of Eq. (\[mnminus\]).

To avoid the presence of the square gradient term in Eq. (\[mnminus\]), and the complications of its, somewhat arbitrary, discretization, we resort to a different change of variables. After $n=\exp h$, $$\begin{aligned}
\partial_t n &=& \nabla^2 n +(a+1)n +bn^{3-p} +n \eta, \nonumber \\
P_{st} &\sim& {1 \over n^{2-a}} \exp\bigg[{b n^{2-p} \over 2-p}-{\n\over n}\bigg].
\label{mnplus}\end{aligned}$$ These equations are simpler than (\[mnminus\]) and (\[pmnminus\]), but at the cost that $\n$ is no longer an order parameter as, at the transition point $\n \to \infty$ rather than going to $0$. Furthermore, Landau expansions only make sense when the order parameter vanishes at the critical point, making the whole approach inconsistent. One possible way to circumvent this problem is to monitor $m \equiv 1/n$, and study $\tilde{P}_{st}(m) dm =P_{st}(n) dn$, but given the non-Gaussian nature of the probability distribution the substitution $\n = 1/\langle m \rangle$ is likely to be incorrect, and there seem to be no safe way to proceed.

Summing up, non-trivial mean-field approaches detect some problems with the model under consideration, and are not able to predict a phase correct phase diagram. A sound mean-field approximation, needs therefore to be found. Notice that none of these problems occur 