Abstract

In recent years, several experiments have highlighted a new type of diffusion anomaly, which was called Brownian yet non-Gaussian diffusion. In systems displaying this behavior, the mean squared displacement of the diffusing particles grows linearly in time, like in a normal diffusion, but the distribution of displacements is non-Gaussian. In situations when the convergence to Gaussian still takes place at longer times, the probability density of the displacements may show a persisting peak around the distribution’s mode, and the pathway of convergence to the Gaussian is unusual. One of the theoretical models showing such a behavior corresponds to a disordered system with local diffusion coefficients slowly varying in space. While the standard pathway to Gaussian, as proposed by the Central Limit Theorem, would assume that the peak, under the corresponding rescaling, smoothens and lowers in course of the time, in the model discussed, the peak, under rescaling, narrows and stays sharp. In the present work, we discuss the nature of this peak. On a coarse-grained level, the motion of the particles in the diffusivity landscape is described by continuous time random walks with correlations between waiting times and positions. The peak is due to strong spatiotemporal correlations along the trajectories of diffusing particles. Destroying these correlations while keeping the temporal structure of the process intact leads to the decay of the peak. We also note that the correlated CTRW model reproducing serial correlations between the waiting times along the trajectory fails to quantitatively reproduce the shape of the peak even for the decorrelated motion, while being quite accurate in the wings of the PDF. This shows the importance of high-order temporal correlations for the peak’s formation.

Graphical abstract

中文翻译：

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相关扩散率景观中的随机游走

摘要

近年来，一些实验强调了一种新型的扩散异常，称为布朗非高斯扩散。在显示这种行为的系统中，扩散粒子的均方位移随时间线性增长，就像在正常扩散中一样，但位移的分布是非高斯的。在收敛到高斯分布的时间仍然较长的情况下，位移的概率密度可能会在分布模式周围显示出持续的峰值，并且收敛到高斯分布的路径是不寻常的。显示这种行为的理论模型之一对应于局部扩散系数在空间中缓慢变化的无序系统。虽然中心极限定理提出的高斯标准路径假设峰值在相应的重新缩放下随着时间的推移而平滑和降低，但在讨论的模型中，峰值在重新缩放下变窄并保持尖锐。在目前的工作中，我们讨论这个峰的性质。在粗粒度层面上，扩散景观中粒子的运动是通过连续时间随机游走以及等待时间和位置之间的相关性来描述的。该峰值是由于沿扩散粒子轨迹的强烈时空相关性造成的。在保持过程的时间结构完整的同时破坏这些相关性会导致峰值的衰减。我们还注意到，再现沿轨迹的等待时间之间的序列相关性的相关 CTRW 模型即使对于去相关运动也无法定量地再现峰值的形状，而在 PDF 的翼中却相当准确。这表明高阶时间相关性对于峰形成的重要性。

图形概要