  Understanding the structural evolution of granular systems is a long-standing
problem. A recently proposed theory for such dynamics in two dimensions
predicts that steady states of very dense systems satisfy detailed-balance. We
analyse analytically and numerically the steady states of this theory in
systems of arbitrary density and report the following. 1. We discover that all
such dynamics almost certainly possess only one physical steady state, which
may or may not satisfy detailed balance. 2. We show rigorously that, if a
detailed balance solution is possible then it is unique. The above two results
correct an erroneous conjecture in the literature. 3. We show rigorously that
the detailed-balance solutions in very dense systems are globally stable,
extending the local stability found for these solutions in the literature. 4.
In view of recent experimental observations of robust detailed balance steady
states in very dilute cyclically sheared systems, our results point to a
self-organisation of process rates in dynamic granular systems.
