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 t 2 . 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 t 5 / 2 . This result is due to the colored noise of incompressible fluids, resulting in a nonzero first moment for the thermal force when conditioned on nonzero initial velocities. These results are a step toward the unraveling of nonequilibrium dynamics of fluids.
Boynewicz et al. (2026) studied this question.