Spatio-temporal correlations of local relaxation rates in glassy systems

We numerically study the equilibrium dynamics of a 3D binary hard-sphere (HARD) system and a 3D Kob-Andersen Lennard-Jones (KALJ) system. We provide a quantitative description of dynamical heterogeneity in terms of fluctuations in the local relaxation rates. We extract a recently defined two-point correlator, s(q,t), that probes those fluctuations. The time decay of χ^φ₂(t) = lim_q→0 s(q,t) is characterized by a certain time scale τ_ex, which we identify as the exchange time of the slow and fast regions, or equivalently the lifetime of the heterogeneities. We calculate the exchange time τ_ex as a function of either the temperature (for KALJ) or packing fraction (for HARD). We find that, in HARD systems, close to the glass transition, τ_ex becomes an order of magnitude longer than the α-relaxation time τ_α and τ_ex/τ_α increases with packing fractions. However, in KALJ systems, τ_ex is approximately twice τ_α and τ_ex/τ_α has no significant dependency on temperature. We also measure the two-point correlation length ξ^φ₂(t) of the relaxation rates. We find that near the glass transition ξ^φ₂(t) grows approximately logarithmically with time up to a maximum at time around 2τ_α , and it decays at long times.