Probability Distribution of Time-Averaged MSD

by D. S. Grebenkov

Laboratory of Condensed Matter Physics,
CNRS - Ecole Polytechnique, F-91128 Palaiseau France

Email: denis.grebenkov@polytechnique.edu

Abstract:

We present a theoretical study of the time-averaged mean square 
displacement (TAMSD) that 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 free Brownian motion, an exact formula for the Laplace transform of 
the probability density of the TAMSD is derived. From this formula, one 
can deduce the first four moments of the TAMSD, the asymptotic behavior 
of the probability density and its accurate approximation by a generalized 
Gamma distribution. This approximation seems to be generic for Gaussian 
processes, as illustrated numerically for fractional Brownian motion.