Slow relaxation due to optimization and restructuring: Solution on a hierarchical lattice

Motivated by the large strain shear of loose granular materials, we introduced a model that consists of consecutive optimization and restructuring steps leading to a self-organization of a density field. The extensive connections to other models of statistical physics are discussed. We investigate our model on a hierarchical lattice that allows an exact asymptotic renormalization treatment. A surprisingly close analogy is observed between the simulation results on the regular and the hierarchical lattices. The dynamics is characterized by the breakdown of ergodicity, by unusual system size effects in the development of the average density, as well as by the age distribution, the latter showing multifractal properties.