Exact enumeration of the Critical States in the Oslo Model
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.
- Publication:
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arXiv e-prints
- Pub Date:
- March 2002
- DOI:
- 10.48550/arXiv.cond-mat/0203260
- arXiv:
- arXiv:cond-mat/0203260
- Bibcode:
- 2002cond.mat..3260C
- Keywords:
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- Condensed Matter
- E-Print:
- 11 pagers, 4 figures, 6 files in all. Apologies for equations in abstract. Submitted to PRL