Magnitude distribution of earthquakes: Two fractal contact area distribution

TL;DR

This paper investigates the distribution of contact areas between fractal surfaces, revealing power law behavior for some fractals and Gaussian for others, which may relate to earthquake energy distribution.

Contribution

It introduces a study of contact area distributions between fractal surfaces, highlighting universal scaling and different distribution types based on fractal models.

Findings

Power law decay in contact area distribution for Cantor sets and gaskets.

Gaussian distribution observed for percolation fractals.

Universal finite size scaling behavior across models.

Abstract

The `plate tectonics' is an observed fact and most models of earthquake incorporate that through the frictional dynamics (stick-slip) of two surfaces where one surface moves over the other. These models are more or less successful to reproduce the well known Gutenberg-Richter type power law in the (released) energy distribution of earthquakes. During sticking period, the elastic energy gets stored at the contact area of the surfaces and is released when a slip occurs. Therefore, the extent of the contact area between two surfaces plays an important role in the earthquake dynamics and the power law in energy distribution might imply a similar law for the contact area distribution. Since, fractured surfaces are fractals and tectonic plate- earth's crust interface can be considered to have fractal nature, we study here the contact area distribution between two fractal surfaces. We consider the overlap set of two self-similar fractals, characterised by the same fractal dimensions, and look for their distribution. We have studied numerically the specific cases of both regular and random Cantor sets in one dimension and gaskets and percolation fractals in two dimension. We find that in all the cases the distributions show an universal finite size scaling behavior. The contact area distributions have got a power law decay for both regular and random Cantor sets and also for gaskets. However, for percolation clusters the distribution shows Gaussian variation.