Particle Contacts Generate Fractional Density Relaxation

Abstract

Dense-liquid relaxation evolves from local particle collisions to cooperative structural rearrangements. While hard-sphere kinetics determines an early $t^{3/2}$ fractional decay in density correlation functions, collective theories describe the subsequent structural relaxation. A central open question has been how short-time contact physics supplies an exact starting point for the memory kernel governing later times without being modified by subsequent many-body rearrangements. Here we resolve this problem for a broad class of reversible Brownian systems. We prove that hard particle contacts act as reflecting boundaries in configuration space, uniquely dictating the amplitude of the leading $t^{3/2}$ density relaxation. Mechanistically, diffusion samples a contact boundary layer of thickness $O(\sqrt{t})$, which combines with the local density response to produce the fractional signal. We derive an explicit surface formula expressing this amplitude in terms of equilibrium contact probability, normal mobility, and density sensitivity. For monodisperse hard spheres, this yields an exact, fit-free prediction determined entirely by static structure $S(k)$, radial contact value $g(\sigma^+)$, and short-time diffusion $D_0$. Extending the construction, we determine the corresponding normalization for soft interfaces and prove via a Gram--Schur projection hierarchy that regular collective variables leave the leading contact amplitude strictly invariant. The resulting formulation connects microscopic collision kinetics directly to caging and glass-like structural relaxation, providing an exact microscopic boundary condition for scattering experiments, molecular simulations, and memory-kernel reconstructions.