Two-step relaxation and the breakdown of the Stokes-Einstein relation in glass-forming liquids

It is well known that glass-forming liquids exhibit a number of anomalous dynamical phenomena, most notably a two-step relaxation in the self-intermediate scattering function and the breakdown of the Stokes-Einstein (SE) relation, as they are cooled toward the glass transition temperature. While these phenomena are generally ascribed to dynamic heterogeneity, specifically to the presence of slow- and fast-moving particles, a quantitative elucidation of the two-step relaxation and the violation of the SE relation in terms of these concepts has not been successful. In this work, we propose a classification of particles according to the rank order of their displacements (from an arbitrarily defined origin of time), and we divide the particles into long-distance (LD), medium-distance, and short-distance (SD) traveling particle groups. Using molecular-dynamics simulation data of the Kob-Andersen model, we show quantitatively that the LD group is responsible for the fast relaxation in the two-step relaxation process in the intermediate scattering function, while the SD group gives rise to the slow (α) relaxation. Furthermore, our analysis reveals that τ_{α} is controlled by the SD group, while the ensemble-averaged diffusion coefficient D is controlled by both the LD and SD groups. The combination of these two features provides a natural explanation for the breakdown in the SE relation at low temperature. In addition, we find that the α-relaxation time, τ_{α}, of the overall system is related to the relaxation time of the LD particles, τ_{LD}, as τ_{α}=τ_{0}exp(Ωτ_{LD}/k_{B}T).