Slow Relaxation in a Constrained Ising Spin Chain: a Toy Model for Granular Compaction

TL;DR

This paper analyzes slow, inverse logarithmic relaxation in a constrained Ising spin chain and maps it onto a granular compaction model, providing analytical insights into the dynamics of both systems.

Contribution

It introduces a toy model linking spin chain dynamics with granular compaction, revealing a new mechanism for inverse logarithmic relaxation.

Findings

Magnetization approaches -1 extremely slowly in inverse logarithmic time.

Granular density grows following an inverse logarithmic law.

Model predictions align with recent experimental observations.

Abstract

We present detailed analytical studies on the zero temperature coarsening dynamics in an Ising spin chain in presence of a dynamically induced field that favors locally the `-' phase compared to the `+' phase. We show that the presence of such a local kinetic bias drives the system into a late time state with average magnetization m=-1. However the magnetization relaxes into this final value extremely slowly in an inverse logarithmic fashion. We further map this spin model exactly onto a simple lattice model of granular compaction that includes the minimal microscopic moves needed for compaction. This toy model then predicts analytically an inverse logarithmic law for the growth of density of granular particles, as seen in recent experiments and thereby provides a new mechanism for the inverse logarithmic relaxation. Our analysis utilizes an independent interval approximation for the particle and the hole clusters and is argued to be exact at late times (supported also by numerical simulations).