Conditioning Hydrodynamic Brownian Motion

Due to technical gains in the last decade, experimental access to the instantaneous velocity of a trapped microsphere has confirmed predictions for equilibrium correlation functions of a Brownian particle far below the particle's momentum relaxation time. In this study, we utilize these experimental methods to go beyond the standard equilibrium statistical functions. We trap dielectric spheres in a single beam optical tweezer in a liquid and track their position well below the corresponding momentum relaxation time via split beam detection of the trapping beam. In contrast to past studies, we condition our experimental trajectories to average only over specific initial velocities. In the case the initial velocity of the Brownian particle is close to zero, we observe a super-ballistic power law scaling of t^(5/2) that is unique to incompressible Newtonian fluids. This analysis is then extended to the case of finite, well defined initial velocities and compared to an extended theory for the mean squared displacement. Our results emphasize the interplay between the non-Markovian thermal force and the history of the Brownian particle. The result is experimental verification for an initial value theory of Brownian motion in an incompressible Newtonian fluid.