of time, and a functional of the fields and their conjugate momenta. Closed form solutions are thus difficult to construct, which is of course also the case for the Langevin equation under a similar approximation. Numerical solutions of the this mean field Fokker Planck equation can nevertheless be easily obtained.

As a starting point let us again consider the overdamped case first. Mean field approximations correspond in general to Gaussian probability distributions. In the overdamped case, this can only lead to $$\begin{aligned}
P_{MF} [\phi_k,t] = {\cal N}_k(t) \exp{\left[ 
- \phi_{-k} {A_k^2(t) \over 2} \phi_k \right]},\end{aligned}$$ where the normalization ${\cal N}(t)$ is a function of $A_k^2(t)$ $$\begin{aligned}
{\cal N}_k(t) = A_k(t)/\sqrt{\pi},\end{aligned}$$ and must therefore be time-dependent. This does not of course introduce any additional dynamical freedom.

The mean-field Fokker-Planck equation for $P_{MF} [\phi_k,t]$ translates into an equation for the coefficients $A_k(t)$ of the form $$\begin{aligned}
&& \eta {d A_k \over dt} = \left[ k^2 + m^2 
+ 3 \lambda G(t) -A_k^2 T \right] A_k, \label{MF1} \\
&& G(t) = \int {d^D p\over (2 \pi)^D} {1 \over A_p^2}. \label{MF2}\end{aligned}$$ Clearly solving the Fokker-Planck equation in this mean-field approximation requires the solution of a system of coupled ordinary differential equations, one for every 