Velocity Autocorrelation Functions of Hard-Sphere Fluids: Long-Time Tails upon Undercooling

Molecular dynamics simulations are employed to obtain the velocity autocorrelation function (VAF) for hard spheres, spanning a wide range of volume fractions from dilute to high-density metastable fluids. For all volume fractions below freezing, Alder's classical positive $3/2$ long-time tail is observed. For volume fractions from 0.45 to 0.48 the VAF becomes negative, before becoming positive and decaying with the positive $3/2$ long-time tail. At the freezing volume fraction (0.494) the Alder $3/2$ tail is not observed. At higher volume fractions a negative tail with an exponent of $5/2$ emerges, which coincides with the long-time tail of a Lorentz gas.