Scaling Theory for the Frictionless Unjamming Transition

We develop a scaling theory of the unjamming transition of soft frictionless disks in two dimensions by defining local areas, which can be uniquely assigned to each contact. These serve to define local order parameters, whose distribution exhibits divergences as the unjamming transition is approached. We derive scaling forms for these divergences from a mean-field approach that treats the local areas as non interacting entities, and demonstrate that these results agree remarkably well with numerical simulations. We find that the asymptotic behaviour of the scaling functions arises from the geometrical structure of the packing while the overall scaling with the compression energy depends on the force law. We use the scaling forms of the distributions to determine the scaling of the total grain area $A_G$, and the total number of contacts $N_C$.