a coarse graining is realized, in order to get an adequate and tractable master equation. We provide an exhaustive validation of the technique and we discuss different coarse-graining over the phase space. Finally, Sec. \[Conclusions\] shows our conclusions.

The Q2R model. {#model}
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For simplicity we shall consider a regular two dimensional lattice with $N=L^2  $ nodes, in which each node is only seen by its four closest neighbors (the von Neuman neighborhood), finally we use periodic boundary conditions. Each node $k$ possesses a discrete value $x_k$ that may take a value +1 or -1. The Q2R model, introduced by Vichniac [@vichniac], is based upon the following two step rule:

$$x^{t+1}_k = x^{t-1}_k \, \phi\left( \sum_{i\in V_k} x^{t}_{i} \right),$$ where the function $\phi$ is such that $\phi(s=0) = -1$ and $\phi(s) = +1$ if $s\neq 0$. In the sum $V_k$ corresponds to the von Neuman neighbor in the site $k$. The reversibility follows directly from the inverse relation $
 x^{t-1}_k = x^{t+1}_k \, \phi\left( \sum_{i\in V_k} x^{t}_{i} \right),$ which is the backward rule (notice that $  \phi\left( \sum_{i\in V_k} x^{t}_{i} \right)^2 =1 $ in all cases).

This two step rule may be naturally re-written as an one step rule by 