  After extensive quasi-static shearing, dense dry granular flows attain a
steady-state condition of porosity and deviatoric stress, even as particles are
continually rearranged. The paper considers two-dimensional flow and derives
the probability distributions of two topological measures of particle
arrangement---coordination number and void valence---that maximize topological
entropy. By only considering topological dispersion, the method closely
predicts the distribution of void valences, as measured in discrete element
(DEM) simulations. Distributions of coordination number are also derived by
considering packings that are geometrically and kinetically consistent with the
particle sizes and friction coefficient. A cross-entropy principle results in a
distribution of coordination numbers that closely fits DEM simulations.
