Extreme value statistics and thermodynamics of earthquakes: aftershock sequences

The Gutenberg-Richter magnitude-frequency law takes into account the minimum detectable magnitude, and treats aftershocks as if they were independent and identically distributed random events. A new magnitude-frequency relation is proposed which takes into account the magnitude of the main shock, and the degree to which aftershocks depend on the main shock makes them appear clustered. In certain cases, there can be two branches in the order-statistics of aftershock sequences: for energies below threshold, the Pareto law applies and the asymptotic distribution of magnitude is the double-exponential distribution, while energies above threshold follow a one-parameter beta distribution, whose exponent is the cluster dimension, and the asymptotic Gompertz distribution predicts a maximum magnitude. The 1957 Aleutian Islands aftershock sequence exemplifies such dual behavior. A thermodynamics of aftershocks is constructed on the analogy between the non-conservation of the number of aftershocks and that of the particle number in degenerate gases.