Hidalgo-Soria M, Barkai E, Burov S
Department of Physics, Institute of Nanotechnology and Advanced Materials, Bar-Ilan University, Ramat-Gan 5290002, Israel.
Department of Physics, Bar-Ilan University, Ramat-Gan 5290002, Israel.
Entropy (Basel). 2021 Feb 17;23(2):231. doi: 10.3390/e23020231.
We study a two state "jumping diffusivity" model for a Brownian process alternating between two different diffusion constants, D+>D-, with random waiting times in both states whose distribution is rather general. In the limit of long measurement times, Gaussian behavior with an effective diffusion coefficient is recovered. We show that, for equilibrium initial conditions and when the limit of the diffusion coefficient D-⟶0 is taken, the short time behavior leads to a cusp, namely a non-analytical behavior, in the distribution of the displacements P(x,t) for x⟶0. Visually this cusp, or tent-like shape, resembles similar behavior found in many experiments of diffusing particles in disordered environments, such as glassy systems and intracellular media. This general result depends only on the existence of finite mean values of the waiting times at the different states of the model. Gaussian statistics in the long time limit is achieved due to ergodicity and convergence of the distribution of the temporal occupation fraction in state D+ to a δ-function. The short time behavior of the same quantity converges to a uniform distribution, which leads to the non-analyticity in P(x,t). We demonstrate how super-statistical framework is a zeroth order short time expansion of P(x,t), in the number of transitions, that does not yield the cusp like shape. The latter, considered as the key feature of experiments in the field, is found with the first correction in perturbation theory.
我们研究了一个用于布朗过程的两态“跳跃扩散率”模型，该布朗过程在两个不同的扩散常数(D_+\gt D_-)之间交替，且在两种状态下都有随机等待时间，其分布相当一般。在长时间测量的极限情况下，可恢复具有有效扩散系数的高斯行为。我们表明，对于平衡初始条件以及当取扩散系数(D_-\to0)的极限时，短时间行为会导致位移分布(P(x,t))在(x\to0)时出现一个尖点，即非解析行为。直观上，这个尖点或帐篷状形状类似于在许多无序环境中扩散粒子的实验中发现的类似行为，例如玻璃态系统和细胞内介质。这个一般结果仅取决于模型不同状态下等待时间的有限平均值的存在。由于遍历性以及状态(D_+)中时间占据分数分布收敛到一个狄拉克δ函数，在长时间极限下实现了高斯统计。同一量的短时间行为收敛到均匀分布，这导致了(P(x,t))中的非解析性。我们展示了超统计框架是(P(x,t))在跃迁次数方面的零阶短时间展开，它不会产生尖点状形状。后者被认为是该领域实验的关键特征，是在微扰理论的一阶修正中发现的。
