Power spectrum of mass and activity fluctuations in a sandpile

Abstract

We consider a directed Abelian sandpile on a strip of size 2×n, driven by adding a grain randomly at the left boundary after every T timesteps. We establish the exact equivalence of the problem of mass fluctuations in the steady state and the number of zeros in the ternary-base representation of the position of a random walker on a ring of size 3ⁿ. We find that while the fluctuations of mass have a power spectrum that varies as 1/f for frequencies in the range 3⁻²ⁿ≪f≪1/T, the activity fluctuations in the same frequency range have a power spectrum that is linear in f.