Traveling density wave models for earthquakes and driven threshold systems

Driven threshold systems are now used to model sandpiles, earthquakes, magnetic depinning transitions, integrate-and-fire neural networks, and driven foams. We analyze a physically motivated model which has many of the same properties as the hard threshold models, but in which all of the nonequilibrium physics is obtained from a Lyapunov functional. The ideas apply to mean-field systems, and lead to a number of predictions, including scaling exponents and metastable lifetimes for nucleating droplets. The former predictions are supported, for example, by data observed for earthquake fault systems. An interesting consequence of the model is that time appears as a scaling field, leading to temporal scaling laws similar to those observed in nature.