Ergodicity recovery of random walk in heterogeneous disordered media*

Abstract

Significant and persistent trajectory-to-trajectory variance are commonly observed in particle tracking experiments, which have become a major challenge for the experimental data analysis. In this theoretical paper we investigate the ergodicity recovery behavior, which helps clarify the origin and the convergence of trajectory-to-trajectory fluctuation in various heterogeneous disordered media. The concepts of self-averaging and ergodicity are revisited in the context of trajectory analysis. The slow ergodicity recovery and the non-Gaussian diffusion in the annealed disordered media are shown as the consequences of the central limit theorem in different situations. The strange ergodicity recovery behavior is reported in the quenched disordered case, which arises from a localization mechanism. The first-passage approach is introduced to the ergodicity analysis for this case, of which the central limit theorem can be employed and the ergodicity is recovered in the length scale of diffusivity correlation.

Non-Gaussian Normal Diffusion in Low Dimensional Systems

“…In Refs. [28][29][30] the present approach has been extended also to microscopically non-Gaussian diffusive processes [where the ∆x distribution of Eq. (2) does not apply].…”

“…The constants A and B have then be determined by normalizing p β (δ t ) to one and ensuring that its second moment yields δ 2 t = 2D for any value of the free parameter β. In view of its derivation, the heuristic distribution (17) may apply also to the transients of microscopically non-Gaussian diffusion models [28][29][30]. The fitting parameter β is allowed to vary with t; it assumes values in the range 1 ≤ β ≤ 2 for leptokurtic distributions (positive excess kurtosis) and β ≥ 2 for platykurtic distributions (negative excess kurtosis).…”

Quantum walk under coherence non-generating channels*

“…It is fair to say CW becomes one of the most significant models in the theories of probability and statistics. [1][2][3][4][5][6][7][8][9][10][11] Its most popular version is that a walker moves on a line in a discrete manner and that the walker flips a coin to decide the direction of the next step. In detail, if the result is the head, the walker will move its position (initially at x = 0) to the position x = 1; if the result is the tail, the walker will move its position to the position x = −1.…”