  We present results of numerical studies of the Landau-Ginzburg dynamics of
the order parameter in one-dimensional models inspired by the condensed matter
analogues of cosmological phase transitions. The main goal of our work is to
show that, as proposed by one of us \cite{Zurek85b}, the density of the
frozen-out topological defects is set by the competition between the quench
rate --- the rate at which the phase transition is taking place --- and the
relaxation rate of the order parameter. In other words, the characteristic
domain size, which determines the typical separation of topological defects in
the new broken symmetry phase, is of the order of the correlation length at the
instant at which the relaxation timescale of the order parameter equals the
time remaining to the phase transition. In estimating the size of topological
domains, this scenario shares with the original Kibble mechanism the idea that
topological defects will form along the boundaries of independently selected
regions of the new broken symmetry vacuum. However, it derives the size of such
domains from non-equilibrium aspects of the transition (quench rate), as
opposed to Kibble's original proposal in which their size was estimated from
the Ginzburg temperature above which thermally activated symmetry restoration
can occur.
