We analyze the non-equilibrium dynamics of the O(N) Phi^4 model in the large
N limit and for states of large energy density. The dynamics is dramatically
different when the energy density is above the top of the tree level potential
V_0 than when it is below it.When the energy density is below V_0, we find that
non-perturbative particle production through spinodal instabilities provides a
dynamical mechanism for the Maxwell construction. The asymptotic values of the
order parameter only depend on the initial energy density and all values
between the minima of the tree level potential are available, the asymptotic
dynamical `effective potential' is flat between the minima. When the energy
density is larger than V_0, the evolution samples ergodically the broken
symmetry states, as a consequence of non-perturbative particle production via
parametric amplification. Furthermore, we examine the quantum dynamics of phase
ordering into the broken symmetry phase and find novel scaling behavior of the
correlation function. There is a crossover in the dynamical correlation length
at a time scale t_s \sim \ln(1/lambda). For t < t_s the dynamical correlation
length \xi(t) \propto \sqrt{t} and the evolution is dominated by spinodal
instabilities, whereas for t>t_s the evolution is non-linear and dominated by
the onset of non-equilibrium Bose-Einstein condensation of long-wavelength
Goldstone bosons.In this regime a true scaling solution emerges with a non-
perturbative anomalous scaling length dimension z=1/2 and a dynamical
correlation length \xi(t) \propto (t-t_s). The equal time correlation function
in this scaling regime vanishes for r>2(t-t_s) by causality. For t > t_s the
equal time correlation function falls of as 1/r. A semiclassical but stochastic
description emerges for time scales t > t_s.Comment: Minor improvements, to appear in Phys. Rev. D. Latex file, 48 pages,
12 .ps figure
