Jamming transition of kinetically-constrained models in rectangular systems

TL;DR

This paper provides a theoretical analysis of the jamming transition in rectangular systems for kinetically-constrained models, revealing how aspect ratio influences critical vacancy density and transition behavior.

Contribution

It introduces analytical formulas for the jamming transition in rectangular systems, highlighting the impact of aspect ratio on critical vacancy density and transition nature.

Findings

Critical vacancy density decreases algebraically with width in long channels.

In square systems, the vacancy density decreases logarithmically with system length.

Analytical results agree with numerical data for small to moderate system sizes.

Abstract

We theoretically calculate the average fraction of frozen particles in rectangular systems of arbitrary dimensions for the Kob-Andersen and Fredrickson-Andersen kinetically-constrained models. We find the aspect ratio of the rectangle's length to width, which distinguishes short, square-like rectangles from long, tunnel-like rectangles, and show how changing it can effect the jamming transition. We find how the critical vacancy density converges to zero in infinite systems for different aspect ratios: for long and wide channels it decreases algebraically $v_{c}\sim W^{-1/2}$ with the system's width W, while in square systems it decreases logarithmically $v_{c}\sim1/\ln L$ with length L. Although derived for asymptotically wide rectangles, our analytical results agree with numerical data for systems as small as $W\approx10$.