A universal field equation for dispersive processes in heterogeneous media |
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Authors: | John H Cushman Moongyu Park Monica Moroni Natalie Kleinfelter-Domelle Daniel O��Malley |
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Institution: | (1) Department of Earth and Atmospheric Sciences, Purdue University, West Lafayette, IN 47907, USA;(2) Department of Mathematics, Purdue University, West Lafayette, IN 47907, USA;(3) Department of Mathematics, University of Alabama at Huntsville, Huntsville, AL 35899, USA;(4) Department of Hydraulics, Transports and Roads, University La Sapienza, Rome, Italy;(5) Department of Mathematics, St. Mary’s College, Notre Dame, IN 46556, USA |
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Abstract: | When formulated properly, most geophysical transport-type process involving passive scalars or motile particles may be described
by the same space–time nonlocal field equation which consists of a classical mass balance coupled with a space–time nonlocal
convective/dispersive flux. Specific examples employed here include stretched and compressed Brownian motion, diffusion in
slit-nanopores, subdiffusive continuous-time random walks (CTRW), super diffusion in the turbulent atmosphere and dispersion
of motile and passive particles in fractal porous media. Stretched and compressed Brownian motion, which may be thought of
as Brownian motions run with nonlinear clocks, are defined as the limit processes of a special class of random walks possessing
nonstationary increments. The limit process has a mean square displacement that increases as tα+1 where α > −1 is a constant. If α = 0 the process is classical Brownian, if α < 0 we say the process is compressed Brownian while
if α > 0 it is stretched. The Fokker–Planck equations for these processes are classical ade’s with dispersion coefficient
proportional to tα. The Brownian-type walks have fixed time step, but nonstationary spatial increments that are Gaussian with power law variance.
With the CTRW, both the time increment and the spatial increment are random. The subdiffusive Fokker–Planck equation is fractional
in time for the CTRW’s considered in this article. The second moments for a Levy spatial trajectory are infinite while the
Fokker–Planck equation is an advective–dispersive equation, ade, with constant diffusion coefficient and fractional spatial
derivatives. If the Lagrangian velocity is assumed Levy rather than the position, then a similar Fokker–Planck equation is
obtained, but the diffusion coefficient is a power law in time. All these Fokker–Planck equations are special cases of the
general non-local balance law. |
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