Emergence of quasi-units in the one dimensional Zhang model

Abstract

We study the Zhang model of sandpile on a one dimensional chain of length $L$, where a random amount of energy is added at a randomly chosen site at each time step. We show that in spite of this randomness in the input energy, the probability distribution function of energy at a site in the steady state is sharply peaked, and the width of the peak decreases as $L^{-1/2}$ for large $L$. We discuss how the energy added at one time is distributed among different sites by topplings with time. We relate this distribution to the time-dependent probability distribution of the position of a marked grain in the one dimensional Abelian model with discrete heights. We argue that in the large $L$ limit, the variance of energy at site $x$ has a scaling form $L^{-1}g(x/L)$, where $g(\xi )$ varies as $\log (1/\xi )$ for small $\xi$, which agrees very well with the results from numerical simulations.

7 pages, 3 figures, RevTex4

References in corpus (1)

• Anomalous scaling in the Zhang model