An entropic explanation for Gutenberg-Richter scaling
We develop a simple explanation for Gutenberg-Richter (G-R) size scaling of earthquakes on a single fault. We discretize the fault and consider all possible contiguous ruptures at that level of discretization. In this static model, we assume that slip scales with rupture length, and that the rupture rates at each point along the fault are consistent with an a priori long-term slip rate. These simple assumptions define an (under-determined) non-negative least-squares inverse problem. Each solution to this inverse problem is a set of earthquake rates that matches the slip-rate constraint. We use a Markov Chain Monte Carlo (MCMC) algorithm to uniformly sample the solution space assuming constant slip rates along the fault. At finer discretizations, deviations from G-R behavior decrease, which is consistent with an entropic pressure towards G-R solutions. When the fault is discretized into 10 or more segments, random solutions found by the MCMC algorithm have G-R size scaling, even though there are trivial solutions that, for example, have earthquakes of only one size. This is because there are simply far more solutions that have G-R scaling; as the problem size increases, the strong degeneracy of GR solutions results in other solutions becoming improbably rare. Also, the entropically favored G-R distribution has a b -value of approximately 1, which agrees with measured b -values in real earthquake catalogs.