Langevin equation with fluctuating diffusivity: A two-state model
Langevin equation with fluctuating diffusivity: A two-state model
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DOI：
10.1103/physreve.94.012109
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发表时间：
2016-07-08
影响因子：
2.4
通讯作者：
Yamamoto, Eiji
中科院分区：
文献类型：
--
作者：
Miyaguchi, Tomoshige;Akimoto, Takuma;Yamamoto, Eiji
最近，在许多单粒子跟踪实验中，已经报道了反常的亚扩散、老化和扩散系数的分散，尽管这些行为的起源仍然是难以捉摸的。在这里，作为一个模型来描述这样的现象，我们研究了朗之万方程的扩散率波动之间的快速和缓慢的状态。也就是说，扩散率遵循二分随机过程。我们假设这两个态的逗留时间分布服从幂律。结果表明，对于非平衡系综，系综平均均方位移（MSD）表现出瞬态亚扩散。相反，时间平均的MSD显示正常扩散，但有效扩散系数瞬时显示老化行为。传播算子在短时间内是非高斯的，在长时间内收敛到高斯分布;对于某些参数值，这种收敛到高斯分布的速度非常慢。对于平衡系综，系综平均和时间平均的MSD只显示正常的扩散，因此我们不能检测到任何痕迹的波动扩散与这些MSD。因此，作为一种替代方法来表征波动的扩散率，相对标准偏差（RSD）的时间平均的MSD被利用，它示出的RSD表现出缓慢的松弛作为签名的波动的扩散率的长时间的相关性。此外，它示出的RSD有关的传播子的非高斯参数。为了得到这些理论结果，我们发展了一个两态更新理论作为分析工具。
Recently, anomalous subdiffusion, aging, and scatter of the diffusion coefficient have been reported in many single-particle-tracking experiments, though the origins of these behaviors are still elusive. Here, as a model to describe such phenomena, we investigate a Langevin equation with diffusivity fluctuating between a fast and a slow state. Namely, the diffusivity follows a dichotomous stochastic process. We assume that the sojourn time distributions of these two states are given by power laws. It is shown that, for a nonequilibrium ensemble, the ensemble-averaged mean-square displacement (MSD) shows transient subdiffusion. In contrast, the time-averaged MSD shows normal diffusion, but an effective diffusion coefficient transiently shows aging behavior. The propagator is non-Gaussian for short time and converges to a Gaussian distribution in a long-time limit; this convergence to Gaussian is extremely slow for some parameter values. For equilibrium ensembles, both ensemble-averaged and time-averaged MSDs show only normal diffusion and thus we cannot detect any traces of the fluctuating diffusivity with these MSDs. Therefore, as an alternative approach to characterizing the fluctuating diffusivity, the relative standard deviation (RSD) of the time-averaged MSD is utilized and it is shown that the RSD exhibits slow relaxation as a signature of the long-time correlation in the fluctuating diffusivity. Furthermore, it is shown that the RSD is related to a non-Gaussian parameter of the propagator. To obtain these theoretical results, we develop a two-state renewal theory as an analytical tool.