Exact enumeration of the Critical States in the Oslo Model

Details Exact enumeration of the Critical States in the Oslo Model Exact enumeration of the Critical States in the Oslo Model

2002-03-12

arxiv.org/abs/cond-mat/0203260v2

Physics

Condensed Matter

11 pagers, 4 figures, 6 files in all. Apologies for equations in abstract. Submitted to PRL

Scientific paper

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.

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