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 $1/f^{\alpha }$ behavior with $\alpha \sim 1$ for the boundary drive, and the cutoff time varies non-linearly. For conservative local dynamics, the cutoff time shows a power-law growth $T\sim L^{\lambda }$ that differs from an exponential form $\sim \exp (\mu L)$ 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.

arXiv comment15 pages, 16 figures

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