Nature of barriers determine first passage times in heterogeneous media

TL;DR

This paper provides an exact analysis of how the nature of barriers affects first passage times in heterogeneous media, revealing non-monotonic behavior for energetic barriers and monotonic increase for entropic barriers, validated through experiments and simulations.

Contribution

It offers the first exact solutions for first passage times in heterogeneous media with patterned obstacles, distinguishing effects of energetic versus entropic barriers.

Findings

First passage times vary non-monotonically with energetic barriers.

First passage times increase monotonically with entropic barriers.

Non-monotonic behavior extends to super-diffusive transport, confirmed by simulations.

Abstract

Intuition suggests that passage times across a region increases with the number of barriers along the path. Can this fail depending on the nature of the barrier? To probe this fundamental question, we exactly solve for the first passage time in general d-dimensions for diffusive transport through a spatially patterned array of obstacles - either entropic or energetic, depending on the nature of the obstacles. For energetic barriers, we show that first passage times vary non-monotonically with the number of barriers, while for entropic barriers it increases monotonically. This non-monotonicity for energetic barriers further reflects in the behaviour of effective diffusivity as well. We then design a simple experiment where a robotic bug navigates a heterogeneous environment through a spatially patterned array of obstacles to validate our predictions. Finally, using numerical simulations, we show that this non-monotonic behaviour for energetic barriers is general and extends to even super-diffusive transport.