Random neighbor theory of the Olami-Feder-Christensen earthquake model

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Abstract
We derive the exact equations of motion for the random neighbor version of the Olami-Feder-Christensen earthquake model in the infinite-size limit. We solve them numerically and compare them with simulations of the model for large numbers of sites. We find perfect agreement. But we do not find any scaling or phase transitions, except in the conservative limit. This is in contradiction to claims by Lise and Jensen [Phys. Rev. Lett. 76, 2326 (1996)] based on approximate solutions of the same model. It indicates again that scaling in the Olami-Feder-Christensen model is only due to partial synchronization driven by spatial inhomogeneities. Finally, we point out that our method can be used also for other self-organized criticality models and treat in detail the random neighbor version of the Feder-Feder model.