# Measurement of the Brownian motion diffusion coefficient: Displacement-distribution versus mean-squared-displacement trajectory technique

### Abstract

We investigate the possible difference in the diffusion coefficient *D* value when measured from the variance of the normal displacement-distribution (*D*_{var}), and from the slope of the linear mean-squared-displacement (MSD) against sampling time interval, (*D*_{MSD}). We considered the pure Brownian motion (BM) of solid 2μm spheres in water, with a viscosity of 0.89mPa-s at 25°C, observed over 60s every 0.1s. For each generated trajectory, *D*_{var} and *D*_{MSD} were calculated and compared against the theoretical *D* of 0.245 μm^{2}/s. For the *D*_{MSD} measurements, the effect of the number of fitting points, *f*, used for plotting the MSD graph and the number of displacement steps, *s*, for computing each MSD per 𝜏 were studied. We have observed that *D*_{MSD} accuracy increases with *f* (*D*_{MSD} is within 5% accuracy at *f* > 30). We have also found that *D*_{var} is more accurate (to within one percent) and precise than *D*_{MSD}, regardless of the *f* and *s* parameters. Our results indicate that using the variance of the displacement distribution may lead to more accurate and precise *D* measurements of BM trajectories.

*Proceedings of the Samahang Pisika ng Pilipinas*

**38**, SPP-2020-3D-03 (2020). URL: https://paperview.spp-online.org/proceedings/article/view/SPP-2020-3D-03.