First passage times and distances along critical curves

TL;DR

This paper introduces a model for anomalous transport along self-similar curves in inhomogeneous environments, using the stochastic Loewner equation to generate fractal curves and analyze first passage probabilities.

Contribution

It presents a novel approach to modeling particle transport along fractal curves with tunable fractal dimension using the stochastic Loewner equation.

Findings

Probability distributions' variance increases with fractal dimension

Distributions exhibit non-monotonic skewness

Tails decay faster than exponential, contrasting fractional dynamics predictions

Abstract

We propose a model for anomalous transport in inhomogeneous environments, such as fractured rocks, in which particles move only along pre-existing self-similar curves (cracks). The stochastic Loewner equation is used to efficiently generate such curves with tunable fractal dimension $d_f$. We numerically compute the probability of first passage (in length or time) from one point on the edge of the semi-infinite plane to any point on the semi-circle of radius $R$. The scaled probability distributions have a variance which increases with $d_f$, a non-monotonic skewness, and tails that decay faster than a simple exponential. The latter is in sharp contrast to predictions based on fractional dynamics and provides an experimental signature for our model.