Diffusion and dynamical transition in hierarchical systems

We have recently found a phase transition in the dynamics of a particle that diffuses in the presence of a hierarchical array of barriers. The transition occurs as R, a parameter that controls the relative size of the barriers, is varied. For R>$R_c$, the system is diffusive; as R→$R_c^{+}$ the diffusion constant vanishes. For R<$R_c$, the system is subdiffusive; the mean-square displacement grows slower than linearly with time, with an exponent that varies continuously with R. Analytic as well as numerical renormalization-group methods are introduced and used to solve some specific models; however, the tools developed here are applicable to more general situations as well. In one dimension we present, in addition to a renormalization-group solution for the asymptotic behavior, several new results; in particular, scaling arguments that predict transient behavior as well. We show that the asymptotic regime is approached algebraically, on time scales that obey a law of a Vogel-Fulcher form near the transition. Higher-dimensional systems that also have such a transition are introduced and studied. Finally, possible connections to many-body systems such as dynamic spin models and glassy materials are suggested and discussed.