Steady State and Relaxation Spectrum of the Oslo Rice-pile

TL;DR

This paper analyzes the steady state and relaxation spectrum of the one-dimensional Oslo rice-pile model, revealing its eigenstructure and correlation properties, and generalizes findings to other abelian critical height models.

Contribution

It determines the exact steady state and spectral properties of the Oslo rice-pile model, linking it to abelian distributed processors and analyzing its relaxation dynamics.

Findings

W has only one eigenvalue 1, others are zero

Connected correlations vanish for time differences greater than n

Exact steady state is explicitly determined

Abstract

We show that the one-dimensional Oslo rice-pile model is a special case of the abelian distributed processors model. The exact steady state of the model is determined. We show that the time evolution operator W for the system satisfies the equation W^{n+1} = W^n where n = L(L+1)/2 for a pile with L sites. This is used to prove that W has only one eigenvalue 1 corresponding to the steady state, and all other eigenvalues are exactly zero. Also, all connected time-dependent correlation functions in the steady state of the pile are exactly zero for time difference greater that n. Generalization to other abelian critical height models where the critical thresholds are randomly reset after each toppling is briefly discussed.