Asymptotic behavior of a stationary silo with absorbing walls (Q1599591)

The authors study the nearest neighbor one-dimensional uniform \(q\)-model of force fluctuations in bead packs, 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 coordinate the role of spatial coordinate. At each layer (time), the weight supported by each grain is a random variable of mean one (its own weight) plus the sum of random fractions of the weighs supported by the nearest neighboring grains at the previous layer. The fraction of the weight given to the right neighbor of the successive layer is a uniform random variable in \([0,1]\), independent of everything. The remaining weight is given to the left neighbor. In the boundaries, a uniform fraction of the weight leans on the wall of the silo. This corresponds to absorbing boundary conditions. For this model the authors show that there exists a unique invariant measure. As the silo grows, the model selects exactly one of the regimes at each macroscopic place.