On the Omori Law in the Physics of Earthquakes

Abstract

This paper proposes phenomenological equations that describe various aspects of aftershock evolution: elementary master equation, logistic equation, stochastic equation, and nonlinear diffusion equation. The elementary master equation is a first-order differential equation with a quadratic term. It is completely equivalent to Omori’s law. The equation allows us to introduce the idea of proper time of earthquake source “cooling down” after the main shock. Using the elementary master equation, one can pose and solve an inverse problem, the purpose of which is to measure the deactivation coefficient of an earthquake source. It has been found for the first time that the deactivation coefficient decreases with increasing magnitude of the main shock. The logistic equation is used to construct a phase portrait of a dynamical system simulating the evolution of aftershocks. The stochastic equation can be used to model fluctuation phenomena, and the nonlinear diffusion equation provides a framework for understanding the spatiotemporal distribution of aftershocks. Earthquake triads, which are a natural trinity of foreshocks, main shock, and aftershocks, are considered. Examples of the classical triad, the mirror triad, the symmetrical triad, as well as the Grande Terremoto Solitario, which can be considered as an anomalous symmetrical triad, are given. Prospects for further development of the phenomenology of earthquakes are outlined.