Generalized Langevin Equation for Tracer Diffusion in Atomic Liquids

We derive the time-evolution equation that describes the Brownian motion of labeled individual tracer particles in a simple model atomic liquid (i.e., a system of $N$ particles whose motion is governed by Newton's second law, and interacting through spherically symmetric pairwise potentials). We base our derivation on the generalized Langevin equation formalism, and find that the resulting time evolution equation is formally identical to the generalized Langevin equation that describes the Brownian motion of individual tracer particles in a colloidal suspension in the absence of hydrodynamic interactions. This formal dynamic equivalence implies the long-time indistinguishability of some dynamic properties of both systems, such as their mean squared displacement, upon a well-defined time scaling. This prediction is tested here by comparing the results of molecular and Brownian dynamics simulations performed on the hard sphere system.