  We examine the nonequilibrium dynamics of a self-interacting $\lambda\phi^4$
scalar field theory. Using a real time formulation of finite temperature field
theory we derive, up to two loops and $O(\lambda^2)$, the effective equation of
motion describing the approach to equilibrium. We present a detailed analysis
of the approximations used in order to obtain a Langevin-like equation of
motion, in which the noise and dissipation terms associated with quantum
fluctuations obey a fluctuation-dissipation relation. We show that, in general,
the noise is colored (time-dependent) and multiplicative (couples nonlinearly
to the field), even though it is still Gaussian distributed. The noise becomes
white in the infinite temperature limit. We also address the effect of
couplings to other fields, which we assume play the r\^ole of the thermal bath,
in the effective equation of motion for $\phi$. In particular, we obtain the
fluctuation and noise terms due to a quadratic coupling to another scalar
field.
