Estimating the free-space diffusion coefficient of trapped particles

We show that observing the trajectories of confined particles in a thermal equilibrium state yields an estimate on the free-space diffusion coefficient. For generic trapping potentials and interactions between particles, the estimate comes in the form of a lower bound on the true diffusion coefficient. For non-interacting particles in parabolic trapping potentials, which approximately describes many experimental situations, the estimate is asymptotically exact. This allows to determine the diffusion coefficient from an equilibrium measurement, as opposed to a direct observation of diffusion, which necessarily starts from a non-equilibrium state. We explicitly demonstrate that the estimate remains quantitatively accurate in the presence of weak interactions and non-parabolic corrections.