Hierarchical Model of Slow Constrained Dynamics

We introduce a simple hierarchically constrained model of slow relaxation. The configurational energy has a simple form as there is no coupling among the spins defining the system; the associated stationary distribution is an equilibrium, Gibbsian one. However, due to the presence of hierarchical constraints in the dynamics the system is found to relax to its equilibrium distribution in an extremely slow fashion when suddenly cooled from an initial temperature T0 to a final one Tf. The relaxation curve in that case can be fit by a stretched-exponential curve. On the other hand, the relaxation function is found to be exponential when Tf>T0, with characteristic times depending on both Tf and T0, with characteristic times obeying an Arrhenius law. Numerical results as well as some analytical studies are presented. In particular, we introduce a simple equation that captures the essence of the slow relaxation.