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The European Physical Journal Special Topics

, Volume 216, Issue 1, pp 57–71 | Cite as

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

  • Denis Boyer
  • David S. Dean
  • Carlos Mejía-MonasterioEmail author
  • Gleb Oshanin
Regular Article

Abstract

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.

Keywords

Brownian Motion European Physical Journal Special Topic Maximum Likelihood Estimator Brownian Particle Standard Brownian Motion 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

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Copyright information

© EDP Sciences and Springer 2013

Authors and Affiliations

  • Denis Boyer
    • 1
  • David S. Dean
    • 2
  • Carlos Mejía-Monasterio
    • 3
    • 4
    Email author
  • Gleb Oshanin
    • 5
  1. 1.Instituto de Física, Universidad Nacional Autónoma de MéxicoMexico CityMexico
  2. 2.Université de Bordeaux and CNRS, Laboratoire Ondes et Matière d’Aquitaine (LOMA)TalenceFrance
  3. 3.Laboratory of Physical Properties, Technical University of MadridMadridSpain
  4. 4.Department of Mathematics and StatisticsUniversity of HelsinkiHelsinkiFinland
  5. 5.Laboratoire de Physique Théorique de la Matière Condensée (UMR CNRS 7600)Université Pierre et Marie Curie/CNRSParis Cedex 5France

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