  As the matter produced in a relativistic heavy ion collision cools through
the QCD phase transition, the dynamical evolution of the chiral condensate will
be driven out of thermal equilibrium. As a prelude to analyzing this evolution,
and in particular as a prelude to learning how rapid the cooling must be in
order for significant deviations from equilibrium to develop, we present a
detailed analysis of the time-evolution of an idealized region of disoriented
chiral condensate. We set up a Langevin field equation which can describe the
evolution of these (or more realistic) linear sigma model configurations in
contact with a heat bath representing the presence of other shorter wavelength
degrees of freedom. We first analyze the model in equilibrium, paying
particular attention to subtracting ultraviolet divergent classical terms and
replacing them by their finite quantum counterparts. We use known results from
lattice gauge theory and chiral perturbation theory to fix nonuniversal
constants. The result is a theory which is ultraviolet cutoff independent and
that reproduces quantitatively the expected equilibrium behavior of the quantum
field theory of pions and sigma fields over a wide range of temperatures.
Finally, we estimate the viscosity $\eta(T)$, which controls the dynamical
timescale in the Langevin equation, by requiring that the timescale for DCC
decay agrees with previous calculations. The resulting $\eta(T)$ is larger than
that found perturbatively. We also determine the temperature below which the
classical field Langevin equation ceases to be a good model for the quantum
field dynamics.
