Jamming and metastability in one dimension: from the kinetically constrained Ising chain to the Riviera model

TL;DR

This paper explores the dynamics and metastability of one-dimensional kinetically constrained models, comparing the well-understood Ising chain with the recently introduced Riviera model, highlighting differences in solvability and attractor properties.

Contribution

It provides new insights into the attractors of the Riviera model through enumeration and simulations, contrasting it with the analytically solvable kinetically constrained Ising chain.

Findings

Riviera model exhibits similar metastability to Ising chains.

Riviera model lacks the shielding property, complicating analysis.

Extensive simulations reveal detailed attractor structures.

Abstract

The Ising chain with kinetic constraints provides many examples of totally irreversible zero-temperature dynamics leading to metastability with an exponentially large number of attractors. In most cases, the constrained zero-temperature dynamics can be mapped onto a model of random sequential adsorption. We provide a brief didactic review, based on the example of the constrained Glauber-Ising chain, of the exact results on the dynamics of these models and on their attractors that have been obtained by means of the above mapping. The Riviera model introduced recently by Puljiz et al. behaves similarly to the kinetically constrained Ising chains. This totally irreversible deposition model however does not enjoy the shielding property characterising models of random sequential adsorption. It can therefore neither be mapped onto such a model nor (in all likelihood) be solved by analytical means. We present a range of novel results on the attractors of the Riviera model, obtained by means of an exhaustive enumeration for smaller systems and of extensive simulations for larger ones, and put these results in perspective with the exact ones which are available for kinetically constrained Ising chains.