https://doi.org/10.1140/epjst/e2013-01729-y
Regular Article
Optimal least-squares estimators of the diffusion constant from a single Brownian trajectory
1 Instituto de Física, Universidad Nacional Autónoma de México, D.F. 04510, Mexico
2 Université de Bordeaux and CNRS, Laboratoire Ondes et Matière d'Aquitaine (LOMA), UMR 5798, 33400 Talence, France
3 Laboratory of Physical Properties, Technical University of Madrid, Av. Complutense s/n, 28040 Madrid, Spain
4 Department of Mathematics and Statistics, University of Helsinki, PO Box 68, 00014 Helsinki, Finland
5 Laboratoire de Physique Théorique de la Matière Condensée (UMR CNRS 7600), Université Pierre et Marie Curie/CNRS, 4 place Jussieu, 75252 Paris Cedex 5, France
a e-mail: carlos.mejia@helsinki.fi
Received:
30
November
2012
Revised:
6
December
2012
Published online:
31
January
2013
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.
© EDP Sciences, Springer-Verlag, 2013
