Optimal least-squares estimators of the diffusion constant from a single Brownian trajectory

Denis Boyer (1) , David S. Dean (2) , Carlos Mejía-Monasterio (3, 4) , Gleb Oshanin (5)

1 Instituto de Fisica
2 LOMA - Laboratoire Ondes et Matière d'Aquitaine
3 Laboratory of Physical Properties
4 Department of Mathematics and Statistics [Helsinki]
5 LPTMC - Laboratoire de Physique Théorique de la Matière Condensée

Modern developments in microscopy and image processing are revolutionising areas of physics, chemistry, and biology as nanoscale objects can be tracked with unprecedented accuracy. However, the price paid for having a direct visualisation of a single particle trajectory with high temporal and spatial resolution is a consequent lack of statistics. This naturally calls for reliable analytical tools which will allow one to extract the properties specific to a statistical ensemble from just a single trajectory. In this article we briefly survey different analytical methods currently used to determine the ensemble average diffusion coefficient from single particle data and then focus specifically on weighted least-squares estimators, seeking the weight functions for which such estimators are ergodic. Finally, we address the question of the effects of disorder on such estimators.

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

Denis Boyer, David S. Dean, Carlos Mejía-Monasterio, Gleb Oshanin. Optimal least-squares estimators of the diffusion constant from a single Brownian trajectory. The European Physical Journal. Special Topics, 2013, 216 (1), pp.57-71. ⟨10.1140/epjst/e2013-01729-y⟩. ⟨hal-00817656⟩