Escape from heterogeneous diffusion

Abstract

Many physical processes depend on the time it takes a diffusing particle to find a target. Though this classical quantity is now well-understood in various scenarios, little is known if the diffusivity depends on the location of the particle. For such heterogeneous diffusion, an ambiguity arises in interpreting the stochastic process, which reflects the well-known Itô versus Stratonovich controversy. Here we analytically determine the mean escape time and splitting probabilities for an arbitrary heterogeneous diffusion in an arbitrary three-dimensional domain with small targets that can be perfectly or imperfectly absorbing. Our analysis reveals general principles for how search depends on heterogeneous diffusion and its interpretation (e.g. Itô, Stratonovich, or kinetic). An intricate picture emerges in which, for instance, increasing the diffusivity can decrease, not affect, or even increase the escape time. Our results could be used to determine the appropriate interpretation for specific physical systems.

Figures (2)

• Figure 1: A particle diffuses with a space-dependent diffusivity inside a general three-dimensional domain with small targets on its boundary.
• Figure 2: Comparison between theory (curves) and stochastic simulations (markers). Simulated particles start in the center of a unit cube domain $\Omega$ with $N=2$ targets with common radius $a=0.01$ located at the center of the left and right boundaries. The diffusivity is $D(x)=D(x^{(1)},x^{(2)},x^{(3)})=0.1+10x^{(1)}$, where $x^{(1)}=0$ and $x^{(1)}=1$ define the left and right boundaries. The MFPTs in the top panel are multiplied by $4a$ for perfect targets and $\kappa\pi a^2$ for imperfect targets. The imperfect targets have reactivity $\kappa=1$data_availability.