mode $k$. This is similar in effort to the solution of the Langevin mean-field equation with the advantage that the stochastic average is already taken into account. This latter step corresponds to the functional integral necessary to compute $G(t)$ in the Fokker-Planck picture, which can be given in closed form because $P_{MF}$ is Gaussian.

The canonical (mean-field) distribution is of course a static solution of Eqs (\[MF1\]-\[MF2\]) where $A_k$ obeys $$\begin{aligned}
A_k^2 = {k^2 + m^2 \over T} + 3 {\lambda \over T}  \int {d^D p\over (2 \pi)^D} {1 \over A_p^2},
\label{GapMF}\end{aligned}$$ It is then clear that for vanishing interactions $\lambda=0$, the free Boltzmann propagator is the solution of Eq. (\[GapMF\]). In general because the mean-field contribution is momentum independent Eq. (\[GapMF\]) are equivalent to the ’gap’ equation $$\begin{aligned}
A_k^2 &&= {k^2 + m^2 + \Delta m^2  \over T}, \\
\Delta m^2 &&= 3 \lambda \int {d^D p\over (2 \pi)^D} {T \over k^2 +m^2 + \Delta m^2}.\end{aligned}$$

For the second order Fokker-Planck equation the most general Gaussian function of the field and momentum modes is of the form $$\begin{aligned}
&&P_{MF} [\pi_k,\phi_k,t] = \\
&& {\cal N}_k(t)\exp \left\{ - \left[ {A^2_k \over 2} 
\phi_k \phi_{-k} + {B^2_k \over 2} \pi_k \pi_{-k} 