2025/09/24 by Raphaël Blumenfeld, Blumenfeld, Raphael · 1 citation
Mathematics · #Disordered Systems and Neural Networks (cond-mat.dis-nn) #FOS: Physical sciences #Mathematical Physics (math-ph) #Point processes and geometric inequalities #Soft Condensed Matter (cond-mat.soft)
paper · pdf · doi:10.48550/arxiv.2509.20132
openalex publication_date 2025/09/24 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
Predicting theoretically the highest density, which a disordered packing of discs can achieve, has been a long-standing unresolved problem. Such predictions are hindered by two difficulties - the dependence of the density on the packing procedure and ensuring disorder. A theory that overcomes these difficulties has been developed recently for mono-disperse disc packing~\citeBl21. However, to minimise order, experiments and numerical simulations often use two-size discs and a prediction of the highest possible packing fraction, ϕRCP, for these packings is arguably more useful. This problem is more complex because in such packings, ϕRCP is not a number but a function of the sizes ratio, D, and concentrations, p, of the disc types. A disorder-guaranteeing theory is formulated here to derive ϕRCP(p,D) under some assumptions, using the concept of the cell order distribution. Exact upper and lower bounds on the densest disordered packing fraction are also derived.