Optimal estimates of the diffusion coefficient of a single Brownian trajectory

Denis Boyer (1) , David S. Dean (2) , Carlos Mejía-Monasterio (3) , Gleb Oshanin (4)

1 Instituto de Fisica
2 LOMA - Laboratoire Ondes et Matière d'Aquitaine
3 Laboratory of Physical Properties
4 LPTMC - Laboratoire de Physique Théorique de la Matière Condensée

Modern developments in microscopy and image processing are revolutionizing areas of physics, chemistry and biology as nanoscale objects can be tracked with unprecedented accuracy. The goal of single particle tracking is to determine the interaction between the particle and its environment. The price paid for having a direct visualization of a single particle is a consequent lack of statistics. Here we address the optimal way of extracting diffusion constants from single trajectories for pure Brownian motion. It is shown that the maximum likelihood estimator is much more efficient than the commonly used least squares estimate. Furthermore we investigate the effect of disorder on the distribution of estimated diffusion constants and show that it increases the probability of observing estimates much smaller than the true (average) value.

Mots clés: Brownian motion; Path-integral methods

Domaines: Mécanique statistique [cond-mat.stat-mech]

Denis Boyer, David S. Dean, Carlos Mejía-Monasterio, Gleb Oshanin. Optimal estimates of the diffusion coefficient of a single Brownian trajectory. Physical Review E : Statistical, Nonlinear, and Soft Matter Physics, 2012, 85 (3), pp.031136. ⟨10.1103/PhysRevE.85.031136⟩. ⟨hal-00696477⟩