Passive Random Walkers and River-like Networks on Growing Surfaces

TL;DR

This paper introduces passive random walkers on growing surfaces to explore topological features and scaling properties, revealing river-like networks and coalescence behaviors specific to KPZ-type surface growth.

Contribution

It presents a novel method using passive walkers to directly measure surface growth exponents and uncover topological structures in fluctuating surfaces.

Findings

Passive walkers reveal river-like network structures in KPZ surfaces.

Walker coalescence behavior differs between KPZ and Edwards-Wilkinson growth.

Walker dynamics relate to shock waves in Burgers equation and passive scalar turbulence.

Abstract

Passive random walker dynamics is introduced on a growing surface. The walker is designed to drift upward or downward and then follow specific topological features, such as hill tops or valley bottoms, of the fluctuating surface. The passive random walker can thus be used to directly explore scaling properties of otherwise somewhat hidden topological features. For example, the walker allows us to directly measure the dynamical exponent of the underlying growth dynamics. We use the Kardar-Parisi-Zhang(KPZ) type surface growth as an example. The word lines of a set of merging passive walkers show nontrivial coalescence behaviors and display the river-like network structures of surface ridges in space-time. In other dynamics, like Edwards-Wilkinson growth, this does not happen. The passive random walkers in KPZ-type surface growth are closely related to the shock waves in the noiseless Burgers equation. We also briefly discuss their relations to the passive scalar dynamics in turbulence.