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.
Categories: cond-mat