+ C^2_k {\rm Re}(\pi_k \phi_{-k}) \right] \right\}.
\nonumber \end{aligned}$$ The normalization is now a function of all $A^2_k,B^2_k,C^2_k$ $$\begin{aligned}
{\cal N}_k = {1 \over \pi}\sqrt{A^2_k B^2_k - C^4_k}. \end{aligned}$$ The equations of motion for $A^2_k,B^2_k$ and $C^2_k$ then follow from the second order Fokker-Planck equation: $$\begin{aligned}
&& {d A_k \over dt} =  {C_k \over A_k} \left[ k^2 + m^2 + 3 \lambda G(t) - \eta T C_k^2 \right] \\
&& {d B_k \over dt} = \eta B_k \left[ 1 - T B_k^2 \right]  \\
&& {d C_k \over dt} = {\eta C_k \over 2} \left[ 1 - 2 T B_k^2 \right] \nonumber \\
&& \qquad \qquad -{1\over 2 C_k} \left[A_k^2 -  \left(
k^2 + m^2 + 3 \lambda G(t) \right) B_k^2 \right],\end{aligned}$$ where $$\begin{aligned}
G(t) = \int {d^D p \over (2 \pi )^D} {B_p^2 \over A_p^2 B_p^2 - C_p^4}.\end{aligned}$$ This system has as static solutions the mean-field canonical distribution namely $$\begin{aligned}
&&A_k^2 = {k^2 +m^2 +\Delta m^2 (T) \over T}, \\
&&B_k^2 = 1/T = \beta \\
&&C_k^2 = 0,\end{aligned}$$ where the equation for $A_k$ is simply (\[GapMF\]). Other static solutions exist though. Additional stationary distributions distinct from thermal equilibrium also occur in the microcanonical evolution of the class of 