Power law relations in earthquakes from microscopic to macroscopic scales

Understanding the physics of earthquakes is a crucial step towards improving the prediction accuracy of earthquakes. Scale invariance or fractal features are often reported in earthquakes, such as the size distribution of earthquakes, the spatial distribution of hypocenters, and the frequency of aftershocks. Here we assess whether other key parameters and quantities involved in earthquakes also conform to the power law. By analyzing a large amount of data collected from the laboratory experiments and field monitoring of earthquakes, we find that the crack density on the two sides of small scale fracture or large scale fault decreases with increasing distance following the power law, and the crack number-crack length distribution is also scale invariant like natural faults. Besides, the earthquake b-value is found to decrease with increasing stress in a power law in the brittle regime of the Earth's crust. The friction coefficient for dry fault and gouges or for partially saturated gouges decreases with the increasing effective normal stress in a power law. The stress dependency of b-value and friction coefficient is dictated by different mechanisms. Our findings will advance the understanding of earthquake physics, and will enable us to better model, predict and conduct hazard assessment of earthquakes.