Power spectrum of mass and activity fluctuations in a sandpile

TL;DR

This paper analyzes the power spectrum of mass and activity fluctuations in a directed sandpile model, revealing a 1/f spectrum for mass fluctuations and a linear spectrum for activity fluctuations within specific frequency ranges.

Contribution

It establishes an exact equivalence between mass fluctuations in the sandpile and the zeroes in a ternary representation of a random walk, providing new analytical insights.

Findings

Mass fluctuations exhibit a 1/f power spectrum.

Activity fluctuations have a power spectrum linear in frequency.

The results apply within a specific frequency range related to system size and driving period.

Abstract

We consider a directed abelian sandpile on a strip of size $2\times n$, driven by adding a grain randomly at the left boundary after every $T$ time-steps. We establish the exact equivalence of the problem of mass fluctuations in the steady state and the number of zeroes in the ternary-base representation of the position of a random walker on a ring of size $3^n$. We find that while the fluctuations of mass have a power spectrum that varies as $1/f$ for frequencies in the range $ 3^{-2n} \ll f \ll 1/T$, the activity fluctuations in the same frequency range have a power spectrum that is linear in $f$.