Correlations and critical behavior of the $q$ model

The $q$ model is a random walk model used to describe the flow of stress in a stationary granular medium. Here we derive the exact horizontal and vertical correlation functions for the $q$ model in two dimensions. We show that close to a critical point identified in earlier work these correlation functions have a universal scaling form reminiscent of thermodynamic critical phenomena. We determine the form of the universal scaling function and the associated critical exponents $\nu$ and $z$.

Figure 1

Upper panel: The $q$ model in two dimensions (reproduced from Ref. [8]). The beads are assumed to be arranged in a square lattice. Each bead is assumed to be supported by its nearest neighbors in the layer below. A random fraction $f$ of the load is transferred to the neighbor at left, the remainder $(1-f)$ to the neighbor to the right. The fractions vary from bead to bead and are assumed to be independent identically distributed random variables. $t$ specifies the horizontal layer occupied by a bead. The horizontal layers are numbered by consecutive integers. The position of a bead within a horizontal layer is given by $n$. $n$ is even in a layer with even $t$ and odd, for odd $t$. Thus the horizontal separation of two beads in the same layer must be an even integer. Similarly the horizontal separation for two beads in layers with an even vertical separation must also be an even integer. Lower panel: The distribution of load in layer $t=100$ computed numerically for a single realization of the $q$ model with the random fractions $f$ drawn from the singular distribution. A uniform load is applied to the top layer $t=1$. It is assumed that there are 100 beads in each layer and periodic boundary conditions have been applied. The gray bar shows the horizontal correlation length appropriate to the singular $q$ model at this depth derived in Sec. 2b.