Observation of super-ballistic Brownian dynamics in liquid

Brownian motion is a foundational physical process characterized by a
mean-squared displacement that scales linearly in time in thermal
equilibrium, known as diffusion. At short times, the mean squared
displacement becomes ballistic, scaling as t2. This effect was predicted
by Einstein in 1907 and recently observed experimentally. We report that
this picture is only true on average; by conditioning specific initial
velocities, we predict theoretically and confirm by experiment that the
mean squared displacement becomes super-ballistic, with a power scaling
law of t5/2. This result is due to the colored noise of incompressible
fluids, resulting in a non-zero first moment for the thermal force when
conditioned on non-zero initial velocities. These results are a first step
towards unraveling the non-equilibrium dynamics of fluids.