Aftershocks and fluctuating diffusivity

TL;DR

This paper models earthquake aftershocks using a hierarchical Fokker-Planck framework, revealing that load state diffusion fluctuations lead to a logarithmic-time aging phenomenon and the Omori-Utsu law as a relaxation process.

Contribution

It introduces a novel Fokker-Planck-based approach to describe aftershock dynamics with fluctuating diffusivity, highlighting aging effects in the load state evolution.

Findings

The load state diffusion coefficient fluctuates as a slow variable.

The subsystem exhibits temporal invariance in logarithmic time.

Aging phenomena emerge in the load state dynamics.

Abstract

The Omori-Utsu law shows the temporal power-law-like decrease of the frequency of earthquake aftershocks and, interestingly, is found in a variety of complex systems/phenomena exhibiting catastrophes. Now, it may be interpreted as a characteristic response of such systems to large events. Here, hierarchical dynamics with the fast and slow degrees of freedom is studied on the basis of the Fokker-Planck theory for the load-state distribution to formulate the law as a relaxation process, in which diffusion coefficient in the space of the load state is treated as a fluctuating slow variable. The evolution equation reduced from the full Fokker-Planck equation and its Green's function are analyzed for the subdynamics governing the load state as the fast degree of freedom. It is shown that the subsystem has the temporal translational invariance in the logarithmic time, not in the conventional time, and consequently the aging phenomenon appears.