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

摘要

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)similar to S(tau)exp(-S/S-o) and the system potential energy distribution from the elastic Hamiltonian has the form n(E)similar to E(v)exp(-E/theta), both gamma distributions. Here n(S) reduces to the Gutenberg-Richter frequency-magnitude law, with slope b similar to tau in the limit that the correlation length xi, 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 I degrees of freedom, where v=(l-2)/2 and theta is proportional to the mean energy per site (E) over bar. We examine the effect of the local energy conservation factor beta 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 beta. The thermal parameter theta varies systematically between the leaf springs and the connecting springs, and is proportional to (E) over bar as predicted. For heterogeneous faults, tau similar to 1 stays relatively constant, consistent with field observation, and S-o increases with increasing beta or decreasing heterogeneity. In contrast, smooth faults produce a systematic decrease in tau with respect to beta and S-o remains relatively constant. For high beta 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. [References: 51]