Quantum and classical diffusion in small-world networks

TL;DR

This paper investigates quantum diffusion on small-world networks, revealing a transition from linear to logarithmic diffusion time with increasing rewiring probability, indicating rapid diffusion characteristic of small-world structures.

Contribution

It provides a numerical analysis of quantum diffusion on small-world networks, highlighting the transition in diffusion times and comparing quantum and classical diffusion behaviors.

Findings

Diffusion time scales as N in regular networks.

Diffusion time scales as log N in small-world networks.

Quantum diffusion is faster on small-world networks.

Abstract

We study numerically quantum diffusion of a particle on small-world networks by integrating the time-dependent Schr\"odinger equation with a localized initial state. The participation ratio, which corresponds to the number of visited sites in the case of classical diffusion, as a function of time is measured and the corresponding diffusion time $\tau$ is computed. In a local regular network, i.e., in the network with the rewiring probability $p=0$, the diffusion time depends on the network size $N$ as $\tau \sim N$, while the behavior $\tau \sim \log N$ is observed as $p$ becomes finite. Such fast diffusion of a particle on a complex network suggests that the small-world transition is also the fast-world transition from a dynamic point of view. The classical diffusion behavior is also studied and compared with the quantum behavior.