Static heterogeneity generates apparent universality in first-passage bursty dynamics

TL;DR

This study shows that apparent universality in bursty activity patterns, like those in human behavior, can arise from static heterogeneity rather than intrinsic scale-free dynamics, demonstrated through molecular diffusion experiments.

Contribution

It demonstrates that observed power-law bursty dynamics with exponent around 1 can result from static heterogeneity, not true scale invariance, in a physical diffusion system.

Findings

Inter-pulse time distributions follow a tempered power law with alpha ~ 1.

Heterogeneity causes the apparent power-law scaling within a finite time window.

Simulations confirm heterogeneity, not scale invariance, explains the observed bursty behavior.

Abstract

Processes involving bursts of activity separated by quiescent periods occur across diverse systems and scales. In human dynamics, these phenomena have been described by power-law inter-event time distributions, $P(t)\sim t^{-\alpha}$, with putative universality classes $\alpha=1$ and $\alpha=\frac{3}{2}$ having been proposed. Whether the observed $\alpha = 1$ scaling reflects intrinsic scale-free dynamics or instead emerges from heterogeneous underlying rates has been debated at length. We address this question in a canonical physical system for first-passage dynamics: two-dimensional molecular diffusion detected by the tip of a scanning tunnelling microscope. The resulting inter-pulse time distributions exhibit the same apparent truncated power-law form reported for human activities such as email communication, web browsing, and library loans. Maximum-likelihood estimation and model comparison decisively favor a Kohlrausch-Williams-Watts--tempered power law, $P(t)\propto t^{-\alpha}\exp\left(-(t/t_c)^\beta\right)$, with $\alpha \sim 1$. Kinetic Monte Carlo simulations reproduce this behavior, showing that the apparent $\alpha \sim 1$ scaling is confined to a finite time window and arises from tip-induced spatial heterogeneity, not scale invariance.