Statistical analysis of random trajectories of vibrated disks: towards a macroscopic realization of Brownian motion

TL;DR

This study demonstrates a macroscopic physical system that mimics Brownian motion through vibrated disks, providing a platform for testing statistical analysis methods used in microscopic particle tracking.

Contribution

It introduces a macroscopic experimental setup for Brownian motion and analyzes its statistical properties, bridging the gap between microscopic theory and macroscopic realization.

Findings

Distribution of increments is nearly Gaussian with minor deviations.

Velocity auto-correlation function shows short-term positive and negative correlations.

Empirical and theoretical mean square displacement distributions are compared.

Abstract

We propose a macroscopic realization of planar Brownian motion by vertically vibrated disks. We perform a systematic statistical analysis of many random trajectories of individual disks. The distribution of increments is shown to be almost Gaussian, with slight deviations at large increments caused by inter-disk collisions. The velocity auto-correlation function takes both positive and negative values at short lag times but rapidly vanishes. We compare the empirical and theoretical distributions of time averaged mean square displacements and discuss distinctions between its mean and mode. These well-controlled experimental data can serve for validating statistical tools developed for the analysis of single-particle trajectories in microbiology.