Finite current stationary states of random walks on one-dimensional lattices with aperiodic disorder

TL;DR

This paper investigates how different types of aperiodic disorder affect the stationary states and drift velocities of random walks on one-dimensional lattices, revealing size-dependent behaviors and structural transitions.

Contribution

It introduces a scaling analysis of stationary states in random walks with aperiodic disorder, highlighting the effects of Thue-Morse, paperfolding, and Rudin-Shapiro sequences on diffusion and drift.

Findings

Thue-Morse sequence leads to extended stationary distribution with size-independent drift velocity.

Paperfolding and Rudin-Shapiro sequences exhibit size-dependent structural transitions under finite current.

Finite current can break hierarchical and fractal structures, resulting in extended distributions in large systems.

Abstract

Stationary states of random walks with finite induced drift velocity on one-dimensional lattices with aperiodic disorder are investigated by scaling analysis. Three aperiodic sequences, the Thue-Morse (TM), the paperfolding (PF), and the Rudin-Shapiro (RS) sequences, are used to construct the aperiodic disorder. These are binary sequences, composed of two symbols A and B, and the ratio of the number of As to that of Bs converges to unity in the infinite sequence length limit, but their effects on diffusional behavior are different. For the TM model, the stationary distribution is extended, as in the case without current, and the drift velocity is independent of the system size. For the PF model and the RS model, as the system size increases, the hierarchical and fractal structure and the localized structure, respectively, are broken by a finite current and changed to an extended distribution if the system size becomes larger than a certain threshold value. Correspondingly, the drift velocity is saturated in a large system while in a small system it decreases as the system size increases.