One-dimensional Kac model of dense amorphous hard spheres

TL;DR

This paper introduces a one-dimensional chain model of dense hard spheres to analytically study glass and jamming transitions, revealing characteristic length scales and critical exponents consistent with mean-field spin glass models.

Contribution

The paper presents a novel one-dimensional chain model of dense hard spheres and analytically derives glass transition densities and length scales in the infinite-dimensional limit.

Findings

Divergence of length scales characterized by exponents -1/4 and -1.

Transition densities computed using replica liquid theory.

Model aligns with mean-field spin glass behavior.

Abstract

We introduce a new model of hard spheres under confinement for the study of the glass and jamming transitions. The model is an one-dimensional chain of the $d$-dimensional boxes each of which contains the same number of hard spheres, and the particles in the boxes of the ends of the chain are quenched at their equilibrium positions. We focus on the infinite dimensional limit ($d \to \infty$) of the model and analytically compute the glass transition densities using the replica liquid theory. From the chain length dependence of the transition densities, we extract the characteristic length scales at the glass transition. The divergence of the lengths are characterized by the two exponents, $-1/4$ for the dynamical transition and $-1$ for the ideal glass transition, which are consistent with those of the $p$-spin mean-field spin glass model. We also show that the model is useful for the study of the growing length scale at the jamming transition.