Time averages and their statistical variation for the Ornstein-Uhlenbeck process: Role of initial particle distributions and relaxation to stationarity
Time averages and their statistical variation for the Ornstein-Uhlenbeck process: Role of initial particle distributions and relaxation to stationarity
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DOI：
10.1103/physreve.98.022134
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发表时间：
2018-08-28
影响因子：
2.4
通讯作者：
Metzler, Ralf
中科院分区：
文献类型：
--
作者：
Cherstvy, Andrey G.;Thapa, Samudrajit;Metzler, Ralf
谐波约束下扩散的遍历性如何？对于在不同初始条件下进行受限运动的热鼓动颗粒，系综位移和时间平均位移的差异有多大？本文研究了一般的Ornstein-Uhlenbeck （OU）过程的这些问题，并推导了二阶矩和四阶矩的解析表达式。这些量词与越来越多的使用光学陷阱的单粒子跟踪实验特别相关。对于固定的起始位置，我们讨论集合平均的定义。我们还量化了平衡和非平衡初始粒子分布对集合和时间平均位移的弛豫特性和新出现的非等效性的影响（即使在长轨迹的极限下）。我们推导了遍历性破断参数的解析表达式，该参数量化了在整个滞后时间范围内，平衡和非平衡初始粒子位置的单个时间平均轨迹的振幅散射。我们的分析预测与抛物势中朗之万方程的计算机模拟结果非常吻合。我们还检验了爱因斯坦关系对u粒子的系综力矩和时间平均力矩的有效性。讨论了一些物理系统，其中我们揭示的松弛和非遍历特征可以被观察到。
How ergodic is diffusion under harmonic confinements? How strongly do ensemble- and time-averaged displacements differ for a thermally-agitated particle performing confined motion for different initial conditions? We here study these questions for the generic Ornstein-Uhlenbeck (OU) process and derive the analytical expressions for the second and fourth moment. These quantifiers are particularly relevant for the increasing number of single-particle tracking experiments using optical traps. For a fixed starting position, we discuss the definitions underlying the ensemble averages. We also quantify effects of equilibrium and nonequilibrium initial particle distributions onto the relaxation properties and emerging nonequivalence of the ensemble- and time-averaged displacements (even in the limit of long trajectories). We derive analytical expressions for the ergodicity breaking parameter quantifying the amplitude scatter of individual time-averaged trajectories, both for equilibrium and outof-equilibrium initial particle positions, in the entire range of lag times. Our analytical predictions are in excellent agreement with results of computer simulations of the Langevin equation in a parabolic potential. We also examine the validity of the Einstein relation for the ensemble- and time-averaged moments of the OU-particle. Some physical systems, in which the relaxation and nonergodic features we unveiled may be observable, are discussed.