Persistence of a particle in the Matheron-de Marsily velocity field

TL;DR

This paper analytically demonstrates that the particle's longitudinal position in a layered random velocity field behaves as a fractional Brownian motion with a dimension-dependent Hurst exponent, leading to specific persistence decay laws.

Contribution

It provides an exact analytical connection between the particle's motion in the Matheron-de Marsily model and fractional Brownian motion, clarifying the persistence decay behavior across dimensions.

Findings

Persistence decays as a power law with exponent d/4 for d<2.

For d≥2, persistence decays as t^{-1/2} with a logarithmic correction at d=2.

The Hurst exponent varies with dimension, being 1-d/4 for d<2 and 1/2 for d>2.

Abstract

We show that the longitudinal position $x(t)$ of a particle in a $(d+1)$-dimensional layered random velocity field (the Matheron-de Marsily model) can be identified as a fractional Brownian motion (fBm) characterized by a variable Hurst exponent $H(d)=1-d/4$ for $d<2$ and $H(d)=1/2$ for $d>2$. The fBm becomes marginal at $d=2$. Moreover, using the known first-passage properties of fBm we prove analytically that the disorder averaged persistence (the probability of no zero crossing of the process $x(t)$ upto time $t$) has a power law decay for large $t$ with an exponent $\theta=d/4$ for $d<2$ and $\theta=1/2$ for $d\geq 2$ (with logarithmic correction at $d=2$), results that were earlier derived by Redner based on heuristic arguments and supported by numerical simulations (S. Redner, Phys. Rev. E {\bf 56}, 4967 (1997)).