Two-scale scenario of rigidity percolation of sticky particles

TL;DR

This paper investigates the nature of the rigidity percolation transition in sticky particle systems, revealing a two-scale coexistence that indicates a continuous transition despite apparent discontinuities in cluster size distributions.

Contribution

It introduces a two-scale scenario for rigidity percolation, challenging the traditional single-scale view and deriving a generalized hyperscaling relation for short-range attractive systems.

Findings

Large system size limit shows diverging cluster sizes.

Discontinuous $P(s)$ does not imply a discontinuous transition.

Two distinct length scales coexist at the transition.

Abstract

In the presence of attraction, the jamming transition of packings of frictionless particles corresponds to the rigidity percolation. When the range of attraction is long, the distribution of the size of rigid clusters, $P(s)$, is continuous and shows a power-law decay. For systems with short-range attractions, however, $P(s)$ appears discontinuous. There is a power-law decay for small cluster sizes, followed by a low probability gap and a peak near the system size. We find that this appearing ``discontinuity'' does not mean that the transition is discontinuous. In fact, it signifies the coexistence of two distinct length scales, associated with the largest cluster and smaller ones, respectively. The comparison between the largest and second largest clusters indicates that their growth rates with system size are rather different. However, both cluster sizes tend to diverge in the large system size limit, suggesting that the jamming transition of systems with short-range attractions is still continuous. In the framework of the two-scale scenario, we also derive a generalized hyperscaling relation. With robust evidence, our work challenges the former single-scale view of the rigidity percolation.