Local fluctuations in the relaxation rate in glassy systems: probing dynamical heterogeneity and its exchange time

We numerically study the equilibrium dynamics of a glass-forming binary hard-sphere mixture in three dimensions, for different packing fractions. We provide a quantitative description of dynamical heterogeneity in terms of fluctuations in the local relaxation rates. We extract a newly defined two-point correlator, χ₂^Φ(t), that probes those fluctuations. The time decay of χ₂^Φ(t) is characterized by a certain time scale τ_ex, which we identify as the exchange time of the slow and fast regions. We calculate the exchange time τ_ex as a function of packing fraction, and we find that in the vicinity of the glass transition, τ_ex is an order of magnitude longer then the α-relaxation time, τ_α.