Grebenkov Denis S
Laboratoire de Physique de la Matière Condensée (UMR 7643), CNRS-Ecole Polytechnique, Palaiseau, France.
Phys Rev E Stat Nonlin Soft Matter Phys. 2011 Jun;83(6 Pt 1):061117. doi: 10.1103/PhysRevE.83.061117. Epub 2011 Jun 14.
The characterization of a stochastic process from its single random realization is a challenging problem for most single-particle tracking techniques which survey an individual trajectory of a tracer in a complex or viscoelastic medium. We consider two quadratic functionals of the trajectory: the time-averaged mean-square displacement (MSD) and the time-averaged squared root mean-square displacement (SRMS). For a large class of stochastic processes governed by the generalized Langevin equation with arbitrary frictional memory kernel and harmonic potential, the exact formulas for the mean and covariance of these functionals are derived. The formula for the mean value can be directly used for fitting experimental data, e.g., in optical tweezers microrheology. The formula for the variance (and covariance) allows one to estimate the intrinsic fluctuations of measured (or simulated) time-averaged MSD or SRMS for choosing the experimental setup appropriately. We show that the time-averaged SRMS has smaller fluctuations than the time-averaged MSD, in spite of much broader applications of the latter one. The theoretical results are successfully confirmed by Monte Carlo simulations of the Langevin dynamics. We conclude that the use of the time-averaged SRMS would result in a more accurate statistical analysis of individual trajectories and more reliable interpretation of experimental data.
对于大多数在复杂或粘弹性介质中追踪示踪剂个体轨迹的单粒子追踪技术而言，从单个随机实现中表征随机过程是一个具有挑战性的问题。我们考虑轨迹的两个二次泛函：时间平均均方位移（MSD）和时间平均均方根位移（SRMS）。对于由具有任意摩擦记忆核和谐波势的广义朗之万方程所支配的一大类随机过程，我们推导了这些泛函的均值和协方差的精确公式。均值公式可直接用于拟合实验数据，例如在光镊微流变学中。方差（和协方差）公式使人们能够估计测量（或模拟）的时间平均MSD或SRMS的固有波动，以便适当地选择实验装置。我们表明，尽管时间平均MSD有更广泛的应用，但时间平均SRMS的波动比时间平均MSD小。通过朗之万动力学的蒙特卡罗模拟成功证实了理论结果。我们得出结论，使用时间平均SRMS将导致对个体轨迹进行更准确的统计分析，并对实验数据进行更可靠的解释。
