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

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

2003-10-14

arxiv.org/abs/cond-mat/0310295v1

Phys. Rev. E 68, 050101(R) (2003).

Physics

Condensed Matter

Statistical Mechanics

4 pages Revtex, 1 .eps figure included, to appear in PRE Rapid Communication

Scientific paper

10.1103/PhysRevE.68.050101

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)).

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