-N , M_1 ), \,  I_2 =[ M_1,M_2 ),\,  \dots I_{K-1} =[ M_{K-2} ,\,  M_{K-1} ), \, I_K =[ M_{K-1},N ]$. In this context, we denote the original distribution function $\rho_{t}(M)$ by a discrete vector of dimension $K$, that is : $ \rho_{t}(M)\to  {\bm f}_{t} = (f_1,f_2,\dots f_K)$. Finally, we obtain a coarse-grained like master equation [@nicolis; @et] $$\begin{aligned}
{\bm f}_{t+1} = \hat W^\dagger {\bm f}_{t},
\label{MasterEqn}\end{aligned}$$ where $\hat W$ is the probability transition matrix defined via the following conditional probability:$$w_{ik} = P(M_{t+1} \in I_i | M_t \in I_k)=  \frac{P(M_{t+1} \in I_i \cap M_t \in I_k)}{P(M_t \in I_k)}.$$ Here $M_{t}$ is at interval $I_k$ at time $t$, and $M_{t+1}$ will be at interval $I_i$ at $t+1$. Finally, $\hat W^\dagger$ denotes the transpose of $\hat W$, and, the matrix $\hat W$ does not depend on time, which is a feature of a Markov process.

The coarse-graining method is schematized in the following Fig. \[Scheme\].

![ \[Scheme\] The distribution $\rho_t(M) $ at a time $t$ is schematized in the left distribution. The fraction inside the interval $I_k$, is distributed, after the evolution into a new distribution $\rho_{t+1}(M)$ schematized in the left diagram. The normalized distribution provides the $i$-th element of the $i$-th 