The problem of the silos

We propose a stochastic model to simulate the stress of granular media in two dimensional silos. The vertical coordinate plays the role of time, and the horizontal coordinates the role of space. The process is a probabilistic cellular automaton with state space $\R^{\{1,\dots,N\}}$. Under this dynamics each weight is increased by a random variable of mean one and distributed among the two nearest neighbors. We consider two rules. In the first one the fraction of the weight given to the right neighbor is a uniform random variable in $[0,1]$ independent of everything. The remaining weight is given to the left neighbor. For this model we show that there exists an invariant measure with a quadratic profile of mean weights. In the second model the state space is $\Z^{\{1,\dots,N\}}$. Each grains has a mean-one Poisson weight. Each unit of weight decides independently to go to the right or left neighbor. In this case we prove that the invariant measure is a product of Poisson random variables with the same profile as in the first case. In this case, the variance of the stress on the wall of the silo is of the order of $N$. The approach is absolutely elementary.