Bao Jing-Dong, Wang Xiang-Rong, Liu Wu-Ming
Department of Physics, Beijing Normal University, Beijing 100875, China.
Department of Physics, The Hong Kong University of Science and Technology, Kowloon, Hong Kong, China.
Phys Rev E. 2021 Mar;103(3-1):032136. doi: 10.1103/PhysRevE.103.032136.
We investigate ergodic time scales in single-particle tracking by introducing a covariance measure Ω(Δ;t) for the time-averaged relative square displacement recorded in lag-time Δ at elapsed time t. The present model is established in the generalized Langevin equation with a power-law memory function. The ratio Ω(Δ;Δ)/Ω(Δ;t) is shown to obey a universal scaling law for long but finite times and is used to extract the effective ergodic time. We derive a finite-time-averaged Green-Kubo relation and find that, to control the deviations in measurement results from ensemble averages, the ratio Δ/t must be neither too small nor close to unity. Our paper connects the experimental self-averaging property of a tracer with the theoretic velocity autocorrelation function and sheds light on the transition to ergodicity.
我们通过引入协方差度量Ω(Δ;t)来研究单粒子追踪中的遍历时间尺度，该协方差度量用于衡量在经过时间t时滞后时间Δ内记录的时间平均相对平方位移。本模型是在具有幂律记忆函数的广义朗之万方程中建立的。结果表明，对于长但有限的时间，Ω(Δ;Δ)/Ω(Δ;t)服从通用标度律，并用于提取有效遍历时间。我们推导了有限时间平均的格林 - 库博关系，并发现，为了控制测量结果与系综平均值的偏差，Δ/t的比值既不能太小也不能接近1。我们的论文将示踪剂的实验自平均性质与理论速度自相关函数联系起来，并阐明了向遍历性的转变。
