
expansion of the quark-gluon plasma produced in a high energy hadron or heavy
ion collision, using the $O(4)$ linear sigma model in the mean field
approximation. Imposing boost invariant initial conditions at an initial proper
time $\tau_0$ and starting from an approximate equilibrium configuration, we
investigate the possibility of formation of disoriented chiral condensate
during the expansion. In order to create large domains of disoriented chiral
condensates low-momentum instabilities have to last for long enough periods of
time. Our simulations show no instabilities for an initial thermal
configuration. For some of the out-of-equilibrium initial states studied, the
fluctuation in the number of particles with low transverse momenta become large
at late proper times.
