Distribution of the least-squares estimators of a single Brownian trajectory diffusion coefficient
| hal.structure.identifier | Instituto de Fisica | |
| hal.structure.identifier | Centro de Ciencias de la Complejidad | |
| dc.contributor.author | BOYER, Denis | |
| hal.structure.identifier | Laboratoire Ondes et Matière d'Aquitaine [LOMA] | |
| dc.contributor.author | DEAN, David S. | |
| hal.structure.identifier | Laboratory of Physical Properties | |
| hal.structure.identifier | Department of Mathematics and Statistics [Helsinki] | |
| dc.contributor.author | MEJIA-MONASTERIO, Carlos | |
| hal.structure.identifier | Laboratoire de Physique Théorique de la Matière Condensée [LPTMC] | |
| dc.contributor.author | OSHANIN, Gleb | |
| dc.date.created | 2013-01-18 | |
| dc.date.issued | 2013 | |
| dc.identifier.issn | 1742-5468 | |
| dc.description.abstractEn | In this paper we study the distribution function $P(u_{\alpha})$ of the estimators $u_{\alpha} \sim T^{-1} \int^T_0 \, \omega(t) \, {\bf B}^2_{t} \, dt$, which optimise the least-squares fitting of the diffusion coefficient $D_f$ of a single $d$-dimensional Brownian trajectory ${\bf B}_{t}$. We pursue here the optimisation further by considering a family of weight functions of the form $\omega(t) = (t_0 + t)^{-\alpha}$, where $t_0$ is a time lag and $\alpha$ is an arbitrary real number, and seeking such values of $\alpha$ for which the estimators most efficiently filter out the fluctuations. We calculate $P(u_{\alpha})$ exactly for arbitrary $\alpha$ and arbitrary spatial dimension $d$, and show that only for $\alpha = 2$ the distribution $P(u_{\alpha})$ converges, as $\epsilon = t_0/T \to 0$, to the Dirac delta-function centered at the ensemble average value of the estimator. This allows us to conclude that only the estimators with $\alpha = 2$ possess an ergodic property, so that the ensemble averaged diffusion coefficient can be obtained with any necessary precision from a single trajectory data, but at the expense of a progressively higher experimental resolution. For any $\alpha \neq 2$ the distribution attains, as $\epsilon \to 0$, a certain limiting form with a finite variance, which signifies that such estimators are not ergodic. | |
| dc.language.iso | en | |
| dc.publisher | IOP Publishing | |
| dc.rights.uri | http://creativecommons.org/licenses/by-nc/ | |
| dc.subject.en | Brownian motion | |
| dc.subject.en | data mining (theory) | |
| dc.subject.en | single molecule | |
| dc.subject.en | diffusion | |
| dc.title.en | Distribution of the least-squares estimators of a single Brownian trajectory diffusion coefficient | |
| dc.type | Article de revue | |
| dc.identifier.doi | 10.1088/1742-5468/2013/04/P04017 | |
| dc.subject.hal | Physique [physics]/Matière Condensée [cond-mat]/Mécanique statistique [cond-mat.stat-mech] | |
| dc.identifier.arxiv | 1301.4374 | |
| bordeaux.journal | Journal of Statistical Mechanics: Theory and Experiment | |
| bordeaux.page | P04017 (1-24) | |
| bordeaux.volume | 2013 | |
| bordeaux.issue | 4 | |
| bordeaux.peerReviewed | oui | |
| hal.identifier | hal-00825399 | |
| hal.version | 1 | |
| hal.popular | non | |
| hal.audience | Internationale | |
| hal.origin.link | https://hal.archives-ouvertes.fr//hal-00825399v1 | |
| bordeaux.COinS | ctx_ver=Z39.88-2004&rft_val_fmt=info:ofi/fmt:kev:mtx:journal&rft.jtitle=Journal%20of%20Statistical%20Mechanics:%20Theory%20and%20Experiment&rft.date=2013&rft.volume=2013&rft.issue=4&rft.spage=P04017%20(1-24)&rft.epage=P04017%20(1-24)&rft.eissn=1742-5468&rft.issn=1742-5468&rft.au=BOYER,%20Denis&DEAN,%20David%20S.&MEJIA-MONASTERIO,%20Carlos&OSHANIN,%20Gleb&rft.genre=article |
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