Emergence of quasi-units in the one dimensional Zhang model

Details Emergence of quasi-units in the one dimensional Zhang model Emergence of quasi-units in the one dimensional Zhang model

2007-11-19

arxiv.org/abs/0711.3021v1

Phys. Rev. E 77, 031122 (2008)

Physics

Condensed Matter

Statistical Mechanics

7 pages, 3 figures, RevTex4

Scientific paper

10.1103/PhysRevE.77.031122

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.

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