Relaxation dynamics of non-Brownian spheres below jamming

TL;DR

This study investigates the relaxation dynamics of non-Brownian spheres below jamming across multiple dimensions, revealing a logarithmic divergence of relaxation time with system size and challenging previous assumptions about the relation between relaxation and viscosity.

Contribution

It uncovers the finite-size and volume fraction effects on relaxation time, providing a new understanding of criticality and divergence behavior below jamming in various dimensions.

Findings

Relaxation time diverges logarithmically with system size below jamming.

In mean-field, relaxation time diverges at jamming with a numerically determined critical exponent.

The relation between relaxation dynamics and shear viscosity breaks down in large finite systems.

Abstract

We numerically study the relaxation dynamics and associated criticality of non-Brownian frictionless spheres below jamming in spatial dimensions $d=2$, $3$, $4$, and $8$, and in the mean-field Mari-Kurchan model. We discover non-trivial finite-size and volume fraction dependences of the relaxation time associated to the relaxation of unjammed packings. In particular, the relaxation time is shown to diverge logarithmically with system size at any density below jamming, and no critical exponent can characterise its behaviour approaching jamming. In mean-field, the relaxation time is instead well-defined: it diverges at jamming with a critical exponent that we determine numerically and differs from an earlier mean-field prediction. We rationalise the finite $d$ logarithmic divergence using an extreme-value statistics argument in which the relaxation time is dominated by the most connected region of the system. The same argument shows that the earlier proposition that relaxation dynamics and shear viscosity are directly related breaks down in large systems. The shear viscosity of non-Brownian packings is well-defined in all $d$ in the thermodynamic limit, but large finite-size effects plague its measurement close to jamming.