A U niversal Field Equation for Dispersive Processes

1

A Universal Field Equation for Dispersive Processes

J. H. Cushman, Purdue University

Former Students

M. Park, University of AlabamaN. Kleinfelter-Domelle, Brown University M. Moroni, University of Rome, I B. Stroud, Private IndustryM. 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*, α) 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 y=0 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 y=0 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 y=0 for reduced chemical potential, μ = -11.7, reduced temperature, T= 1.0, wall registration, α = 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.

9

A Gedankin Experiment

11

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

( ) (0)ˆ ( , ) i t itG t e e k X k Xk

( , ) ( ( ( ) (0)))G t t x x X X

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 Ĝ

ˆ ˆ( , ) ( , )tiG t e G t k Xk k

( ) ( ) ( )tt t t X X X

( ) (0)ˆ ( , ) i t iG t e e k X k Xk

13


For suitably well behaved g(t) we have

00

( ) ( )t

t

g gK g t d

t t

Letting G´ in the last slide be g above, then upon differentiating and assuming dP/dt=0 where P is underlying probability and G'(k,0)=0 we obtain the following equation in Fourier space

0

ˆˆ ˆˆ( ) ( , ) ( , ) ( , , ) ( , )

tGi t G t K t G t d

t

k v k k k k

/])()([),,( tt ttiet XXkk

where

( ) gK g

Here we’ve introduced the Exponential Differential Displacement:

14


With a little work can be rewritten

1 2ˆˆ ˆ ˆ

ˆGK i i i

G

k D k D k

~( ) (0)

1

( )ˆˆ

i t it e e

G

k X k XaD

~( ) (0)

2

( ) ( )ˆˆ

i t it e e t

G

k X k Xv vD

0

ˆˆ ˆˆ( ) ( , ) ( , ) ( , , ) ( , )

tGi t G t K t G t d

t

k v k k k k

ˆ ( , )K k

15


Now, invert into real space and obtain

ddtGt

ddtGttGt

tG

t

yx

R

x

t

R

xx

0

2

0

1

),(),,(

),(),,(),(),(

3

3

yyxyD

yyxyDxvx

where Di' is the inverse transform of (Di'Δ)^.

This is an equation for dispersion without the need for scale

constraints.

16

At equilibrium the first and second terms are zero

17

Small wave vector limit-retrieving the classical ADE

Assume constant mean velocity so thatdepends only on k and τ making

ddtGt

ddtGttGt

tG

t

yx

R

x

t

R

xx

0

2

0

1

),(),,(

),(),,(),(),(

3

3

yyxyD

yyxyDxvx

a convolution. Using a Taylor series expansion about |k|=0, on G^ it can be shown that

1

~21 22

0

1ˆ ( ) (0)2

D

D t

v v X X

If the system is stationary, then the mean square displacement can be replaced by twice velocity covariance.

/])()([),,( tt ttiet XXkk

19

Anomalous Dispersion in Porous Media

3-D Porous Media PTV Experiments

20

Porous media experimental set-up

Particle Tracking Velocimetry (2D-PTV)

.

High power lamp

Thermostatic vessel

Acquisition system

Pump

CCD camera

CCD camera

21

Three-Dimensional Particle Tracking

Velocimetry

22

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


ddtGt

ddtGttGt

tG

t

yx

R

x

t

R

xx

0

2

0

1

),(),,(

),(),,(),(),(

3

3

yyxyD

yyxyDxvx

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, k=2/d, compared with the velocity covariance .

29

CTRW Master Equation for Porous Media

0 0

( , )( , ) ( , ) ( , ) ( , )

t tC tt t C t dt t t C t dt

t

s s

ss s s s s s

1( , ) and ( , )

2t t

s s

v s s s ss

0

( , ) 1( , )( ) ( , ) ( , ) ( )( ) : ( , ) .

2

tC tdt t t C t t t C t

t

s

ss s s s s s s s s s s s

ddtGt

ddtGttGt

tG

t

yx

R

x

t

R

xx

0

2

0

1

),(),,(

),(),,(),(),(

3

3

yyxyD

yyxyDxvx

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

33

Bacterial mortor and drive train

• courtesy of David DeRosier,

Brandeis university

34

Microbe vs. Levy Motion

E coli swimming Berg, Phys Today, Jan 2000. / Levy motion with α=1.2

35

Lévy versus Brownian motion

fractal dimension of the trace of Lévy motion (including

Brownian motion) =

DISTANCE FROM ORIGIN

DIS

TA

NC

E F

RO

M O

RIG

IN

= 1.7

= 2

-20

-15

-10

-5

0

5

10

15

20

25

0 10 20 30 40

36

Levy trajectory dispersion

ˆ( ) ( ) ( , )M

S

G i M d G t k s s k

( , )M

GD G t

t

v x

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

ˆ ( , ) ( , )d

i

R

G t e G t d k xk x x

A particle which follows an α-stable Levy motion has a transition density given by the following equation (for a divergencefree velocity field)

37


2

ˆ( ) ( ) ( , )

ˆ( ) ( ) ( , )

ˆ( ) ( , )

M

S

i i j j

S

i ij j

G i M d G t

ik i s s M d ik G t

ik D G t ik

k s s k

k s s k

k k

Rewrite the fractional derivative as:

ddtG

dGDGD

yx

t

R

x

yx

R

xM

d

d

yyxyD

yyxyD

),(),(

)()(

0

2,1)()()(),(21

ssky dMssiDD ji

S

ij

1 : Inverse FT

Now take theInverse Fouriertransform

38


yyxyDvq ddtGG yx

t

Rn

),(),(0

q

t

G

The equation can be written in the divergence form

with convolution-Fickian flux

39

DefineDefine the Finite-Size Lyapunov the Finite-Size Lyapunov Exponent (FSLE)Exponent (FSLE)

or or

aT

rd

a ln1

)(

ae rT ad )(

aa: : is the expected exponential rate two is the expected exponential rate two particles separated by a distance particles separated by a distance r r initially, initially, separate to separate to arar at time at time tt. . TTdd is the expected doubling time, i.e. the is the expected doubling time, i.e. the time it takes two particles to separate from time it takes two particles to separate from rr to to arar..

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.

The Finite-Size Lyapunov Exponent:Determining the exponent for Levy motion

40

A Statistical Mechanical Formulation of the A Statistical Mechanical Formulation of the Finite-Size Lyapunov Exponent Finite-Size Lyapunov Exponent

Assume for simplicity that the system is completely ‘expansive’. Let

<<>> indicate integration over the product space with respect to a joint probability density ,f, defined on (τ) x (τ+t).

Here 1,2 1,3 -1, 1,2 1,3 -1, 1,2 1,2 -1, -1,f ( ( ), ( ), ..., ( ), ( ), ( ),..., ( )) ... N N N N N N N Nr r r r t r t r t dr dr dr dr

is the joint probability particles i and j have separation in at time τ and separation in at time

, , ,,i j i j i jr r dr , , ,,i j i j i jr r dr

, 1,..., 1, 1,..., .t i N j N

41

( | ( ) ( ) |) ( | ( ) ( ) |)1( , , )

( 1) ( | ( ) ( ) |)j l j l

j l j l

y t t xG x y t

N N x

X X X X

X X

A Statistical Mechanical Formulation of A Statistical Mechanical Formulation of the Finite-Size Lyapunov Exponentthe Finite-Size Lyapunov Exponent

0

( , , ) ( , , )G x y t G x y t d

0

( ) ( , ; ) ( , ; ) .aT x E G x ax t tG x ax t dt

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:

42

FSLE for α-stable Lévy Motion Varying α

0.01

0.1

1

10

100

0.001 0.01 0.1 1 10 100

Initial Separation

FS

LE

alpha=1.2 a=1.3

alpha=1.5 a=1.3

alpha=2 a=1.3

Slope -1.2

Slope -1.5

Slope -2

1( ) ln ~

( )aa

x a xT x

44

For the case when (1+α)/2 is an integer (i.e., α=1), 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 largeRe

LV

46

Turbulent mixing layer growth and internal waves

formation: laboratory simulations

48

The flux through the interface between the mixing layer The flux through the interface between the mixing layer and the stable layer plays a fundamental role in and the stable layer plays a fundamental role in characterizing and forecasting the quality of water in characterizing and forecasting the quality of water in stratified lakes and in the upper oceans, and the quality stratified lakes and in the upper oceans, and the quality of air in the atmosphere.of air in the atmosphere.

49

Experimental set-up

51

Convective Layer Growth and Internal waves

52

Only internal waves

53

Vortex Generating Chromatograph

54

The experimental apparatus

Longitudinal section (dimensions in mm)

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.

Flo

w

in

Flo

w

ou

t

300

8

12 4

30

20

8

15

30 60

30

R1 R2 R3 R4 R5 R6 R7 R8

C1 C3 C4 C5 C6 C7 C8

55

The experimental apparatus

Upper and lower view of the apparatus (dimensions in mm)

Flo

w

in Flo

w

ou

t

I7

I6

I5

I4

I3

I2

I1

C1 C2 C3 C4 C5 C6 C7 C 8

I8

300

200

30 20 8

R1 R2 R3 R4 R6 R5 R7 R8

O8

O7

O6

O5

O4

O3

O2

O1

56

Vortex Generating Chromatograph

57

Streamlines. Sectors 1 and 2 are largely stirring regions while Sector 3 is largely a mixing region

58

Locations of passage planes

59

Passage time pdfs

-2

0

2

4

6

8

10

12

14

0 0.5 1 1.5 2Time (s)

pd

f

p(X>x1)

p(X>x2)

p(X>x3)

p(X>x4)

p(X>x5)

p(X>x6)

60

Super Diffusion

61

Richardson’s Super-DiffusionRichardson’s scaling law for turbulent

diffusion:

= the separation of a pair of particles:

It also gives with

2 3| ( ) |r t t

( )r t0

( ) ( )t

r t v s ds

2( )c

D t ct

2( )D t t

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 3

32( , ) exp | | ( )

3b S

tf k t k s M ds

( )v t( ) (0).v t v

3

2( , ) exp | | ( )S

f k t t k s M ds

( )r t( , )bf k t

( ) (0)v t v

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 :

bf

3

32

0

2| | | ( ).

3k b k S

tf z M dz

3

2 2 2 30| ( ) | | | ( , ) | | ( ) |b b b k b k bR

r t r f r t dr f r t t

2

( )2

bb

b b b b

fq

t

tq D t f Q f

2bD t

64

Levy Super-Diffusion Extending Richardson Diffusion

With the results for the material particle with Brownian velocity for motivation, we generalize in the spirit of Mandelbrot’s intermittency.Assume 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.

65

Levy Super-Diffusion Extending Richardson Diffusion

The α-stable Levy processes have these two characteristics (stationary increments and fractal paths). With this as motivation we assume the Lagrangian velocity is α-stable Levy.Let VL(t) be the α -stable Levy velocity. Though the graph of VL is fractal, it can be shown that the graph of rL is not.

66

Levy Super-Diffusion

3

1

ln ( , ) | | ( , , ) ( )1L S

tf k t k s k s M ds

One can show

where ( , , ) 1 sgn( ) tan( / 2)k s i k s

Note when α=2 Richardson’s scaling is obtained.For

with

1 2

3( , ) ( , )

L

x y LR

fq

t

q D y t f x y t dy

3

1 2( , ) ( ) ( ) ( )jk j kSD y t D t ik s s s M ds

( ) cos( / 2)

tD t

67

Microbial Dynamics in Fractal Porous Media

68

Biofilm

69

Introduced Degrader

Degradation Genes on Mobile Element (Plasmid)

Recipient Bacterium, Unable to Degrade

Chromosome

Cell to Cell Contact and Gene Exchange

A.

Recipient Bacterium Now Able to Degrade

B.

C.

or

A. The recipient strain will first be allowed to be established on the glass slide. Experiments will be performed with increasing cell density of the recipient on the slide. Then the optimal recipient density will be used and the donor density will be varied. The donor will be put into the inflow and passed through in varying densities, nutrient status and flow rates.B. Number and duration of cell-to-cell contacts will be determined by direct microscopy and examining the captured images. Image analysis will also reveal the time required for gene transfer and expression. Gene expression will be detected by fluorescence of the GFP. Recipients and donors are differentiated by both morphology and the color they fluoresce.C. Number of attached cells able to degrade 3-chlorobenzoate will be determined by cell fluorescence because of activation of the GFP genes. Determinations will be made whether both cells remain attached to the surface, one cell or no cells after and during gene exchange.

Bacterial Conjugation

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

71

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.

-Microscale velocity is stationary, ergodic, Markov and particle is subject to Levy diffusion

72

Stream tube

(yellow)

Trajectory of microbe

74

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 α -stable Levy process. If α=2, 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 α-stable Levy. The diffusive structure at the mesoscale is dictated by the asymptotics of the microscale process.

75

Upscaling from Microscale to Mesoscale

Microscale SODE:

with stationary, ergodic and Markovian with mean , and L(0) is an αl -stable Levy process which accounts for self-motility.

One can show via a classical Central Limit Theorem (CLT):

(0) (0) (0) (0) (0) (0)

0( ) (0) ( ( )) ( ),

tX t X v X r dr L t 0t

(0) (0) (0)( ) ( ( ))V r v X r(0)

V

(0)( ( ) ( , ))l l lL t S tM t

( 0) ( 0)

1

( ) ( )l

l

nt nt

n

VX n

converges to αl -stable Lévy process in distribution as

(0) (0) (0)(0)

(0) (0) (0)1 1 1 2 11 2 0

( ) ( ) 1 1 ( )( ( ))

l l l

ntl lnt nt nt nt

X r drn n n n

LVX V V

1 2 (0) (0) (1)(0)

0( ( )) ( )

dnt

n r dr t V X BV

(0) (0)(1)(1)

1

( )( )

l

dlnt nt

tn

LL

76

Scale Change

Let

with

One can show and hence

(2) (1)( ) lim ( )t t

Y Y

(1) (1) (1) (1) (1) (1)

0 0

(1) (1)

0

( ) ( ) (0) ( ) (0) ( )

( ) ( )

t t

t

t r dr r dr t t

t r dr

Y V V V V Y

Y V

(2) (1)( ) ( )t tY Y

(2)(2) (1)( ) ( )t t t Y V Y

77

Total Upscaling

Assume macroscale periodicity in the initial data:

for integers and is a particle path in the porous medium.

Let

where

(1)i

(1) (1) (1) (1) (1) (1)( ) ( (0)) ( (0) )t t t V V V(1) (1) (1) (1)

1 2 3( )

(1) (1) (1) (1)1 1[ ] ( (0)) (0)

A BE t dt d

A B

V V

(1) (1) 3( ) { (0 ) (0 ) [0 1] }A A (1) 3( ) { 0 ( ) [0 1] }B B t t

(1) (0)

78

Total Upscale and Upscaled ADELet

One can show if and , then for each ,

For ,with

1v

(1) (1) (1) (1)

1 1

(1) (1) (1) (1) (1)(1) (1)

1 1 1 1 1

( ) ([ ] )( )

( [ ]) ( ) ( )

v

v v v

t tt

t t t t t

VXZ

V Y LV V

1 2 1 1v (1) (2)( ) ( )dt Y t Z

0t

(2) (2)(1) (1)( ) ([ ] ) ( )t t t VX Y

(2) (1)( ) lim ( )t t

X X

79

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:

1 2,1 2L V

( ) ,f

v f D t ft

( ) / cos( / 2),D t t

3

( ) ( ( , )) ( ) ( , )S

f i s M ds f t

with µ0 the mean velocity on the small scale,

and v=[V]+ µ0

80

3

3

3

(2) (2)

(2)

0

2(2) (2) 1

( ) ( ) ( ) ( )

( ) ( )

( ) ( ) ( ) ( ( )) ( )

v

v

R

t

R

jk j kS

D t f t f t d

t f t d d

t D t F i s s s M dsD

x x y

x x y

D y x y y

y x y yD

y y

Main Result (cont.)

The fractional Fokker-Planck equation can be written in the divergence form:

where

Cushman and Moroni (2001):

where

1(2) (2) (2)( ) ( ) vf

f t ft

v D q

1(2) (2) (2) ( ) 1 2vvf t f q v D

81

References Cushman, J.H., Hu, X., Ginn, T.R. Nonequilibrium statistical mechanics of preasymptotic

dispersion. J. Stat. Phys., 75:859-878, 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 13;75-80, 2001

Moroni M., Cushman, J.H. Statistical mechanics with three-dimensional particle tracking velocimetry experiments in the study of anomalous dispersion. II. Experiments, Phys. Fluids;13: 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. 217:159-165.

• 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.

A U niversal Field Equation for Dispersive Processes

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