Emergence of quasi-units in the one dimensional Zhang model

TL;DR

This paper investigates the steady-state energy distribution in a one-dimensional Zhang sandpile model, revealing a sharply peaked distribution with a width decreasing as the system size increases, and relates it to a similar Abelian model.

Contribution

It introduces a detailed analysis of energy distribution and fluctuations in the 1D Zhang model, connecting it to the behavior of a marked grain in the Abelian model, and provides scaling forms and numerical validation.

Findings

Energy distribution at a site is sharply peaked in steady state.

Width of the energy distribution scales as L^{-1/2} for large L.

Variance of energy follows a scaling form with a logarithmic behavior for small positions.

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.