Static heterogeneity generates apparent universality in first-passage bursty dynamics

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, $P(t)\sim t^{-\hat{I}\pm }$, with putative universality classes $\hat{I}\pm =1$ and $\hat{I}\pm =\frac{3}{2}$ having been proposed. Whether the observed $\hat{I}\pm =1$ 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, $P(t)\propto t^{-\hat{I}\pm }\exp \left(-(t/t_c{)}^2\right)$, with $\hat{I}\pm \sim 1$. Kinetic Monte Carlo simulations reproduce this behavior, showing that the apparent $\hat{I}\pm \sim 1$ scaling is confined to a finite time window and arises from tip-induced spatial heterogeneity, not scale invariance.

34 pages (main, including four figures and four tables); 32 pages (Supplementary Text, including 14 figures and 1 table)