Two Diverging Length Scales in the Structure of Jammed Packings

Abstract

At densities higher than the jamming transition for athermal, frictionless repulsive spheres we find two distinct length scales, both of which diverge as a power law as the transition is approached. The first, ξ_{Z}, is associated with the two-point correlation function for the number of contacts on two particles as a function of the particle separation. The second, ξ_{f}, is associated with contact-number fluctuations in subsystems of different sizes. On scales below ξ_{f}, the fluctuations are highly suppressed, similar to the phenomenon of hyperuniformity usually associated with density fluctuations. The exponents for the divergence of ξ_{Z} and ξ_{f} are different and appear to be different in two and three dimensions.

Can a Large Packing be Assembled from Smaller Ones?

“…Finding differences between 2d and 3d, we also consider 4d and mean-field variants of the jamming model. These appear to be consistent with results from 3d, suggesting that the the upper critical dimension is below three [7].To summarize, in this paper (i) we introduce an entirely new procedure (in the context of jamming) for finite size scaling analysis, by comparing subsystems of a large packing with periodic packings of the same size; (ii) we identify a new length scale, in addition to the ones reported in [6]; (iii) we provide accurate measurements of this new length scale in dimension d = 2, 3, 4 (note that previous calculations of jamming length scales could only obtain accurate results for 2d systems); (iv) thanks to this improved finite size scaling analysis, we obtain strong quantitative evidence that d u ∼ 2 is the upper critical dimension; (v) we analyze models with hypostatic jamming and find that the corresponding suppression of fluctuations does not take place. Our analysis thus shows that anomalous contact fluctuations survive in mean-field like models, and are crucially related to isostaticity.We begin by defining the jamming model, in which overlapping particles of radius R i interact via a harmonic potential:Here, r ij is the distance between the centers of the particles and the Heaviside step function, Θ (x), insures that only overlapping particles interact.…”

“…1). Recently it has been found that the contact statistics possess unusual long range correlations near the transition [6]. We therefore focus on contact fluctuations to compare the two ensembles.While we expect convergence of the two ensembles for large enough systems, the system size V * for which these converge appears to be in many cases well beyond that accessible via simulations.…”

“…1). Recently it has been found that the contact statistics possess unusual long range correlations near the transition [6]. We therefore focus on contact fluctuations to compare the two ensembles.…”

“…To summarize, in this paper (i) we introduce an entirely new procedure (in the context of jamming) for finite size scaling analysis, by comparing subsystems of a large packing with periodic packings of the same size; (ii) we identify a new length scale, in addition to the ones reported in [6]; (iii) we provide accurate measurements of this new length scale in dimension d = 2, 3, 4 (note that previous calculations of jamming length scales could only obtain accurate results for 2d systems); (iv) thanks to this improved finite size scaling analysis, we obtain strong quantitative evidence that d u ∼ 2 is the upper critical dimension; (v) we analyze models with hypostatic jamming and find that the corresponding suppression of fluctuations does not take place. Our analysis thus shows that anomalous contact fluctuations survive in mean-field like models, and are crucially related to isostaticity.…”

“…We now briefly review the metrics and the results of Ref. [6] for characterizing the fluctuations. For a packing of N particles excluding the rattlers, we define Z i to be the number of particles in contact with particle i, Z = 1 N N i=1 Z i the average contact number in a given packing, and δZ i = Z i −Z the deviation from the average.…”