Macroscopic Brownian Motion on a Chaotic Fluid Interface

TL;DR

This work presents a macroscopic tabletop analogue of Brownian motion by placing a millimetric disk on a chaotically forced fluid interface generated by Faraday waves. The authors model the dynamics with an underdamped Langevin equation $m \dot{\mathbf{v}} + γ \mathbf{v} = \mathbf{F}(t)$, reformulated as $τ_c \dot{\mathbf{v}} + \mathbf{v} = \sqrt{2D}\,\boldsymbol{η}(t)$, and derive analytic expressions for the velocity autocorrelation function $VACF(Δt)$ and the mean-squared displacement $MSD(Δt)$. From 12 video trials, they extract $τ_c = 0.26 ± 0.034$ s and $D = 1.9 ± 0.24$ mm$^2$/s, with observed VACF and MSD agreeing with theory. The work provides an accessible, open-resource platform for teaching and exploring the crossover between ballistic and diffusive motion and connects tactile fluid dynamics with standard Brownian-motion theory, with extensions to confinement, tracer behavior, and active baths.

Abstract

Brownian motion is the erratic motion of an object due to collisions with the fluid in which it is immersed. In this work, we detail a tabletop laboratory demonstration of underdamped Brownian motion wherein a macroscopic particle resting on a driven fluid interface exhibits ballistic motion at short times and diffusive motion at long times. We observe the trajectory of a millimetric disk driven by a field of chaotic Faraday waves excited by a shaker. The crossover from ballistic to diffusive motion occurs at time and length scales experimentally accessible through particle tracking of a video recorded with a standard phone camera. Along with representative data, we provide a complete assembly guide, and operating procedure for students so that the experiment can be readily applied in the classroom. The tabletop setup can also be adapted for other student projects and active research topics relating to particle motion on a vibrating fluid interface.

Figures (4)

• Figure 1: (a) Photograph of a floating disk of radius $R=6$ mm on an air-water interface of Faraday waves forced at $f=180$ Hz, visualized with a reflective color pattern.harris2017visualization (b) Schematic of the experimental setup. A 3D-printed bath filled with water is hot glued to a speaker. As the amplitude of speaker oscillations is increased by increasing the volume, waves form on the surface that mimic a random 2D forcing, moving the disk around the bath. (c) Photograph of the experiment setup.
• Figure 2: (a) The four qualitative regimes of Faraday wave behavior demonstrated in a small square bath with sides of 4 cm at $f=60$ Hz: (1) still, (2) standing, (3) erratic, and (4) drop emission. (b) The forcing lengthscale as a function of frequency for water in a dish of depth $H= 0.5$ cm. We choose disks that are larger than the forcing lengthscale, i.e. $R > L_F \equiv \lambda_F /2$. The experimental conditions presented herein are indicated by the black dot.
• Figure 3: (a) The PDF of the $x$ and $y$ components of the velocity over 12 independent trials, with the best-fit Gaussian (black dashed line). The speed of the particle $|\mathbf{v}| = \sqrt{v_x^2+v_y^2}$ follows a Rayleigh distribution (black dashed line). The mean is marked with the gray dashed line.
• Figure 4: (a) The velocity auto-correlation function (VACF) as a function of time. The fitted value of $\tau_c$ (gray dashed line) captures the exponential decay of the VACF. (b) The mean-squared displacement (MSD) as a function of the time. In both cases, independent trials are shown in solid gray and the mean across the 12 trials is shown in blue. The best-fit curve (black dashed line) for both the VACF (Eq. \ref{['eq: VACF']}) and MSD (Eq. \ref{['eq: MSD']}) using the fitted value for the crossover time $\tau_c$ and resultant diffusion constant $D$ is overlaid. In (b), the limiting cases for ballistic (green) and diffusive (red) motion are shown from Eq. \ref{['eq:approxs']}.