Numerical Simulation of the Perrin - like Experiments

Abstract

A simple model of random Brownian walk of a spherical mesoscopic particle in viscous liquids is proposed. The model can be both solved analytically and simulated numerically. The analytic solution gives the known Eistein-Smoluchowski diffusion law $<r^2> = Dt$ where the diffusion constant $D$ is expressed by the mass and geometry of a particle, the viscosity of a liquid and the average effective time between consecutive collisions of the tracked particle with liquid molecules. The latter allows to make a simulation of the Perrin experiment and verify in detailed study the influence of the statistics on the expected theoretical results. To avoid the problem of small statistics causing departures from the diffusion law we introduce in the second part of the paper the idea of so called Artificially Increased Statistics (AIS) and prove that within this method of experimental data analysis one can confirm the diffusion law and get a good prediction for the diffusion constant even if trajectories of just few particles immersed in a liquid are considered.