| We describe analytical calculations and simulations of the quasi-one-dimensional (Q1-D) model for glassy dynamics. In the Q1-D models, hard rods undergo single-file diffusion through a series of narrow channels connected by J intersections. The topology of the model is specified by J, the maximum number of rods in each middle channel K, and the number of rods N. We assume that the rods cannot turn at the intersections, and thus there is a single, continuous route through the system. This model displays hallmarks of glassy dynamics including caging behavior and subdiffusion, rapid growth in the structural relaxation time and collective particle rearrangements.;The mean-square displacement Sigma(t) for the Q1-D model displays four dynamical regimes: 1) short-time diffusion Sigma( t) ∼ t, 2) a plateau Sigma(t) ∼ t0 caused by caging behavior, 3) single-file diffusion characterized by anomalous scaling Sigma(t) ∼ t0.5 at intermediate times, and 4) a crossover to long-tine diffusion Sigma(t) ∼ t for times that grow with the system size. We develop a general procedure for calculating analytically the structural relaxation time tD, beyond which the system undergoes long-time diffusion, as a function of density and system topology. The method involves several steps: 1) uniquely defining the set of microstates for the system and transitions among them, 2) constructing networks of connected microstates and identifying minimal, closed, directed loops that give rise to structural relaxation, 3) calculating the probabilities for obtaining each of the microstates that form the closed loops and for transitioning from one microstate to another, and 4) using these probabilities to deduce the dependence of tD on packing fraction. We find that to obeys power-law scaling tD ∼ (&phis; g-&phis;)-alpha, where &phis; g (the packing fraction corresponding to complete kinetic arrest) and alpha depend on the system topology, and can be calculated exactly. The analytical calculations are supported quantitatively by Monte Carlo simulations. |
