Finite size effects in the microscopic critical properties of jammed configurations: A comprehensive study of the effects of different types of disorder

TL;DR

This study investigates finite-size effects on the distributions of forces and gaps in jammed systems, confirming mean-field predictions in most models and revealing longer-range correlations in gaps than forces.

Contribution

It provides a systematic analysis of finite-size scaling of force and gap distributions in jamming, validating mean-field theory predictions and exploring the effects of different types of disorder.

Findings

Finite-size effects are more pronounced in gap distributions than in force distributions.

Mean-field predictions agree well with numerical results in most models.

A secondary linear tail regime in distributions is linked to near-isostaticity.

Abstract

Jamming criticality defines a universality class that includes systems as diverse as glasses, colloids, foams, amorphous solids, constraint satisfaction problems, neural networks, etc. A particularly interesting feature of this class is that small interparticle forces ($f$) and gaps ($h$) are distributed according to nontrivial power laws. A recently developed mean-field (MF) theory predicts the characteristic exponents of these distributions in the limit of very high spatial dimension, $d\rightarrow\infty$ and, remarkably, their values seemingly agree with numerical estimates in physically relevant dimensions, $d=2$ and $3$. These exponents are further connected through a pair of inequalities derived from stability conditions, and both theoretical predictions and previous numerical investigations suggest that these inequalities are saturated. Systems at the jamming point are thus only marginally stable. Despite the key physical role played by these exponents, their systematic evaluation has yet to be attempted. Here, we carefully test their value by analyzing the finite-size scaling of the distributions of $f$ and $h$ for various particle-based models for jamming. Both dimension and the direction of approach to the jamming point are also considered. We show that, in all models, finite-size effects are much more pronounced in the distribution of $h$ than in that of $f$. We thus conclude that gaps are correlated over considerably longer scales than forces. Additionally, remarkable agreement with MF predictions is obtained in all but one model, namely near-crystalline packings. Our results thus help to better delineate the domain of the jamming universality class. We furthermore uncover a secondary linear regime in the distribution tails of both $f$ and $h$. This surprisingly robust feature is understood to follow from the (near) isostaticity of our configurations.