Emergence of quasiunits in the one-dimensional Zhang model

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 ln(1/xi) for small xi , which agrees very well with the results from numerical simulations.