Slow relaxation in granular compaction

TL;DR

This paper models the slow, logarithmic relaxation of granular material density during vibration using a stochastic adsorption-desorption process, explaining experimental observations of gradual densification.

Contribution

It introduces a simple stochastic model that captures the inverse logarithmic relaxation behavior in granular compaction, linking microscopic rearrangements to macroscopic density evolution.

Findings

Density approaches final value logarithmically over time

Model reproduces experimental relaxation law

Reveals the role of local rearrangements in slow dynamics

Abstract

Experimental studies show that the density of a vibrated granular material evolves from a low density initial state into a higher density final steady state. The relaxation towards the final density value follows an inverse logarithmic law. We propose a simple stochastic adsorption-desorption process which captures the essential mechanism underlying this remarkably slow relaxation. As the system approaches its final state, a growing number of beads have to be rearranged to enable a local density increase. In one dimension, this number grows as $N=\rho/(1-\rho)$, and the density increase rate is drastically reduced by a factor $e^{-N}$. Consequently, a logarithmically slow approach to the final state is found $\rho_{\infty}-\rho(t)\cong 1/\ln t$.