Statistical physics of earthquakes: Comparison of distribution exponents for source area and potential energy and the dynamic emergence of log-periodic energy quanta

Abstract

We investigate the relationship between the size distribution of earthquake rupture area and the underlying elastic potential energy distribution in a cellular automaton model for earthquake dynamics. The frequency-rupture area distribution has the form n(S)~Sτ exp(-S/S(o)) and the system potential energy distribution from the elastic Hamiltonian has the form n(E)~E(v) exp(-E/θ), both gamma distributions. Here n(S) reduces to the Gutenberg-Richter frequency-magnitude law, with slope b~τ, in the limit that the correlation length ξ, related to the characteristic source size S(O), tends to infinity. The form of the energy distribution is consistent with a statistical mechanical model with l degrees of freedom, where v=(l-2)/2 and θ is proportional to the mean energy per site E. We examine the effect of the local energy conservation factor β and the degree of material heterogeneity (quenched disorder) on the distribution parameters, which vary systematically with the controlling variables. The inferred correlation length increases systematically with increasing material homogeneity and with increasing β. The thermal parameter θ varies systematically between the leaf springs and the connecting springs, and is proportional to E as predicted. For heterogeneous faults, τ~1 stays relatively constant, consistent with field observation, and S(o) increases with increasing β or decreasing heterogeneity. In contrast, smooth faults produce a systematic decrease in τ with respect to β and S(o) remains relatively constant. For high β approximately log-periodic quanta emerge spontaneously from the dynamics in the form of modulations on the energy distribution. The output energy for both types of fault shows a transition from strongly quasi-periodic temporal fluctuations for strong dissipation, to more chaotic fluctuations for more conservative models. Only strongly heterogeneous faults show the small fluctuations in energy strictly required by models of self-organized criticality.