Broken ergodicity and the geometry of rugged landscapes

We study the nature and mechanisms of broken ergodicity (BE) in specific random walk models corresponding to diffusion on random potential surfaces, in both one dimension and high dimension. Using both rigorous results and nonrigorous methods, we confirm several aspects of the standard BE picture and show that others apply in one dimension, but need to be modified in higher dimensions. These latter aspects include the notions that a fixed temperature confining barriers increase logarithmically with time, that " components" are necessarily bounded regions of state space which depend on the observational time scale, and that the system continually revisits previously traversed regions of state space. We examine our results in the context of several experiments, and discuss some implications of our results for the dynamics of disordered and/or complex systems.