A Universal Field Equation for Dispersive Processes

1
A Universal Field Equation for Dispersive
Processes

• J. H. Cushman, Purdue University
• Former Students
• M. Park, University of Alabama
• N. Kleinfelter-Domelle, Brown University
• M. Moroni, University of Rome, I
• B. Stroud, Private Industry
• M. Schoen, Berlin Technical

2
Anomalous Diffusion in Slit Nanopores

• Computational Statistical Mechanics Quasi-static
  experiments

3
Slit Pore Expansion
Each frame of the animation shows a single
configuration generated in a Monte Carlo
simulation in the grand-canonical isostrain
ensemble (µ, T, h, a) for a fixed pore width.
Played in succession, the frames show
configurations for the quasistatic pore
expansion.  It is important to note that the
animation does not portray the time evolution of
an expanding pore. Periodic boundary conditions
are imposed in the plane of the pore. The
configurations shown represent small sub-regions
of the fluid in an infinite slit pore. The pore
walls are five atoms thick, but only a single
layer of atoms is rendered for each wall with the
remainder of the walls being indicated in
outline. As the pore expands you should observe
that the configurations are highly ordered for
particular wall separations. This order is the
result of freezing of the pore fluid.
                                                  
                              
4
Simulation Cell
                                               
                                                  
                                                  
              Snapshot of a single configuration
in the simulation cell.
5
Thick Nano-Wire Expansion

•                                                 
                                                    
                                                
• Maps of two dimensional slices in the plane y0
  through the three dimensional, ensemble averaged
  particle density for a Lennard-Jones 10-wire
• reduced chemical potential, µ -11.7
• reduced temperature, T 1.0
• wall registration, alpha 0.0
• and reduced pore serpation, h, as indicated.
  Points on the rendered density surfaces are
  displaced towards the viewer in proportion to the
  density.

6
Thin Nano-Wire Expansion

•                                                 
                                                    
                
• Maps of two dimensional slices in the plane y0
  through the three dimensional, ensemble averaged
  particle density for a Lennard-Jones 5-wire
• reduced chemical potential, mu -11.7
• reduced temperature, T 1.0
• wall registration, alpha 0.0
• and reduced pore serpation, h, as indicated.
  Points on the rendered density surfaces are
  displaced towards the viewer in proportion to the
  density.

7
Nano-Wire and Slit Pore Expansion
                                               
                                                  
                                        Maps of
two dimensional slices through the three
dimensional, ensemble averaged particle density
in the plane y0 for reduced chemical potential,
µ -11.7, reduced temperature, T 1.0, wall
registration, a 0.0 and reduced pore
serpation, h, for (from left to right) a Lennard
Jones 5-wire, 10-wire, and an infinite-wire (slit
pore). Points on the rendered density surfaces
are displaced towards the viewer in proportion to
the density.
8
Anomalous Diffusion When the mean-free-path of
a molecule is on the same order as a
characteristic dimension of the pore, the
classical Fickian expression relating the
diffusive flux to the activity gradient breaks
down. That is, both diffusivity and the related
viscosity become wave vector and frequency
dependent.
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A Gedankin Experiment

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• Panel b is especially interesting as it shows
  sub-diffusion. Here the mean-square displacement
  scales with d 0.64 as opposed to the Brownian
  limit of d 1.

12
General Dispersion theory without scale
constraints
G is the conditional probability of finding a
particle at x at time t given it was initially at
X(0). If all particles are identical and are
released from the origin, then G represents the
concentration. The Fourier transform of G is G
13
General Dispersion theory without scale
constraints
For suitably well behaved g(t) we have
where
Here weve introduced the Exponential
Differential Displacement
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General Dispersion theory without scale
constraints

• With a little work can be rewritten

15
General Dispersion theory without scale
constraints

• Now, invert into real space and obtain

where Di' is the inverse transform of
(Di'?). This is an equation for dispersion
without the need for scale constraints.
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At equilibrium the first and second terms are
zero
17
Small wave vector limit-retrieving the classical
ADE

• Assume constant mean velocity so that
• depends only on k and t making

a convolution. Using a Taylor series expansion
about k0, on G it can be shown that
If the system is stationary, then the mean
square displacement can be replaced by twice
velocity covariance.
18
CTRW Master Equation for Porous Media
19
Anomalous Dispersion in Porous Media

• 3-D Porous Media PTV Experiments

20
Porous media experimental set-up Particle
Tracking Velocimetry (2D-PTV)
21
Three-Dimensional Particle Tracking Velocimetry
22
Porous Media Flow
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Porous Media Flow
24
Baricenters from the coupled cameras (Ex. 1)
Baricenters recognized on the image acquired by
camera  1 (21741 baricenters)
Baricenters recognized on the image acquired by
camera  2 (16740 baricenters)
25
3D-PTV 3-D Trajectory reconstruction from two
2-D projected trajectories
26
3 dimensional view of trajectories longer than 6
seconds
27
General Dispersion theory without scale
constraints
This equation for dispersion is just as
applicable in porous media as in the nano-film.
28
Generalized dispersion coefficient
Generalized dispersion coefficient in the
transverse directions, k2p/d, compared with the
velocity covariance .
29
CTRW Master Equation for Porous Media
30
Anomalous Dispersion of Motile Bacteria
31
Bacterial Swimming
Videos courtesy of Howard Berg, Molecular and
Cellular Biology, Harvard University
32
Particle tracking of microbes
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Bacterial mortor and drive train

• courtesy of David DeRosier, Brandeis university

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Microbe vs. Levy Motion
E coli swimming Berg, Phys Today, Jan 2000. /
Levy motion with a1.2
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Lévy versus Brownian motion
a 2
fractal dimension of the trace of Lévy motion
(including Brownian motion) a
DISTANCE FROM ORIGIN
a 1.7
DISTANCE FROM ORIGIN
36
Levy trajectory dispersion
A particle which follows an a-stable Levy motion
has a transition density given by the following
equation (for a divergence free velocity field)
where the general (asymmetric) fractional
derivative is defined in Fourier space by the
following with M the mixing measure, S the
d-dimensional unit sphere
37
Levy trajectory dispersion
38
Levy trajectory dispersion
The equation can be written in the divergence form
with convolution-Fickian flux
39
The Finite-Size Lyapunov ExponentDetermining
the exponent for Levy motion
Define the Finite-Size Lyapunov Exponent (FSLE)
The FSLE describes the exponential divergence of
two trajectories that start a specified distance,
r, apart and depends on that initial starting
scale as well as the threshold ratio, a.
or
la is the expected exponential rate two
particles separated by a distance r initially,
separate to ar at time t. Td is the expected
doubling time, i.e. the time it takes two
particles to separate from r to ar.
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A Statistical Mechanical Formulation of the
Finite-Size Lyapunov Exponent
Assume for simplicity that the system is
completely expansive. Let
ltlt?gtgt indicate integration over the product space
with respect to a joint probability density ,f,
defined on ?(t) x ? (tt). Here
is the joint probability particles i and j have
separation in at time t and separation in
at time
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A Statistical Mechanical Formulation of the
Finite-Size Lyapunov Exponent
where G(x,y,t) is the probability of a particle
going from separation x to separation y in time
t. Therefore the a-time is
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FSLE for a-stable Lévy Motion Varying a
Slope -1.2
Slope -1.5
Slope -2
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• For the case when (1a)/2 is an integer
  (i.e., a1), the summation P(2,1)?2/16

45
Turbulent Flows

• Fluid Flow Eulerian approach
• Lagrangian approach
• (1) Laminar Flow Re small
• (2) Turbulent Flow Re very large

46
Turbulent mixing layer growth and internal waves
formation laboratory simulations
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Penetrative convection in lakes
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The flux through the interface between the mixing
layer and the stable layer plays a fundamental
role in characterizing and forecasting the
quality of water in stratified lakes and in the
upper oceans, and the quality of air in the
atmosphere.
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Experimental set-up
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Convective Layer Growth and Internal waves
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Only internal waves
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Vortex Generating Chromatograph
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The experimental apparatus
A vortex-generating chomatograph is a device
constructed from a sequence of closed parallel
cylindrical tubes welded together in plane. The
complex is sliced down its lateral mid-plane and
the lower half is shifted laterally and then
fixed relative to the upper half.
Longitudinal section (dimensions in mm)
55
The experimental apparatus
Flow out
Flow in
Upper and lower view of the apparatus
(dimensions in mm)
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Vortex Generating Chromatograph
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Streamlines. Sectors 1 and 2 are largely stirring
regions while Sector 3 is largely a mixing region
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Locations of passage planes
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Passage time pdfs
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Super Diffusion
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Richardsons Super-Diffusion

• Richardsons scaling law for turbulent diffusion
• the separation of a pair of
  particles
• It also gives
  with

62
Brownian Velocity Processes and Richardson
Turbulence

• Take the velocity, v, to be a Brownian process,
  or to take advantage of its stationary
  increments, replace by
• The characteristic function for
  is
• It can be shown that the characteristic function
  for is

63
Brownian Velocity Processes and Richardson
Turbulence

• Take the Laplacian of with respect to
  k and set k to zero
• Now by Fourier Inversion of

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Levy Super-Diffusion Extending Richardson
Diffusion
With the results for the material particle with
Brownian velocity for motivation, we generalize
in the spirit of Mandelbrots intermittency. Assum
e in an Eulerian frame that the momentum change,
and hence energy dissipation, is concentrated on
a fractal set so that it has an Eulerian fractal
character. Assume the Eulerian fractal velocity
field has homogeneous increments. It is
reasonable then to assume a Lagrangian particle
passing through this Eulerian fractal field will
(i) take on a fractal character of its own (its
graph will be fractal), and (ii)
probabilistically have stationary increments.
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Levy Super-Diffusion Extending Richardson
Diffusion
The a-stable Levy processes have these two
characteristics (stationary increments and
fractal paths). With this as motivation we
assume the Lagrangian velocity is a-stable Levy.
Let VL(t) be the a -stable Levy velocity.
Though the graph of VL is fractal, it can be
shown that the graph of rL is not.
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Levy Super-Diffusion
One can show where

Note when a2 Richardsons scaling is obtained.

• For
• with

67
Microbial Dynamics in Fractal Porous Media
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Biofilm
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Bacterial Conjugation

• Conjugation Gene transfer from a donor to a
  recipient by direct physical contact between cells

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Conjugation Rates of E-Coli

• Conjugation rate
• (transconjugant per recipient)

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Upscaling of Microbial Dynamics in Fractal Media
- fractal functionality on the mesoscale (between
an upper and a lower cut-off) -Below the lower
cut-off (microscale) and above the upper cut-off
(macroscale) the system is non-fractal. -Microscal
e velocity is stationary, ergodic, Markov and
particle is subject to Levy diffusion
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Stream tube (yellow)
Trajectory of microbe
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Upscaling of Microbial Dynamics in Fractal Media
To account for particles that might be
self-motile, such as microbes, the microscale
diffusion is assumed driven by an a -stable Levy
process. If a2, then the process is Brownian
and the model can account for non-motile
particles. On the mesoscale the Eulerian velocity
is assumed fractal with spatially homogeneous
increments, and as in the turbulence problem, we
assume this translates into a fractal Lagrangian
drift velocity with temporally stationary
increments. Hence we assume the drift velocity at
the mesoscale is a-stable Levy. The diffusive
structure at the mesoscale is dictated by the
asymptotics of the microscale process.

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Upscaling from Microscale to Mesoscale

• Microscale SODE
• with stationary,
  ergodic and Markovian with mean , and
  L(0) is an al -stable Levy process which
  accounts for self-motility.
• One can show via a classical Central Limit
  Theorem (CLT)

al -stable Lévy process in distribution as
converges to
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Scale Change

• Let
• with
• One can show
• and hence

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Total Upscaling

• Assume macroscale periodicity in the initial
  data
• for integers and is a particle
  path in the porous medium.
• Let
• where

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Total Upscale and Upscaled ADE

• Let
• One can show if and
  , then for each ,
• For ,
• with

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Main Result

• f(x,t) the density function of X(t) for
  large t.
• Assume that then
• 
  
• where
• and recall the fractal derivative is
  defined by

with µ0 the mean velocity on the small scale,
and vV µ0
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Main Result (cont.)

• The fractional Fokker-Planck equation can be
  written in the divergence form
• where
• Cushman and Moroni (2001)
• where

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References

• Cushman, J.H., Hu, X., Ginn, T.R.
  Nonequilibrium statistical mechanics of
  preasymptotic dispersion.  J. Stat. Phys.,
  75859-878, 1994
• Cushman, J.H., Moroni, M. Statistical
  mechanics with three-dimensional particle
  tracking velocimetry experiments in the study of
  anomalous dispersion. I. Theory, Phys. Fluids
  1375-80, 2001
• Moroni M., Cushman, J.H. Statistical
  mechanics with three-dimensional particle
  tracking velocimetry experiments in the study of
  anomalous dispersion. II. Experiments, Phys.
  Fluids13 81-91,2001
• Parashar, R., and J. H. Cushman (2007) The
  finite-size Lyapunov exponent for Levy motions.
  Physical Review E 76, 017201 1-4.
• Park, M. and J. H. Cushman (2006) On upscaling
  operator-stable Levy motions in fractal porous
  media. J. Comp. Phys. 217159-165.
• Park, M., N. Kleinfelter and J. H. Cushman
  (2006) Renormalizing chaotic dynamics in fractal
  porous media with application to microbe
  motility. Geophysical Research Letters, 33,
  L01401.
• Park, M., N. Kleinfelter and J. H. Cushman
  (2005) Scaling laws and Fokker-Planck equations
  for 3-dimensional porous media with fractal
  mesoscale. SIAM Multiscale Modeling and
  Simulation, 4(4) 1233-1244.
• Park, M., N. Kleinfelter and J. H. Cushman
  (2005) Scaling laws and dispersion equations for
  Levy particles in 1-dimensional fractal porous
  media., Physical Review E, 72, 056305 1-7.
• Kleinfelter, N., M. Moroni, and J. H. Cushman
  (2005) Application of a finite-size Lyapunov
  exponent to particle tracking velocimetry in
  fluid mechanics experiments. Physical Review E,
  72, 056306 1-12.
• Cushman, J. H., M. Park, N. Kleinfelter, and M
  Moroni (2005) Super-diffusion via Levy
  Lagrangian velocity processes. Geophysical
  Research Letters, 32 (19) L19816 1-4.