First passage time properties of diffusion with a broad class of stochastic diffusion coefficients

TL;DR

This paper analyzes how stochastic diffusion coefficients affect first passage times in diffusion processes, revealing that fluctuations can enhance early transport efficiency and exhibit distinct long-term behaviors.

Contribution

It introduces a comprehensive study of FPT properties under broad classes of stochastic DCs, highlighting effects of fluctuations on transport efficiency and long-term distributions.

Findings

Stochastic DCs ensure eventual absorption with probability one.

Early arrival of particles is enhanced by stochastic DCs compared to ensemble averages.

Long-time FPT distribution converges to a Lévy-Smirnov distribution for ergodic DCs.

Abstract

This study investigates the first passage time (FPT) properties of particles with a broad class of positive stochastic diffusion coefficients (DCs), representing diffusion in heterogeneous environments or of particles with conformational fluctuations. We demonstrate that for diffusion in a one-dimensional semi-infinite domain with an absorbing boundary, particles will eventually reach the absorbing boundary with probability one. We also show that a stochastic DC provides higher transport efficiency in an early arrival of particles at the boundary than would be expected under diffusion whose DC is the ensemble average of the stochastic DC. Furthermore, a stochastic DC with a larger supremum exhibits a more efficient transport even if ensemble averages are the same. For ergodic DCs, we show three more properties: the mean FPT diverges, the enhancement of early-arrival efficiency diminishes over long times, and the FPT distribution converges to a L\'evy-Smirnov distribution in the long-time limit. These properties are shown to arise from the convergence of the time-averaged DC to the ensemble average, with the convergence speed determined by the DC's fluctuation time scale. We finally discuss the similarities and differences of FPT properties between three-dimensional diffusion outside a spherical absorbing boundary and the one-dimensional diffusion. Our results indicate that fluctuations in DCs may need to be non-Markov and/or non-ergodic to allow efficient transport of particles to distant targets. Our results also suggest that fluctuations in a DC play an important role, for example, in diffusion-limited reactions triggered by single molecules in physics, chemistry, or biology.