## Slow Relaxation in Granular Compaction

### E. Ben-Naim, J.B.Knight, and E.R.Nowak

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
$\rho_{\infty}-\rho(t)\cong 1/\ln t$.

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