Coarse-grained single-particle dynamics in two-dimensional solids and liquids.

We consider the dynamics of a single tagged particle in a two-dimensional system governed by Lennard-Jones interactions. Previous work based on the Mori-Zwanzig projection operator formalism has shown that the single-particles dynamics can be described via a generalized Langevin equation (GLE) which is exact within the harmonic approximation, that is, for a low-temperature solid [J. M. Deutch and R. Silbey, Phys. Rev. A 3, 2049 (1971)]. In the present work we explore to what an extent the GLE reproduces the effective dynamics under thermodynamic conditions where the harmonic approximation is no longer justified. To this end we compute characteristic time autocorrelation functions for the tagged particle in molecular dynamics simulations of the full system and compare these functions with those obtained from solving the GLE. At low temperatures we find excellent agreement between both data sets. Deviations emerge at higher temperatures which are, however, surprisingly small even in the high-temperature liquid phase.