The disc random packing problem: a disorder criterion and an explicit solution

Predicting the densest random disc packing fraction is an unsolved paradigm problem relevant to a number of disciplines and technologies. One difficulty is that it is ill-defined without setting a criterion for the disorder. Another is that the density depends on the packing protocol and the multitude of possible protocol parameters has so far hindered a general solution. A new approach is proposed here. After formulating a well-posed form of the general protocol-independent problem for planar packings of discs, a systematic criterion is proposed to avoid crystalline hexagonal order as well as further topological order. The highest possible random packing fraction is then derived exactly: $\phi_{RCP}=0.852525...$. The solution is based on the cell order distribution that is shown to: (i) yield directly the packing fraction; (ii) parameterise all possible packing protocols; (iii) make it possible to define and limit all topological disorder. The method is further useful for predicting the highest packing fraction in specific protocols, which is illustrated for a family of simply-sheared packings that generate maximum-entropy cell order distributions.