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.
Categories: cond-mat.stat-mech