  The nonequilibrium dynamics of the chiral phase transition expected during
the expansion of the quark-qluon plasma produced in a high energy hadron or
heavy ion collision is studied in the O(4) linear sigma model to leading order
in a large $N$ expansion. Starting from an approximate equilibrium
configuration at an initial proper time $\tau$ in the disordered phase we study
the transition to the ordered broken symmetry phase as the system expands and
cools. We give results for the proper time evolution of the effective pion
mass, the order parameter $<\sigma>$ as well as for the pion two point
correlation function expressed in terms of a time dependent phase space number
density and pair correlation density. We determine the phase space of initial
conditions that lead to instabilities (exponentially growing long wave length
modes) as the system evolves in time. These instabilities are what eventually
lead to disoriented chiral condensates. In our simulations,we found that
instabilities that are formed during the initial phases of the expansion exist
for proper times that are at most $3 fm/c$ and lead to condensate regions that
do not contain large numbers of particles. The damping of instabilities is a
consequence of strong coupling.
