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Depinning as a coagulation process
We consider a one-dimensional model that describes the depinning of an
elastic string of particles in a strongly pinning, phase-disordered
periodic environment under a slowly increasing force. The evolution
towards depinning occurs by the triggering of avalanches in regions of
activity which are at first isolated, but later grow and merge. For
large system sizes the dynamically critical behavior is dominated by
the coagulation of these active regions. Our analysis and numerical
simulations show that the evolution of the sizes of active regions is
well described by a Smoluchowski coagulation equation, allowing us to
predict correlation lengths and avalanche sizes in terms of certain
moments of the size distribution.