科研之友
微信
新浪微博
Facebook
分享链接
摘要
In this paper we study the distribution function P(u(alpha)) of the estimators u(alpha) similar to T-1 integral(T)(0) omega(t)B(t)(2)dt, which optimize the least-squares fitting of the diffusion coefficient D-f of a single d-dimensional Brownian trajectory B. We pursue here the optimization further by considering a family of weight functions of the form omega(t) = (t(0) + t)(-alpha), where to is a time lag and a is an arbitrary real number, and seeking such values of a for which the estimators most efficiently filter out the fluctuations. We calculate P(u(alpha)) exactly for arbitrary a and arbitrary spatial dimension d, and show that only for alpha = 2 does the distribution P(u(alpha)) converge, as epsilon = t(0)/T -%26gt; 0, to the Dirac delta function centred 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 not equal 2 the distribution attains, as epsilon -%26gt; 0, a certain limiting form with a finite variance, which signifies that such estimators are not ergodic.
- 出版日期2013-4
