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≥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)).