Trapping, percolation, and anomalous diffusion of particles in a two-dimensional random field

Marco Avellaneda, Frank Elliott, Christopher Apelian

Research output: Contribution to journalArticle

Abstract

We analyze from first principles the advection of particles in a velocity field with Hamiltonian H(x, y)=- V1y-- V2x+W1(y)-W2(x), where Wi, i=1, 2, are random functions with stationary, independent increments. In the absence of molecular diffusion, the particle dynamics are very sensitive to the streamline topology, which depends on the mean-to-fluctuations ratio ρ=max(|-V1|/Ū; |-V2|/Ū), with Ū=〈|W1′|21/2=rms fluctuations. Remarkably, the model is exactly solvable for ρ ≥1 and well suited for Monte Carlo simulations for all ρ, providing a nice setting for studying seminumerically the influence of streamline topology on large-scale transport. First, we consider the statistics of streamlines for ρ=0, deriving power laws for pnc(L) and 〈λ(L)〉, which are, respectively, the escape probability and the length of escaping trajectories for a box of size L, L » 1. We also obtain a characterization of the "statistical topography" of the Hamiltonian H. Second, we study the large-scale transport of advected particles with ρ > 0. For 0 <ρ < 1, a fraction of particles is trapped in closed field lines and another fraction undergoes unbounded motions; while for ρ≥ 1 all particles evolve in open streamlines. The fluctuations of the free particle positions about their mean is studied in terms of the normalized variables t-v/2[x(t)-〈x(t)〉] and t-v/2[y(t)-〈(t)〉]. The large-scale motions are shown to be either Fickian (ν=1), or superdiffusive (ν=3/2) with a non-Gaussian coarse-grained probability, according to the direction of the mean velocity relative to the underlying lattice. These results are obtained analytically for ρ ≥ 1 and extended to the regime 0<ρ<1 by Monte Carlo simulations. Moreover, we show that the effective diffusivity blows up for resonant values of {Mathematical expression}) for which stagnation regions in the flow exist. We compare the results with existing predictions on the topology of streamlines based on percolation theory, as well as with mean-field calculations of effective diffusivities. The simulations are carried out with a CM 200 massively parallel computer with 8192 SIMD processors.

Original languageEnglish (US)
Pages (from-to)1227-1304
Number of pages78
JournalJournal of Statistical Physics
Volume72
Issue number5-6
DOIs
StatePublished - Sep 1993

Fingerprint

Anomalous Diffusion
Trapping
Random Field
Streamlines
trapping
topology
diffusivity
Fluctuations
Diffusivity
Topology
SIMD (computers)
Monte Carlo Simulation
parallel computers
molecular diffusion
stagnation point
simulation
trapped particles
Independent Increments
Percolation Theory
advection

Keywords

  • percolation
  • superdiffusion
  • Trapping

ASJC Scopus subject areas

  • Statistical and Nonlinear Physics
  • Physics and Astronomy(all)
  • Mathematical Physics

Cite this

Trapping, percolation, and anomalous diffusion of particles in a two-dimensional random field. / Avellaneda, Marco; Elliott, Frank; Apelian, Christopher.

In: Journal of Statistical Physics, Vol. 72, No. 5-6, 09.1993, p. 1227-1304.

Research output: Contribution to journalArticle

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