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