First passage time distribution in heterogeneity controlled kinetics: going beyond the mean first passage time

TL;DR

This paper derives the full distribution of first passage times in heterogeneous systems, revealing new insights into the different characteristic time scales beyond the average, including short, long, and an overlooked intermediate scale.

Contribution

It provides a rigorous derivation of the complete FPT distribution in heterogeneous systems, highlighting differences from the mean FPT and identifying a new characteristic time scale.

Findings

Typical FPT differs from the mean FPT

Short time behavior relates to direct trajectories

Discovery of a third, intermediate time scale

Abstract

The first passage is a generic concept for quantifying when a random quantity such as the position of a diffusing molecule or the value of a stock crosses a preset threshold (target) for the first time. The last decade saw an enlightening series of new results focusing mostly on the so-called mean and global first passage time (MFPT and GFPT, respectively) of such processes. Here we push the understanding of first passage processes a step further. For a simple heterogeneous system we derive rigorously the complete distribution of first passage times (FPTs). Our results demonstrate that the typical FPT significantly differs from the MFPT, which corresponds to the long time behaviour of the FPT distribution. Conversely, the short time behaviour is shown to correspond to trajectories connecting directly from the initial value to the target. Remarkably, we reveal a previously overlooked third characteristic time scale of the first passage dynamics mirroring brief excursion away from the target.