Memory effects in avalanche dynamics: a key to the statistical properties of earthquakes
Résumé
Many complex systems respond to continuous input of energy by accumulation of stress over time and sudden energy releases in the form of avalanches. Avalanches are paradigmatic non-equilibrium phenomena displaying power law size distribution and involving all the length scales in the system. Conventional avalanche models disregard memory effects and thus miss basic features observed in real systems. Notable examples are aftershocks and the anomalous exponent of the Gutenberg-Richter law which characterize earthquake statistics. We propose a model which accounts for memory effects through the introduction of viscoelastic relaxation at an intermediate time scale. We demonstrate that in the resulting dynamics, coherent oscillations of the stress field emerge spontaneously without fine tuning of any parameter. Remarkably, in two dimensions, which is relevant in seismicity, these oscillations generate instability patterns that produce realistic earthquake dynamics with the correct Gutenberg-Richter exponent.