Reciprocal Space Study of Brownian Yet Non-Gaussian Diffusion of Small Tracers in a Hard-Sphere Glass

The simultaneous presence of normal (Brownian) diffusion and non-Gaussian statistics of particle displacements has been identified as a recurring motif for a broad spectrum of physical and biological systems. While not yet fully understood, it is generally accepted that a key ingredient for observing this Brownian yet non-Gaussian (BNG) diffusion is that the environment hosting the particles appears stationary and homogenous on the small length and time scales, while displaying significant fluctuations on larger distances and/or longer time scales. To date, most of the experimental studies on systems displaying BNG diffusion have been performed in direct space, usually via a combination of optical microscopy and particle tracking to quantify the particle's self-diffusion. Here, we demonstrate that a reciprocal space analysis of the density fluctuations caused by the particle motion as a function of the wave vector enables the investigation of BNG diffusion in situations where single-particle tracking is impossible. To accomplish this aim, we use confocal differential dynamic microscopy (ConDDM) to study the BNG dynamics of diluted sub-resolution tracers diffusing in a glassy matrix of larger hard spheres. We first elucidate the nontrivial connection between the tracer self-diffusion and collective relaxation of the resulting density fluctuations. We find that the experimentally determined intermediate scattering functions are in excellent agreement with the recent predictions of a "diffusing diffusivity" model of BNG diffusion, whose analytical predictions are available only in reciprocal space. Our results show that studying BNG diffusion in reciprocal space can be an invaluable strategy to access the fast, anomalous dynamics occurring at very small scales in crowded environments.