Observation of Super-ballistic Brownian Motion in Liquid

TL;DR

The paper investigates short-time Brownian dynamics of a particle in an incompressible fluid, revealing that non-Markovian hydrodynamic memory and colored thermal noise lead to a super-ballistic MSD scaling when the initial velocity is conditioned. It combines theory and experiment by solving the hydrodynamic generalized Langevin equation in the Laplace domain and validating the conditioned MSD, including the mean trajectory $\langle x(t) \rangle = M v(0) B(t)$, with high-precision optical trapping measurements. A key finding is that the conditioned thermal force and history terms cancel in the mean dynamics due to the fluctuation-dissipation theorem, while fluctuations follow the equilibrium thermal force statistics, yielding a distinct short-time $t^{5/2}$ scaling. This work advances the understanding of nonequilibrium fluid-particle dynamics and demonstrates how equilibrium data can reveal non-Markovian effects when conditioned on specific initial states.

Abstract

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 non-zero first moment for the thermal force when conditioned on non-zero initial velocities. These results are a step towards the unraveling of nonequilibrium dynamics of fluids.

Figures (7)

• Figure 1: Short time Brownian motion detection scheme. (A) Diagram of optical trapping and detection setup. Counter-propagating beams are focused through a microfluidic chamber forming an optical tweezer and trapping the particle. The out-coming infrared beam is re-collimated and split spatially with a D-shaped cut mirror. Each half of the beam is sent to a port on a balanced photodetector which monitors the particle's position. (B) Close up of the microfluidic chamber with z-shaped channel and focused beam passing through. (C) Time trace data with time on the vertical and position on the horizontal axis. Two points are chosen that fall within the conditioning tolerance around 0 velocity and +1 standard deviation of the velocity, respectively. Overlaid on these initial points are sub-traces of other parts of the time trace that fall within the same tolerance along orange lines indicating 1 standard deviation from the mean trajectory.
• Figure 2: MSDs for a Brownian particle for specific initial velocities. (A) MSD built from trajectories with the particle beginning close to rest. The orange circles are experimental data points. The solid black line is the hydrodynamic theory. The blue dashed line corresponds to equation \ref{['two_term_expansion']}. The green dotted line is the $t^{5/2}$ scaling term, which the experimental data collapses onto at short times. The dot-dashed line corresponds to the MSD found with the same velocity conditioning and no particle in the trap. (B) MSD curves for three different initial velocities. The blue squares, orange diamonds, and green circles refer to initial velocities of $0.5$, $1$, and $2$ times the velocity standard deviation respectively. The black lines are the theory. Error bars for both graphs are calculated using the method outlined in the Supplementary Materials.
• Figure 3: Analysis of mean and fluctuations around the mean trajectories for the Brownian particle. (A) Fluctuations around the mean trajectory for 3 different initial velocities. The blue squares, orange diamonds, and green circles correspond to initial velocities with squared values $0.5 \langle v^2 \rangle$, $\langle v^2 \rangle$, $2\langle v^2 \rangle$ respectively. The black line is the full theory and the red dashed line is the short time expansion from equation \ref{['two_term_expansion']}. (B) Trajectories for the Brownian particle with initial velocity squared of $\langle v^2 \rangle$. The black line is the mean trajectory, and the gray dashed lines are individual trajectories. (C) Mean trajectory for different initial velocities. The blue squares, orange diamonds, and green circles correspond to initial velocities with squared values $0.5 \langle v^2 \rangle$, $\langle v^2 \rangle$, $2\langle v^2 \rangle$ respectively. Solid lines are the theory from equation \ref{['mean_trajectory_eq']}. Error bars for panels A and C are calculated using the method outlined in the Supplementary Materials. (D) Distribution of fluctuations around the mean trajectory for an initial velocity squared of $\langle v^2 \rangle$. The orange squares are for a time of 75 $\mu$s and the blue circles for a time of 7.5 $\mu$s. Black lines are Gaussian curves with standard deviation set by the zero velocity MSD.
• Figure 4: Histogram for fluctuations around the mean trajectory. Built for initial speeds between minus one and one velocity standard deviation. The variations are taken 7.5 $\mu$s after the starting trajectory. The variation at these short times is caused by the thermal force, and therefore is a good test for its finer stochastic properties. We find a standard deviation 97% the value predicted by the theory, which could result from the density uncertainty of the sphere or finite differencing effects on the fractal-like time series.
• Figure 5: Comparison of white noise and hydrodynamic Langevin equations. (A) MSDs for hydrodynamic and white noise Langevin equations with different initial conditions. The hydrodynamic curves are blue, while the white noise curves are orange. The solid lines are MSDs taken with thermal initial conditions, while the dashed lines assume an initial velocity of 0. (B) Crossing times as a function of initial velocity. The solid black line is the crossing of the two analytic curves. The orange circles are the experimental times when a crossing occurs. The vertical error bars are set by the time step between data points. The horizontal error bars are set by the width of the velocity bins used in the analysis.