Intricate dynamics of a deterministic walk confined in a strip

TL;DR

This paper investigates the complex dynamics of a deterministic walk in a narrow, randomly filled strip, revealing a transition from ballistic to non-ergodic behavior with power-law distributed excursions and velocity inversions.

Contribution

It introduces a novel model of deterministic walks in confined spaces exhibiting a transition to non-ergodic regimes and analyzes the statistical properties of its complex dynamics.

Findings

Transition from ballistic to non-ergodic motion with increasing width

Power-law distributions of excursion durations and first-passage times

Presence of log-oscillations and wild outliers in certain parameter regimes

Abstract

We study the dynamics of a deterministic walk confined in a narrow two-dimensional space randomly filled with point-like targets. At each step, the walker visits the nearest target not previously visited. Complex dynamics is observed at some intermediate values of the domain width, when, while drifting, the walk performs long intermittent backward excursions. As the width is increased, evidence of a transition from ballistic motion to a weakly non-ergodic regime is shown, characterized by sudden inversions of the drift velocity with a probability slowly decaying with time, as $1/t$ at leading order. Excursion durations, first-passage times and the dynamics of unvisited targets follow power-law distributions. For parameter values below this scaling regime, precursory patterns in the form of "wild" outliers are observed, in close relation with the presence of log-oscillations in the probability distributions. We discuss the connections between this model and several evolving biological systems.