Exact enumeration of the Critical States in the Oslo Model

TL;DR

This paper analytically determines the exact number of recurrent states in the 1D Oslo model, revealing exponential growth with system size and non-ergodic behavior, confirmed by simulations.

Contribution

It provides an exact analytical formula for the number of recurrent states in the Oslo model, a significant advancement over previous numerical estimates.

Findings

Number of recurrent states grows exponentially with system size.

Exact formula matches computer simulations for small sizes.

States in the attractor are not equally probable, indicating non-ergodicity.

Abstract

We determine analytically the number $N_{\mathcal{R}}(L)$ of recurrent states in the 1d Oslo model as a function of system size L. The solution $N_{\mathcal{R}}(L) = \frac{1+\sqrt{5}}{2\sqrt{5}}(\frac{3+\sqrt{5}}{2})^L + \frac{\sqrt{5}-1}{2\sqrt{5}}(\frac{3-\sqrt{5}}{2})^L$ is in exact agreement with the number enumerated in computer simulations for $L = 1 - 10$. For $L \gg 1$, the number of allowed metastable states in the attractor increases exponentially as $N_{\mathcal{R}}(L) \approx c_+ {\lambda}_+^L$, where $\lambda_+ = \frac{3+\sqrt{5}}{2}$ is the golden mean. The system is non-ergodic in the sense that the states in the attractor are not equally probable.