Loop-based compactivity in jammed granular packings: linking topology and geometry.

Edwards' statistical mechanics offers a thermodynamic framework for jammed granular matter, but identifying volume-like measures that capture mesoscopic topology and remain meaningful under shear remains a core challenge. Conventional per-particle Voronoi tessellations average out contact-network topology, limiting resolution of structural disorder and intermittent plastic dynamics. We develop a mesoscopic statistical framework for 2D granular assemblies based on particle contact loops, characterizing microstates jointly by loop order K and normalized loop area V, and derive a loop-based modified compactivity X extended to quasi-static biaxial compression. Discrete element simulations show that both K and V follow exponential distributions, consistent with a maximum-entropy loop-based Edwards ensemble; all entropy measures and mutual information I(K;V) increase monotonically with porosity. The derived compactivity rises steeply and diverges at the random loose packing limit, exhibiting systematically larger values and higher sensitivity to mesoscopic disorder than Voronoi-based measures. Under shear, compactivity converges to a stable critical-state plateau synchronized with stress ratio and porosity. The plateau shows pronounced initial-state and path dependence, demonstrating compactivity is not a strictly path-independent thermodynamic state function under far-from-equilibrium driving. The two measures are complementary: Voronoi compactivity tracks smooth global volumetric evolution, while loop-based compactivity resolves intermittent plastic avalanches as a topological probe. This framework offers a robust topological approach for investigating quasi-thermodynamic behavior and geometry-force coupling in jammed and driven granular matter.