[Paper Review] Residual Diffusivity in Elephant Random Walk Models with Stops

Jiancheng Lyu, Jack Xin+1 more|arXiv (Cornell University)|May 8, 2017

Stochastic processes and statistical mechanics4 references3 citations

TL;DR

This paper introduces a perturbed elephant random walk model with small-probability symmetric steps (stops) to study residual diffusivity in sub-diffusive systems. By analytically deriving exact asymptotics for the first two moments in one and two dimensions, it demonstrates that even in the limit of vanishing perturbations ($\epsilon \downarrow 0$), the effective diffusivity remains strictly positive, confirming residual diffusivity in a wedge-shaped parameter regime.

ABSTRACT

We study the enhanced diffusivity in the so called elephant random walk model with stops (ERWS) by including symmetric random walk steps at small probability $ε$. At any $ε> 0$, the large time behavior transitions from sub-diffusive at $ε= 0$ to diffusive in a wedge shaped parameter regime where the diffusivity is strictly above that in the un-perturbed ERWS model in the $ε\downarrow 0$ limit. The perturbed ERWS model is shown to be solvable with the first two moments and their asymptotics calculated exactly in both one and two space dimensions. The model provides a discrete analytical setting of the residual diffusion phenomenon known for the passive scalar transport in chaotic flows (e.g. generated by time periodic cellular flows and statistically sub-diffusive) as molecular diffusivity tends to zero.

Motivation & Objective

• To model and analyze residual diffusivity in discrete stochastic processes mimicking chaotic fluid flows.
• To investigate how small symmetric random walk steps (stops) induce a transition from sub-diffusive to diffusive behavior in the elephant random walk model.
• To establish analytically that the effective diffusivity remains strictly positive in the limit of vanishing perturbations ($\epsilon \downarrow 0$), demonstrating residual diffusivity.
• To extend the analysis from one to two dimensions and confirm the persistence of residual diffusivity in higher dimensions.

Proposed method

• Introduce a perturbed elephant random walk with a small probability $\epsilon$ of symmetric steps (0) at each time step.
• Define the transition mechanism where the walker either repeats a past step, flips it, or stays put, with probabilities $p$, $q$, and $r$, respectively.
• Use conditional expectation and recurrence relations to derive exact expressions for the first and second moments of position.
• Apply asymptotic analysis of the Gamma function to derive large-time behavior of the second moment $\langle |X_t|^2 \rangle$.
• Generalize the 1D model to 2D by introducing vector-valued steps and using matrix forms for the drift and diffusion components.
• Solve the resulting moment equations using properties of the Gamma function and derive explicit formulas for $\langle |X_t|^2 \rangle$ in terms of parameters $\epsilon$, $r$, $\gamma$, and $\gamma'$.

Experimental results

Research questions

• RQ1 Does the inclusion of small-probability symmetric steps in the elephant random walk model lead to a transition from sub-diffusive to diffusive behavior?
• RQ2 Can residual diffusivity be analytically demonstrated in a solvable discrete model, even as the perturbation strength $\epsilon \to 0$?
• RQ3 What is the asymptotic behavior of the second moment $\langle |X_t|^2 \rangle$ in the perturbed model, and how does it compare to the unperturbed case?
• RQ4 Is the residual diffusivity effect robust in two dimensions, and does it persist in a wedge-shaped parameter region in the $(r, \gamma)$ plane?

Key findings

• The second moment $\langle |X_t|^2 \rangle$ asymptotically grows linearly in time with a slope strictly greater than that of the unperturbed elephant random walk in the $\epsilon \downarrow 0$ limit.
• The effective diffusivity in the perturbed model remains strictly positive in the limit $\epsilon \downarrow 0$, confirming residual diffusivity.
• A wedge-shaped region in the $(r, \gamma)$ parameter plane exists where the enhanced diffusivity exceeds the unperturbed diffusivity in the zero-$\epsilon$ limit.
• The 2D model exhibits the same residual diffusivity behavior as the 1D model, with identical asymptotic scaling of the second moment.
• Explicit formulas for $\langle |X_t|^2 \rangle$ are derived using the Gamma function, showing $\langle |X_t|^2 \rangle = \frac{\epsilon}{(1 - 2\gamma)(\epsilon + r)}t + \text{lower-order terms}$.
• The coefficient of the linear term in $t$ diverges as $\epsilon \downarrow 0$ when $\gamma \to 1/2$, indicating a critical transition in the diffusive regime.