# Asymmetric Random Walk in a One-Dimensional Multi-Zone Environment

## TL;DR

This paper models a one-dimensional asymmetric random walk across multiple zones with fixed transition probabilities, deriving analytical solutions for the probability distribution and mean squared displacement, revealing transient anomalous diffusion.

## Contribution

It introduces an analytical approach to asymmetric random walks in multi-zone environments, accounting for interface effects and validating with numerical simulations.

## Key findings

- Derived probability distribution function considering diffusion and drift.
- Identified transient anomalous diffusion behavior.
- Validated analytical results with numerical simulations.

## Abstract

We consider a random walk model in a one-dimensional environment, formed by several zones of finite width with the fixed transition probabilities. It is also assumed that the transitions to the left and right neighboring points have unequal probabilities. In continuous limit, we derive analytically the probability distribution function, which is mainly determined by a walker diffusion and drift and accounts perturbatively for interface effects between zones. It is used for computing the probability to find a walker in a given space-time point and the time dependence of the mean squared displacement of a walker, which reveals the transient anomalous diffusion. To justify our approach, the probability function is compared with the results of numerical simulations for a three-zone environment.