Simulated dynamical weakening and Abelian avalanches in mean-field driven threshold models

Mean-field coupled lattice maps are used to approximate the physics of driven threshold systems with long-range interactions. However, they are incapable of modeling specific features of the dynamic instability responsible for generating avalanches. Here we present a method of simulating specific frictional weakening effects in a mean-field slider-block model. This provides a means of exploring dynamical effects previously inaccessible to discrete time simulations. This formulation also results in Abelian avalanches, where rupture propagation is independent of the failure sequence. The resulting event size distribution is shown to be generated by the boundary crossings of a stochastic process. This is applied to typical models to explain some commonly observed features.