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

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 $θ=d/4$ for $d<2$ and $θ=1/2$ 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)).