Universality and ambiguity in extremes of anomalous diffusion

TL;DR

This paper investigates the extreme first passage times in various anomalous diffusion models, revealing universal behaviors and model-dependent ambiguities in their physical relevance.

Contribution

It demonstrates that logarithmic decay of fFPT and the speed comparison between subdiffusion and normal diffusion are universal features across models, but their applicability varies with model specifics.

Findings

fFPT decays logarithmically with the number of searchers across models

Subdiffusion can be faster than normal diffusion in these models

Model-specific parameters influence the relevance of universal features

Abstract

Many biophysical processes begin when the fastest searcher finds a target out of many random searchers, which is called an extreme or fastest first passage time (fFPT). In some models, (i) the fFPT vanishes logarithmically as the number of searchers grows, and (ii) the fFPT can be faster for subdiffusive search compared to normal diffusion. Though mathematically rigorous, the relevance of (i) and (ii) to actual physical systems is suspect since their derivations involve searchers which move with unbounded speed. Indeed, we previously proved that the fFPT for searchers with bounded speed converges exponentially to a strictly positive minimal search time as the number of searchers grows. In this paper, we study fFPTs for a broad class of anomalous and normal diffusion models with bounded or unbounded speed. These models include scaled Brownian motion, Riemann-Liouville fractional Brownian motion, and fractional Brownian motion. For all of these models, we show that the fFPT decays logarithmically in the number of searchers and that subdiffusion can be faster than normal diffusion (we further show that superdiffusion can be slower than normal diffusion). In this sense, features (i) and (ii) are rather universal. On the other hand, we show that the parameter regimes in which (i) and (ii) are valid depend on the particulars of the individual model, and thus ambiguities remain in the relevance of these features to specific physical systems.