A Statistical Ensemble for Soft Granular Matter

Work on packings of soft spheres (PRE \textbf{68}, 011306 (2003)) has shown the existence of a Jamming transition and has highlighted the need for a general statistical framework to describe granular packings. This work presents an extension of the formalism proposed by Edwards (Physica A \textbf{157}, 1080 (1989)) to packings of soft particles. We base our analysis on a height formalism developed in two dimensions (PRL \textbf{88}, 115505 (2002)) to extract a topological invariant $\Gamma$, the trace of the global stress tensor, which is conserved under internal rearrangements of the system. Upon assuming a flat measure in $\Gamma$-space, we can derive a canonical distribution of the local $\Gamma$-values in a grain packing. We then check the predictions of this ensemble against distributions of mechanically stable packings of frictionless disks obtained from computer simulations. Work supported by NSF-DMR 0549762.