Energy fluctuations in one dimensional Zhang sandpile model

Abstract

We consider the Zhang sandpile model in one-dimension (1D) with locally conservative (or dissipative) dynamics and examine its total energy fluctuations at the external drive time scale. The bulk-driven system leads to Lorentzian spectra, with a cutoff time T growing linearly with the system size L. The fluctuations show 1/f α behavior with α ∼ 1 for the boundary drive, and the cutoff time varies non-linearly. For conservative local dynamics, the cutoff time shows a power-law growth T ∼ L λ that differs from an exponential form ∼exp(μL) observed for the nonconservative case. We suggest that the local dissipation is not a necessary ingredient of the system in 1D to get the 1/f noise, and the cutoff time can reveal the distinct nature of the local dynamics. We also discuss the energy fluctuations for locally nonconservative dynamics with random dissipation.