Spectral properties of zero temperature dynamics in a model of a compacting granular column

TL;DR

This paper analyzes the spectral properties of a zero-temperature granular column model using a one-dimensional Ising framework, revealing unique degeneracies and the necessity of Jordan forms in its stochastic dynamics.

Contribution

It provides an analytical study of the spectral structure of the dynamics, highlighting the unusual degeneracies and the use of Jordan forms, which are not typical in such models.

Findings

Eigenvalues are highly degenerate at zero temperature.

The generator lacks a spectral expansion and requires Jordan form analysis.

Many properties of the dynamics are established analytically.

Abstract

The compacting of a column of grains has been studied using a one-dimensional Ising model with long range directed interactions in which down and up spins represent orientations of the grain having or not having an associated void. When the column is not shaken (zero 'temperature') the motion becomes highly constrained and under most circumstances we find that the generator of the stochastic dynamics assumes an unusual form: many eigenvalues become degenerate, but the associated multi-dimensional invariant spaces have but a single eigenvector. There is no spectral expansion and a Jordan form must be used. Many properties of the dynamics are established here analytically; some are not. General issues associated with the Jordan form are also taken up.