Title: On the distributions of time averaged weighted Brownian
trajectories
Abstract:
Modern developments in microscopy and image processing are
revolutionizing areas of physics, chemistry, and biology as nanoscale
objects can be tracked with unprecedented accuracy. The goal of
single-particle tracking is to determine the interaction between the
particle and its environment. The price paid for having a direct
visualization of a single particle is a consequent lack of
statistics. Here we address the question of extracting diffusion
constants from single trajectories of $d$-dimensional Brownian motion
using the time averages of a squared trajectory, tempered by some
power-law functions of time. We show that such estimators possess an
ergodic property, i.e., the distribution converges to a delta-function
centered at the ensemble average value of the diffusion coefficient as
the observation time tends to infinity, only for certain weighting
functions. We show that for a certain choice of parameters such
functionals efficiently filter out the fluctuations and provide the
true ensemble average diffusion coefficient with any necessary
precision, but at expense of progressively higher experimental
resolutions. These results are generalized for fractional Brownian
motion, for which we also specify the optimal weighting functions
producing an ergodic behavior, and discuss as well, the influence of
disorder on the distributions of the time-averaged, weighted
least-squares estimators.