Energy fluctuations in one dimensional Zhang sandpile model

TL;DR

This paper studies energy fluctuations in a one-dimensional Zhang sandpile model, revealing how local dissipation and boundary driving influence spectral behavior and cutoff times, with implications for understanding $1/f$ noise.

Contribution

It demonstrates that local dissipation is not essential for $1/f$ noise in 1D sandpile models and characterizes how cutoff times scale with system size under different dynamics.

Findings

Lorentzian spectra with cutoff time $T$ growing linearly with system size $L$.

$1/f^{eta}$ behavior with $eta o 1$ observed for boundary drive.

Different scaling laws for cutoff time in conservative vs. nonconservative dynamics.

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.