  Discrete-element simulations are used to monitor several micro-scale
characteristics within a granular material, demonstrating their convergence
during loading toward the critical state, their stationarity at the critical
state, and the evolution of their disorder toward the critical state.
Convergence, stationarity and disorder are studied in the context of the
Shannon entropy and two forms of Kullback-Leibler relative entropy. Probability
distributions of 20 aspects of micro-scale configuration, force and movement
are computed for three topological objects: particles, voids and contacts. The
probability distributions of these aspects are determined at numerous stages
during quasi-static biaxial compression and unloading. Not only do stress and
density converge to the critical state, but convergence and stationarity are
manifested in all of the micro-scale aspects. The statistical disorder
(entropy) of micro-scale movements and strains generally increases during
loading until the critical state is reached. When the loading direction is
reversed, order is briefly restored, but continued loading induces greater
disorder in movements and strains until the critical state is reached again.
