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Quasi-One Dimensional Models for Glassy Dynamics

We describe numerical simulations and analyses of a quasi-one-dimensional (Q1D) model of glassy dynamics. In this model, hard rods undergo Brownian dynamics through a series of narrow channels connected by $J$ intersections. We do not allow the rods to turn at the intersections, and thus there is a single, continuous route through the system. This Q1D model displays caging behavior, collective particle rearrangements, and rapid growth of the structural relaxation time, which are also found in supercooled liquids and glasses. The mean-square displacement $Σ(t)$ for this Q1D model displays several dynamical regimes: 1) short-time diffusion $Σ(t) \sim t$, 2) a plateau in the mean-square displacement caused by caging behavior, 3) single-file diffusion characterized by anomalous scaling $Σ(t) \sim t^{0.5}$ at intermediate times, and 4) a crossover to long-time diffusion $Σ(t) \sim t$ for times $t$ that grow with the complexity of the circuit. We develop a general procedure for determining the structural relaxation time $t_D$, beyond which the system undergoes long-time diffusion, as a function of the packing fraction $ϕ$ and system topology. This procedure involves several steps: 1) define a set of distinct microstates in configuration space of the system, 2) construct a directed network of microstates and transitions between them, 3) identify minimal, closed loops in the network that give rise to structural relaxation, 4) determine the frequencies of `bottleneck' microstates that control the slow dynamics and time required to transition out of them, and 5) use the microstate frequencies and lifetimes to deduce $t_D(ϕ)$. We find that $t_D$ obeys power-law scaling, $t_D\sim (ϕ^* - ϕ)^{-α}$, where both $ϕ^*$ (signaling complete kinetic arrest) and $α>0$ depend on the system topology.