  We investigate the non-equilibrium properties of an N-component scalar field
theory. The time evolution of the correlation functions for an arbitrary
ensemble of initial conditions is described by an exact functional differential
equation. In leading order in the 1/N expansion the system can be understood in
terms of infinitely many conserved quantities. They forbid the approach to the
canonical thermal distribution. Beyond leading order only energy conservation
is apparent generically. Nevertheless, we find a large manifold of stationary
distributions both for classical and quantum fields. They are the fixed points
of the evolution equation. For small deviations of the correlation functions
from a large range of fixed points we observe stable oscillations. These
results raise the question of if and in what sense the particular fixed point
corresponding to thermal equilibrium dominates the large time behavior of the
system.
