publication . Article . Preprint . 2012

Time-averaged MSD of Brownian motion

Andreanov, Alexei; Grebenkov, Denis;
Open Access
  • Published: 09 May 2012 Journal: Journal of Statistical Mechanics: Theory and Experiment, volume 2,012, page P07001 (eissn: 1742-5468, Copyright policy)
  • Publisher: IOP Publishing
Abstract
We study the statistical properties of the time-averaged mean-square displacements (TAMSD). This is a standard non-local quadratic functional for inferring the diffusion coefficient from an individual random trajectory of a diffusing tracer in single-particle tracking experiments. For Brownian motion, we derive an exact formula for the Laplace transform of the probability density of the TAMSD by mapping the original problem onto chains of coupled harmonic oscillators. From this formula, we deduce the first four cumulant moments of the TAMSD, the asymptotic behavior of the probability density and its accurate approximation by a generalized Gamma distribution.
Subjects
free text keywords: Statistics, Probability and Uncertainty, Statistics and Probability, Statistical and Nonlinear Physics, Mathematical analysis, Reflected Brownian motion, Diffusion process, Brownian motion, Mathematics, Anomalous diffusion, Variance gamma process, Laplace transform, Heavy traffic approximation, Laplace distribution, Condensed Matter - Statistical Mechanics

ϕt(s) = dy1...dymdz0...dzm

R2m+1

(mY−2 Z Zk(t−δ)=yk+1 [1] Weiss G H, 1994 Aspects and Applications of the Random Walk (North-Holland, Amsterdam) [2] Ben-Avraham D and Havlin S, 2000 Diffusion and reaction in disordered systems (Cambridge

University Press) [3] Saxton M J and Jacobson K, 1997 Annu. Rev. Biophys. Biomol. Struct. 26 373 [4] Majumdar S N, 2005 Curr. Sci. 89 2076 [5] Grebenkov D S, 2007 Rev. Mod. Phys. 79 1077 [6] Bouchaud J-P and Potters M, 2000 Theory of Finantial Risks: From Statistical Physics to Risk

Management (Cambridge University Press) [7] Qian H, Sheetz M P, and Elson E L, 1991 Biophys. J. 60 910 [8] Saxton M J, 1993 Biophys. J. 64 1766 [9] Saxton M J, 1997 Biophys. J. 72 1744 [10] Goulian M and Simon S M, 2000 Biophys. J. 79 2188 [11] Toli´c-Norrelykke I M, Munteanu E-L, Thon G, Oddershede L, and Berg-Sorensen K, 2004 Phys.

Rev. Lett. 93 078102 [12] Golding I and Cox E C, 2006 Phys. Rev. Lett. 96 098102 [13] Arcizet D, Meier B, Sackmann E, Ra¨dler J O, and Heinrich D, 2008 Phys. Rev. Lett. 101 248103 [14] Wilhelm C, 2008 Phys. Rev. Lett. 101 028101 [15] Wirtz D, 2009 Ann. Rev. Biophys. 38 301 [16] Metzler R, Tejedor V, Jeon J-H, He Y, Deng W H, Burov S, and Barkai E, 2009 Acta Phys. Pol.

B 40 1315 [17] Feynman R P and Hibbs A R, 1965 Quantum Mechanics and Path Integrals (New York, McGraw-

Hill) [18] Freidlin M, 1985 Functional Integration and Partial Differential Equations, Annals of Mathematics [OpenAIRE]

Studies (Princeton University, Princeton, New Jersey) [19] Bray A J and Moore M A, 1980 J. Phys. C: Solid State Phys. 13 L655 [20] Kirkpatrick T R and Thirumalai D, 1987 Phys. Rev. B 36 5388 [21] Ruben H, 1962 Ann. Math. Stat. 33 542 [22] Ruben H, 1963 Ann. Math. Stat. 34 1582 [23] Robbins H, 1948 Ann. Math. Stat. 19 266 [24] Robbins H and Pitman E J G, 1949 Ann. Math. Stat. 20 552 [25] Pachares J, 1955 Ann. Math. Stat. 26 128 [26] Shah B K and Khatri C G, 1961 Ann. Math. Stat. 32 883 [27] Gurland J, 1953 Ann. Math. Stat. 24 416 [28] Gurland J, 1955 Ann. Math. Stat. 26 122 [29] Gurland J, 1956 Ind. J. Stat. 17 37 [30] Kotz S, Johnson N L and Boyd D W, 1967 Ann. Math. Stat. 38 823 [31] Kotz S, Johnson N L and Boyd D W, 1967 Ann. Math. Stat. 38 838. [32] Duits M H G, Li Y, Vanapalli S A, and Mugele F, 2009 Phys. Rev. E 79 051910 [33] Boyer D and Dean D S, 2011 J. Phys. A: Math. Theor. 44 335003 [34] Boyer D, Dean D S, Mejia-Monasterio C, and Oshanin G, 2012 Phys. Rev. E 85 031136 [35] Voisinne G, Alexandrou A, and Masson J-B, 2010 Biophys. J. 98 596 [36] Grebenkov D S, 2011 Phys. Rev. E 83 061117 [37] Grebenkov D S, 2011 Phys. Rev. E 84 031124 [38] Nakagawa K, 2007 IEEE Trans. Inf. Theory 53 3239 [39] Hoel P G, 1962 Introduction to Mathematical Statistics (John Wiley and Sons, New York)

Powered by OpenAIRE Open Research Graph
Any information missing or wrong?Report an Issue
publication . Article . Preprint . 2012

Time-averaged MSD of Brownian motion

Andreanov, Alexei; Grebenkov, Denis;