We discuss the behavior of statistical models on a novel class of complex “Hanoi” networks. Such modeling is often the cornerstone for the understanding of many dynamical processes in complex networks. Hanoi networks are special because they integrate small-world hierarchies common to many social and economical structures with the inevitable geometry of the real world these structures exist in. In addition, their design allows exact results to be obtained with the venerable renormalization group (RG). Our treatment will provide a detailed, pedagogical introduction to RG. In particular, we will study the Ising model with RG, for which the fixed points are determined and the RG flow is analyzed. We show that the small-world bonds result in non-universal behavior. It is shown that a diversity of different behaviors can be observed with seemingly small changes in the structure of hierarchical networks generally, and we provide a general theory to describe our findings.
The transition into a glassy state of the ensemble of static, mechanically stable configurations of a tapped granular pile is explored using extensive molecular dynamics simulations. We show that different horizontal subregions (“layers”) along the height of the pile traverse this transition in a similar manner but at distinct tap intensities. We supplement the conventional approach based purely on properties of the static configurations with investigations of the grain-scale dynamics by which the tap energy is transmitted throughout the pile. We find that the effective energy that particles dissipate is a function of each particle's location in the pile and, moreover, that its value plays a distinctive role in the transformation between configurations. This internal energy provides a “temperature-like” parameter that allows us to align the transition into the glassy state for all layers, as well as different annealing schedules, at a critical value.
This paper links the nonequilibrium glassy relaxation behavior of otherwise athermal granular materials to those of thermally activated glasses. Thus, it demonstrates a much wider universality among complex glassy materials out of equilibrium. Our three-dimensional molecular dynamics simulations, fully incorporating friction and inelastic collisions, are designed to reproduce experimental behavior of tapped granular piles. A simple theory based on a dynamics of records explains why the typical phenomenology of annealing and aging after a quench should extend to such granular matter, activated by taps, beyond the more familiar realm of polymers, colloids, and magnetic materials that all exhibit thermal fluctuations.