Trapping diffusion model for glass transition and slow dynamics in supercooled liquids

We present the trapping diffusion model for understanding the glass transition and slow dynamics in supercooled liquids in a unified manner. We focus on the stochastic dynamics in the mesoscopic scale, which is described by a trapping-type master equation. The parameters of the master equation are chosen so that the waiting time distribution agrees with the observation for a binary soft sphere by molecular dynamics simulation. We first discuss the memory effect originated from the randomness and explain the coherent medium approximation to be used in the numerical calculation. By analyzing the long time behavior of jump motion, we show that the glass transition is understood as Gaussian to non-Gaussian transition, and that in the glass state the anomalous diffusion is expected because of the divergence of the mean waiting time and hence the intermediate scattering function becomes a stretched exponential function in the long time limit. In a certain range above the glass transition point, the jump motion becomes subanomalous because the fluctuation of the waiting time distribution diverges. Consequently, the intermediate scattering function becomes a stretched exponential function in a certain time window, indicating that the jump motion yields the α-relaxation. We analyze the various singularities of the dynamical structure factor near the static limit. We include the stray motion in the model and show that the crossover between jump motion and stray motion gives rise to the characteristics of the β-relaxation.