Effective dynamics and steady state of an Ising model submitted to tapping processes

TL;DR

This paper models granular media compaction using a one-dimensional Ising model with tapping, deriving an effective dynamics for metastable states, and showing the steady state aligns with Edwards thermodynamics.

Contribution

It introduces an effective dynamics for Ising model metastable states under tapping and demonstrates the steady state matches Edwards thermodynamic theory.

Findings

Steady state probability distribution has a canonical form.

Effective dynamics connects metastable states during tapping.

Spatial correlations in the steady state are characterized.

Abstract

A one-dimensional Ising model with nearest neighbour interactions is applied to study compaction processes in granular media. An equivalent particle-hole picture is introduced, with the holes being associated to the domain walls of the Ising model. Trying to mimic the experiments, a series of taps separated by large enough waiting times, for which the system freely relaxes, is considered. The free relaxation of the system corresponds to a T=0 dynamics which can be analytically solved. There is an extensive number of metastable states, characterized by all the holes being isolated. In the limit of weak tapping, an effective dynamics connecting the metastable states is obtained. The steady state of this dynamics is analyzed, and the probability distribution function is shown to have the canonical form. Then, the stationary state is described by Edwards thermodynamic granular theory. Spatial correlation functions in the steady state are also studied.