One-Dimensional Random Walk in Multi-Zone Environment

TL;DR

This paper analyzes a symmetric one-dimensional random walk in a multi-zone environment with varying transition probabilities, deriving the probability distribution and revealing transient anomalous diffusion behavior.

Contribution

It introduces an analytical approach to study random walks in multi-zone environments with different transition probabilities and diffusion coefficients, highlighting non-Gaussian distributions and anomalous diffusion.

Findings

Probability distribution is non-Gaussian due to zone properties.

Mean squared displacement shows transient anomalous diffusion.

Derived analytical expressions for position and time probabilities.

Abstract

We study a symmetric random walk (RW) in one spatial dimension in environment, formed by several zones of finite width, where the probability of transition between two neighboring points and corresponding diffusion coefficient are considered to be differently fixed. We derive analytically the probability to find a walker at the given position and time. The probability distribution function is found and has no Gaussian form because of properties of adsorption in the bulk of zones and partial reflection at the separation points. Time dependence of the mean squared displacement of a walker is studied as well and revealed the transient anomalous behavior as compared with ordinary RW.