Relaxation times, rheology, and finite size effects

TL;DR

This study uses simulations of soft disks to analyze finite size effects on relaxation times and viscosity near jamming, challenging previous claims of problematic size dependence and suggesting new insights into shear-driven jamming.

Contribution

The paper provides a detailed analysis of finite size effects on relaxation times and viscosity in a jamming model, introducing particle relaxation times and contrasting initial configurations to clarify size dependence issues.

Findings

Finite size effects on relaxation time are less problematic than previously thought.

Differences in initial configurations significantly affect relaxation time measurements.

Pressure-equivalent viscosity shows weak finite size dependence, contrary to earlier claims.

Abstract

We carry out overdamped simulations in a simple model of jamming - a collection of bi-disperse soft core frictionless disks in two dimensions - with the aim to explore the finite size dependence of different quantities, both the relaxation time obtained from the relaxation of the energy and the pressure-equivalent of the shear viscosity. The motivation for the paper is the observation [Nishikawa et al., J. Stat. Phys, 182, 37 (2021)] that there are finite size effects in the relaxation time, $\tau$, that give problems in the determination of the critical divergence, and the claim that this is due to a finite size dependence, $\tau\sim\ln N$, which makes $\tau$ an ill-defined quantity. Beside analyses to determine the relaxation time for the whole system we determine particle relaxation times which allow us to determine both histograms of particle relaxation times and the average particle relaxation times - two quantities that are very useful for the analyses. The starting configurations for the relaxation simulations are of two different kinds: completely random or taken from steady shearing simulations, and we find that the difference between these two cases are bigger than previously noted and that the observed problems in the determination of the critical divergence obtained when starting from random configurations are not present when instead starting the relaxations from shearing configurations. We also argue that the the effect that causes the $\ln N$-dependence is not as problematic as asserted. When it comes to the finite size dependence of the pressure-equivalent of the shear viscosity we find that our data don't give support for the claimed strong finite size dependence, but also that it is at odds with what one would normally expect for a system with a diverging correlation length, and that this calls for a novel understanding of the phenomenon of shear-driven jamming.