Mean-Field Predictions of Scaling Prefactors Match Low-Dimensional Jammed Packings

TL;DR

This paper demonstrates that mean field theory predictions for scaling prefactors in jammed packings are exact in low dimensions, challenging the notion that infinite dimensions are necessary for such results.

Contribution

It provides empirical evidence that mean field predictions for scaling prefactors are accurate in low-dimensional jammed packings, and offers an exact derivation for one of these prefactors.

Findings

Mean field predictions match measured prefactors in dimensions 2-10.

Finite dimensional corrections do not affect the accuracy of these prefactors.

An exact derivation is provided for one scaling prefactor.

Abstract

No known analytic framework precisely explains all the phenomena observed in jamming. The replica theory for glass and jamming is a mean field theory which attempts to do so by working in the limit of infinite dimensions, such that correlations between neighbors are negligible. As such, results from this mean field theory are not guaranteed to be observed in finite dimensions. However, many results in mean field for jamming have been shown to be exact or nearly exact in low dimensions. This suggests that the infinite dimensional limit is not necessary to obtain these results. In this paper, we perform precision measurements of jamming scaling relationships between pressure, excess packing fraction, and number of excess contacts from dimensions 2-10 in order to extract the prefactors to these scalings. While these prefactors should be highly sensitive to finite dimensional corrections, we find the mean field predictions for these prefactors to be exact in low dimensions. Thus the mean field approximation is not necessary for deriving these prefactors. We present an exact, first principles derivation for one, leaving the other as an open question.