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University of Nevada
Reno
The Fractional Advection--Dispersion Equation:
Development and Application
A dissertation submitted in partial fulfillment of the
requirements for the degree of
Doctor of Philosophy in Hydrogeology
by
David Andrew Benson
Stephen W. Wheatcraft, Dissertation Advisor
May 1998
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E 1997, 1998
David Andrew Benson
All Rights Reserved
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i
The dissertation of David Andrew Benson is approved:
Dissertation Advisor
Department Chair
Dean, Graduate School
University of Nevada
Reno
May 1998
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Dedicated to my father, who taught me how to think about the
world,
and to my mother, who taught me how to live in it.
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ACKNOWLEDGEMENTS
No metaphysician ever felt the deficiency of language so much as
the grateful.
-- Charles Caleb Colton, Lacon
Oddly, I must first thank my Master’s advisor, David Huntley.
When I told him I was considering a Ph.D.,without hesitation he
told me to go to UNR and talk to Steve Wheatcraft. I have never
received more sageadvice. I went in March 1993, and I had the
strange and pleasant feeling that I was not only accepted to
theprogram, but I was being actively recruited. I must thank Dr.
John Warwick for his part in that feeling. I’malso glad to thank
Dr. Warwick for financial and philosophic help over the years. I
only wish we had beenin the same building.
From the beginning, Steve Wheatcraft has pushed but never
prodded, taught but never instructed, enthusedbut never gladhanded.
His humility is endless and his door is never closed. This
dissertation was clearlyoutside of my capabilities a few years ago,
but I believed in its virtue because he did. Nobody else on
earthwho could have planted this seed in my head and had it come to
fruition, so I thank him. One of his colleaguessays that every
professor should graduate a total of three Ph.D. students -- one to
continue his work, one toadvance the science, and one to replace
the teacher. I can only say that I am very lucky Steve didn’t
followthis piece of advice, since I am number nine.
The yeoman of my committee was Mark Meerschaert. There is no
question that this document would not existwithout his help. By
shear dumb luck I figured something out about hydrogeology and a
member of my com-mittee is an expert in that subject of
mathematics. I can’t decide whether to name my first child Levy
orMeerschaert. While on the subject of mathematics, I wish to
acknowledge the fantastic courses I took (ormerely sat in on) from
Jeff McGough. I learned more in those classes than any others I
took here at UNR.I hereby officially urge all students at UNR to
rely on the valuable resources in the form of Drs. Meerschaertand
McGough.
The other members of my committee -- Scott Tyler, Britt Jacobson
and Katherine McCall -- did many thingsfor me, not the least of
which was to remind me of all of the things that I don’t know or
understand. I appreci-ate the time they spent helping me.
I sincerely thank all of the students in the Hydrologic Sciences
program. First, the students maintain the highquality of the
program and make all of our degrees more valuable. Second, the
reputation of the programand hard work of the students bring the
best speakers in the world to our campus. I have gained very
muchfrom interaction with visiting speakers. Third, my interaction
with fellow students has added more refine-ment to the ideas
presented in this dissertation than could possibly come frommy
ownhead. Being mysound-ing board is an unenviable chore, so I give
special thanks to fellow students and colleagues Dr. Anne Carey,Dr.
Hongbin Zhan, Dr. Greg Pohll, Maria Dragila, and even Joe
Leising.
I thank the Desert Research Institute (DRI) and the generosity
of Elizabeth Stout for financial support in theform of the George
Burke Maxey fellowship. I also thank the U.S.G.S. and the Mackay
School of Minesfor their generosity in the form of scholarships.
Thanks also to Dave Prudic and Kathryn Hess at the U.S.G.S.for
delivering the Cape Cod data. DRI also paid my salary when I taught
the last hurrah of Geol 785 -- Ground-water Modeling. I’m sure I
learned more from the students -- Rina Schumer, Dave Decker, David
McGraw,and Marija Grabaznjak than I got across to them.
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Many of my old friends kept me in--touch (and in cheap digs!)
during this long process, so I thank Tom, Chris,Greggy and Strato,
Don (thanks for the deal on the scooter, yeah right!), Laura, and
John and Laura.
My mother is the smartest person I know. She should have been
the U.S. ambassador to the U.N., but shechose the difficult path of
being the mother of her children. Her calm and levelheaded support
through theyears has been inspirational. I hope I am able to give
one--tenth as much as I received. I must also shatterthe cliche and
thank my wife’s parents, Doug and Kathy Guinn, for their unflagging
encouragement. Theyare models of thinking, caring citizens.
Finally, I thank my wife Marnee for making so many sacrifices;
for leaving her dearest friends and the sunnybeaches of San
Clemente and postponing her own dreams of higher education. Your
time will come and Iwill remember.
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ABSTRACT
The traditional 2nd--order advection--dispersion equation (ADE)
does not adequately describe the movementof solute tracers in
aquifers. This study examines and re--derives the governing
equation. The analysis startswith a generalized notion of particle
movements, since the second--order equation is trying to impart
Brow-nian motion on a mathematical plume at any time. If particle
motions with long--range spatial correlationare more favored, then
the motion is described by Lévy’s family of α--stable densities.
The new governing(Fokker--Planck) equation of these motions is
similar to the ADE except that the order (α) of the highest
deriv-ative is fractional (e.g., the 1.65th derivative).
Fundamental solutions resemble the Gaussian except that theyspread
proportional to time1/α and have heavier tails. The order of the
fractional ADE (FADE) is shown tobe related to the aquifer velocity
autocorrelation function.
The FADE derived here is used to model three experiments with
improved results over traditional methods.The first experiment is
pure diffusion of high ionic strength CuSO4 into distilled water.
The second experi-ment is a one--dimensional tracer test in a
1--meter sandbox designed and constructed for minimum
hetero-geneity. The FADE, with a fractional derivative of order α =
1.55, nicely models the non--Fickian rate ofspreading and the heavy
tails often explained by reactions or multi--compartment
complexity. The final ex-periment is the U.S.G.S. bromide tracer
test in the Cape Cod aquifer. The order of the FADE is shown tobe
1.6. Unlike theories based on the traditional ADE, the FADE is a
“stand--alone” equation since the disper-sion coefficient is a
constant over all scales.
A numerical implementation is also developed to better handle
the nonideal initial conditions of the CapeCod test. The numerical
method promises to reduce the number of elements in a typical
numerical modelby orders--of--magnitude while maintaining
equivalent scale--dependent spreading that would normally becreated
by very fine realizations of the K field.
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TABLE OF CONTENTS
PAGE
ACKNOWLEDGEMENTS
..........................................................................................
iii
ABSTRACT
................................................................................................................
v
LIST OF FIGURES
......................................................................................................
viii
LIST OF TABLES
...................................................................................................
xi
CHAPTER
1 INTRODUCTION
......................................................................................................
11.1 Notation and Dimensions
.............................................................................
3
2 CLASSICAL THEORY
...........................................................................................
5
2.1 Advection--Dispersion Equation
...................................................................
52.2 Brownian Motion
.......................................................................................
92.3 The Diffusion Equation and Brownian Motion
.......................................... 10
3 STABLE LAWS
....................................................................................................
123.1 Characteristic Functions
.............................................................................
123.2 Stable Distributions (Stable Laws)
..............................................................
233.3 Moments and Quantiles
..............................................................................
28
4 PHYSICAL MODEL
.............................................................................................
20
4.1 Lévy Flights -- Discrete Time
......................................................................
21Lévy Flights -- Continuous Time
........................................................... 26
4.2 Lévy Walks -- Continuous Time Random Walks
......................................... 27Coupled Space--Time
Probability
.......................................................... 28
4.3 Velocity Statistical Properties
.....................................................................
36
5 THE FRACTIONAL ADVECTION--DISPERSION EQUATION
......................... 46
5.1 Fractional Fokker--Planck Equation
..................................................................
465.2 Solutions
..........................................................................................................
53
6 EXPERIMENTS
....................................................................................................
60
6.1 High Concentration Diffusion
.....................................................................
606.2 Laboratory--Scale Tracer Test
.....................................................................
646.3 Cape Cod Aquifer
.......................................................................................
67
A Posteriori Estimation of Parameters
................................................ 70Analytic
Solutions
..............................................................................
72A Priori Estimation of Parameters
...................................................... 75
7 NUMERICAL APPROXIMATIONS
...................................................................
78
7.1 Motivation
................................................................................................
787.2 Finite Differences
.....................................................................................
79
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8 DISCUSSION OF RESULTS
...............................................................................
84
9 CONCLUSIONS AND RECOMMENDATIONS
................................................. 90
9.1 Conclusions
..............................................................................................
909.2 Recommendations
....................................................................................
91
10 REFERENCES
....................................................................................................
92
APPENDICES
...........................................................................................................
96I FORTRAN LISTINGS
...................................................................................
96
I.1 Program SIMSAS.F
..................................................................................
96I.2 Program ENSEM.F
..................................................................................
98I.3 Program AVEGAM.F
...............................................................................
102I.4 Program WEIER.F
...................................................................................
107I.5 Subroutine CFASTD.F
.............................................................................
108I.6 Subroutine DFASTD.F
.............................................................................
111I.7 Program CVX.F
.......................................................................................
114I.8 Program CVT.F
........................................................................................
116I.9 Program FRACDISP.F
..............................................................................
118
II STABLE LÉVY MOTION CALCULATIONS
............................................... 121II.1 The Green’s
Function Chapman--Kolmogorov Equation for
random walks of random duration
....................................................... 121II.2
Exact Solutions for the transformed α--stable densities
....................... 122II.3 Calculation of power--law
transition density Fourier/Laplace
transforms
...........................................................................................
123
III VELOCITY AUTOCOVARIANCE OF LÉVY WALKS
................................ 128III.1 Velocity Autocovariance
for Lévy Walks with
Lower Cutoff
......................................................................................
128III.2 Lévy Walks with Converging Autocovariance
..................................... 131III.3 Autocovariance with
Velocity Proportional to
Lévy Walk Size
...................................................................................
132III.4 Full α--stable density
...........................................................................
132
IV FRACTIONAL DERIVATIVES AND THEIR PROPERTIES
....................... 134
V FINITE DIFFERENCE APPROXIMATION OF THEFRACTIONAL DERIVATIVE
.......................................................................
139
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LIST OF FIGURES
Figure 1.1 Schematic of the techniques used to obtain solutions
to generalizedrandom walks
............................................................................................
2
Figure 2.1 Illustration of the definition of the divergence of
solute flux over many scales.The solid lines denote assumption of
local homogeneity and multi--scale,integer--order (classical)
divergence. Dashed lines denote continuum--heterogene-ity and the
resulting noninteger--order divergence. To reconcile the growth in
theinteger divergence (using current theories) from scale a to b,
the first orderfluctuations v!C! are approximated by DoC with
increasing, spatially local D. ... 6
Figure 3.1 Plots of the distribution function F(x) versus x for
several standard symmetricα--stable distributions using a) linear
scaling and b) probability scaling. TheGaussian normal (α = 2.0)
plots as a straight line using probability scaling for thevertical
axis.
..................................................................................................
15
Figure 3.2 Plots of symmetric α--stable densities showing
power--law, “heavy” tailedcharacter. a) linear axes, and b)
log--log axes. ....................................................
17
Figure 3.3 Expectation of the absolute value of random variable
X with a standard,symmetric, α--stable distribution for 0 < α
< 2. ................................................. 19
Figure 4.1 Lévy flights in two dimensions.
........................................................................
23
Figure 4.2 Numerical approximation of one--dimensional,
continuous--time, randomLévy walks.
..................................................................................................
24
Figure 4.3 Graphs of the Fourier transform of the particle jump
probability (the structurefunction) of a “clustered” walk on a
discrete lattice. For this graph, the latticespacing (Δ) was set to
unit length. Note the good approximation of the completeWeierstrass
function by the exponential function for wave numbers smaller
thanthe inverse of the lattice spacing (i.e. k
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continuous source. For α < 2, the late--time slope on
log--log plots is equal to —α.(c) Half of the scaled instant pulse
breakthrough. For α < 2, the late--time slope onlog--log plots
is equal to —(1+α).
.....................................................................37
Figure 4.9 Graphs of a) the Lévy process, b) the velocity
function and c) joint probabilitydistribution of jump length as a
function of spatial separation. ......................... 38
Figure 4.10 Log--log and linear plots of the analytical and
numerical velocity semivariogramfunctions when the velocity is
modeled as proportional to Lévy walk size. Thenumerical result is
the ensemble mean of 112 realizations of 1000--jump walksusing a
stability index (α) of
1.7.........................................................................
41
Figure 4.11 Log--log and linear plots of the velocity
semivariogram for large and small valuesof ν. The value of α used
in all plots is 1.7. An exponential model, γ =1--exp(3.8ξ) is
plotted for comparison.
..............................................................
42
Figure 4.12 Plot of the scaling prefactor Pα for 1 < α <
2. ....................................................44
Figure 4.13 Maximum expected jump size in discrete standard
Gaussian versus near--GaussianLévy process with index of stability
(α = 1.99). ................................................ 45
Figure 5.1 Integer and fractional derivatives of two simple
power functions. Top row: Integerderivatives of f(x) = x2. Middle
row: Integer derivatives of g(x) = x2.33. Bottomrow: Fractional
derivatives around the point a=0 of g(x) = x2.33.
....................... 47
Figure 5.2 Comparison of the development of spatially symmetric
(dashed lines) andpositively skewed (solid lines) plumes
represented by a) continuous source and b)pulse source. Three
dimensionless elapsed times (0.1, 1.0, and 10) are shown. Asα gets
closer to 2, the skewing diminishes. All curves use α = 1.7 and D =
1.................................................. 54
Figure 6.1 Idealized schematic representation of diffusion via
random walk within a highionic strength, high gradient fluid. The
random walk occurs within a partially--occupied network. The
probability of a walk toward lower concentration (to theright of
the figure) is always higher than into higher concentration, where
moresites are occupied by other solute ions. At high enough
concentrations, the set ofconnected available sites is
non--Euclidean, precluding Fickian diffusion. ......... 61
Figure 6.2 Scaled diffusion profiles from Carey’s (1995)
experiment: a) scaled by thetraditional (Fickian) square root of
time, and b) scaled by time1/α with α = 2.5.The lower curves are
also shifted by a mean flux position of x = 1.97 cm.
..........62
Figure 6.3 Closeup of the low--concentration limb of the scaled
diffusion profiles fromCarey’s (1995) experiment: a) scaled by the
traditional (Fickian) square root oftime, and b) scaled by time1/α
with α = 2.5. The lower curves are also shifted by amean flux
position of x = 1.97 cm.
....................................................................63
Figure 6.4 Schematic view of the experimental sandbox tracer
tests. The flowpathhighlighted by the arrow is analyzed in detail.
...................................................64
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Figure 6.5 Calculated dispersivities versus distance of probe
from source. The flow pathchosen for analysis is shown with the
connecting line. The best--fit dashed lineindicates a fractional
dispersion index (α) of 1.55.
............................................. 66
Figure 6.6 Plot of normalized concentration versus scaled time
for probe 20, test 3 (Burns,1997). A best fit line (implying an
underlying Gaussian profile) is typically usedto calculate the
apparent dispersivity. Compare this data with the
α--stabletheoretical plots in Chapter 3 (Figure 3.1).
........................................................ 67
Figure 6.7 Measured breakthrough “tails” at probes along the
flowpath: a) Rescaled by t1/1.55,b) Rescaled by the traditional
t1/2. Note the strong skewness that separates theleading and
trailing limbs of the plume. Very early and late data show
probenoise.
...............................................................................................................68
Figure 6.8 Comparison of traditional and fractional ADEs with
the data from probe 3 (x = 55cm) in the sandbox test: a) real time,
and b) data tails. Note the large under--prediction of
concentration by the traditional ADE at very early and late time.
..69
Figure 6.9 Aerial view of the Cape Cod Br-- plume. The plume
deviated from travelling dueSouth by approximately 8_ to the East.
Circles are multi--level samplers (MLSs),diamonds are permeameter
core samples, and squares are flowmeter tests. ........ 70
Figure 6.10 Calculated plume variance (Garabedian et al. [1991])
along the direction of meantravel.
...............................................................................................................71
Figure 6.11 Simple analytic models of the Cape Cod plume in
1--D. Symbols are maximumconcentrations along plume centerline.
Solid lines are solutions to the FADEusing D = 0.14 m1.6/d and
classical ADE using asymptotic Fickian D = 0.42 m2/d.a) Early time
data. b) Late--time data. Sample times (in days) are shown
abovepeaks.
...............................................................................................................73
Figure 6.12 Simple analytic models of the Cape Cod plume in
1--D. Symbols are maximumconcentrations along plume centerline.
Solid lines are solutions to FADE andclassical ADE using identical
(early--time) dispersion coefficients of 0.13. a) Earlytime data.
b) Late--time data. Sample times (in days) are shown above peaks.
... 74
Figure 6.13 Semi--log plots of the plume profile modeled (solid
lines) and measured (symbols)at 349 days. (a) Maximum concentration
in the y--z plane and (b) average ofvertical samples from the same
MLS that from which the maximumconcentrations were measured. The
smaller average uses zero for non--detectableconcentration, while
the larger average ignores those data.
................................. 76
Figure 6.14 Theoretical dimensionless velocity semivariogram for
α = 1.4 and α = 1.8. ...... 77
Figure 7.1 Numerical solution of the FADE with α = 1.6 for a
series of times: a) log--logaxes, and b) linear axes. Initial
conditions were a “point source” of unit mass atthe node located at
x0. The solutions follow the scaling law of the analyticsolution.
Note the oscillatory error at the extreme tail ends.
..............................81
Figure 7.2 Comparison of analytic (lines) versus numerical
(symbols) solutions of the FADEwith “point source” initial
condition. In all solutions, D, t, and Δx set to unity. 82
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Figure 7.3 Numerical and analytic solutions of the FADE compared
to Cape Cod Br-- plume:a) linear axes, and b) semi--log axes.
Numerical model used Δx = 1.0 m and Δt =0.1 days. Both models used
α = 1.6 and D = 0.14. Note better fit of thenumerical solution at
13 and 55 days.
................................................................83
Figure 8.1 Comparison of the plume growth predicted by the
traditional ADE (ADE), Gelharand Axness (1983) (GA), the fractional
ADE (FADE), Mercado (1967) stratifiedflow (M), and Wheatcraft and
Tyler (1988) fractal tortuosity model (WT). Theordinate log(Xc2) is
roughly equivalent to estimated plume variance. The GAcurve has
slope 2:1 at a plume’s origin, transitioning to Fickian 1:1 slope
at latetime.
...............................................................................................................85
Figure 8.2 a) Possible values of the velocity parameter (dashed
lines) in Carey’s (1995)diffusion experiment. Probable particle
behavior as a function of increasingconcentration is shown by the
arrow. Variance exponent (VAR ∝ tη) for arrowpath α→ 2 predicted by
b) variance equation (4.51) and c) the propagatorequation (4.45).
......................................................................................87
Figure A2.1 Particle travel distance variance when mixed sk2
terms are included in the small--kapproximation of the
Laplace--Fourier transformed conditional Lévy walkprobability
p(r,t). The dashed line indicates results from simplified form used
byBlumen et al. (1989).
....................................................................................125
Figure A3.1 Pareto distribution with lower cutoff.
..............................................................
128
Figure A4.1 Plots of the functions Γ(x) and 1/Γ(x) for 0 < x
< 4. Note that n! = Γ(n+1). ....136
Figure A5.1 Flux as a power function of gradient. This is the
basis for the numericalapproximation implemented in Chapter 7.
....................................................... 142
LIST OF TABLES
Table 5.1 Error function SERFα(Z) of the symmetric stable
distributions for a range of αfrom 1.6 to 2.0.
......................................................................................56
Table 5.2 Error function SERFα(Z) of the symmetric stable
distributions for a range of αfrom 0.9 to 1.5.
.....................................................................................
58
Table A5.1 Comparison of analytic and numerical fractional
derivatives of power functions.The power function is f(x) = x--α--1
and the derivative is of order α--1. .............. 143
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CHAPTER 1
INTRODUCTION
When n is a positive integer and if p should be a function of x,
the ratio dnp to dxn
can always be expressed algebraically. Now it is asked: what
kind of ratio can bemade if n be a fraction? The difficulty in this
case can easily be understood. For ifn is a positive integer, dn
can be found by continued differentiation. Such a way, how-ever, is
not evident if n is a fraction. But the matter may be expedited
with the helpof the interpolation of series as explained earlier in
this dissertation.
-- Euler (1730)
This study examines the governing equation that is traditionally
used to model the movement of dissolvedsolutes in aquifers. The
classical governing equation is based on the diffusionequation,
whichuses thedefini-tion of divergence. In order for the equation
to be defined, a number of assumptions must remain valid. Pri-mary
among these is that the dispersion of particles due to differences
in velocity should exist as a controlvolume shrinks to zero. Since
the velocity fluctuations only arise from disparate aquifer
material at a scalethat is large compared to the observation or
measurement scale, the classical derivatives in the
advection--dis-persion equation are not well--defined. As a result,
the classical governing equation does not fully explainthe movement
of solutes, and the equation’s parameters are thought to “scale,”
or grow larger, with distance.A great deal of effort, in the form
of hundreds or thousands of articles, has been expended to explain
the scal-ing of parameters. Far less work has been done examining
the structure of the governing equation, especiallythe suitability
of the differential equation itself. At issue is the structure of
the second--order diffusion equa-tion used to model solute
spreading as a plume moves. This equation uses mathematical
operators whosehidden assumptions are violated when used to model
the macroscopic process of solute spreading.
An alternate approach to scaling the parameters is to
reformulate the governing equations. Since the equa-tions are
models of some underlying process, a good starting point is to
generalize the process of solute trans-port to include motions that
deviate significantly from the Brownian motion modeled by the
diffusion equa-tion. This study examines and generalizes the
underlying physical model used to derive the equations ofsolute
movement. The generalized motions lead to a new governing equation
that uses fractional--order, rath-er than the typical
integer--order, derivatives.
Solute transport in subsurface material also can be viewed as a
purely probabilistic problem. This viewpointis intimately tied to
the classical divergence (Eulerian) point of view through a string
of mathematical equiva-lences. Einstein (1908) first explored this
method by assuming that a single microscopicparticle
wascontinu-ously bombarded by other particles, resulting in a
random motion, or a random walk. By taking appropriatelimits
(letting nx and nt of the discrete walks go to zero), he found that
the resulting Green’s function ofthe probability of finding a
particle somewhere in space was a Gaussian (Normal) probability
density. TheGreen’s function solution is used to specifies an
initial condition of a single particle at the origin. If the
mo-tions of a large number of particles are assumed independent,
then the particle probability and the concentra-
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2
tion of a diffusing tracer are interchangeable. One can also
solve the parabolic “diffusion” equation Ct = Cxxand arrive at the
same Gaussian Green’s function for a “spike” of tracer placed at
the origin.
A series of uniqueness arguments leads to the conclusion that a
diffusion equation implies all of the assump-tions of Einstein’s
Brownian motion. The most important of these is that Brownian
motion implies that aparticle’s motion has little or no spatial
correlation, i.e. long walks in the same direction are rare. In
orderto use the ADE for spatially correlated velocity fields, a
correction is used that forces more dispersion thanthe diffusion
equation provides. This is the basis for a scale--dependent,
continuously evolving, effectivediffusion coefficient (the
dispersion tensor) used to better match the spread of real plumes.
Yet the underlyingequation is trying to impart a Gaussian profile
(the Green’s function) on a plume at any moment in time. Aquestion
naturally arises: What are the equations that describe particle
motions with long--range spatial cor-relations? The answer relies
on fractional calculus and a class of probability densities first
described by Lévy.These stable densities are a superset of the
familiar Gaussian and are often called α--stable or
Lévy--stable.
This study endeavors to do four important things. First, it is a
catalogue of many mathematical techniquesand concepts that are
relatively new to the field of hydrology. It is hoped that this
text can serve as a stand--alone repository of information related
to fractional calculus, Lévy’s Stable Laws, and current techniques
inrandom walk studies. For this reason, many derivations that can
be found elsewhere are included. Second,this text seeks to unify
the derivations from various fields of science and mathematics and
provide a standardset of symbology and notation useful to
hydrogeologists and others. A number of errors were found duringthe
translation and re--derivation process. Many articles refer to
erroneous prior results, making an indepen-dent trek through the
literature somewhat arduous. Corrections have been made along with
the “cataloguing”effort. Third, in the derivation of the governing
equation for particle movement in aquifers, this study hasattempted
to unify the techniques, concepts, and results of various prior
studies (Figure 1.2). By starting withan underlying model of random
particle movements that is a superset of traditional Brownian
motion, theend result is a generalization of the concept of
divergence. In the process, a new and correct derivation ofa
fractional ADE is made. Finally, the match between the new
theoretical models and experimental data is
Fokker--Planck(advection--dispersion)
Equation
Generalized random walks:Markov Process
Convolution
Fourier/Laplacetransforms
Instantaneousapproximation
Boundary value problemsLévy stable laws
Fourier/Laplacetransforms
Figure 1.2 Schematic of the techniques used to obtain solutions
to
generalized random walks.
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3
investigated. Some surprises are discovered here, when several
systems that are expected to yield classical,Gaussian behavior are
better described by the new approach.
Specifically, Chapter 2 is a more extensive review of classical
transport theory, including Brownian motion.Chapters 3 and 4 cover
Lévy’s α--stable Laws (probability distributions) and Lévy motions,
which are super-sets of the corresponding Normal Law and Brownian
motion. Using this generalized notion of random mo-tion, the
equations of solute transport are derived, starting with the most
basic assumption of particle transport-- that a future excursion is
unaffected by the previous journey (the Markov property). This
gives rise to theChapman--Kolmogorov equation of the space--time
evolution of a particle’s position probability.
Two different tacks are used to obtain solutions of the
Chapman--Kolmogorov equation (Figure 1.2). Thefirst uses the fact
that a convolution is present and transfers to Fourier/Laplace
space for solutions. The se-cond stays in real space and solves the
instantaneous change in probability resulting in a
(Fokker--Planck)differential equation. Similar to equations of
conservation of mass, the Fokker--Planck equation is a state-ment
of the conservation of probability of a single particle’s
whereabouts. Solutions to the partial differentialequation are most
easily gained via Fourier and Laplace transforms, so the two
methods end up in the sameplace. The two methods generalize the
notion of random walks, and rely on fractional calculus (Chapter
5)and the non--Gaussian (Lévy)α--stable laws. The solution space is
alsobriefly explored in Chapter5. Chapter6 examines two laboratory
and one field experiment to investigate the utility and validity of
the fractionalapproach. Chapter 7 contains an ad hoc numerical
implementation of the new fractional equation. Becausemany of the
theories in this dissertation are relatively new to the field of
hydrogeology, a large number ofrecommendations for future work are
listed in Chapter 9.
1.1 Notation and Dimensions
α stability exponent in Levy’s stable distributions (also order
of fractional space derivative).
β skewness parameter in Levy’s stable distributions: --1 ≤ β≤
1.
γ(h) semivariogram at a separation of (h).
Γ the Gamma function.
δ(x -- a) Dirac delta function centered at x = a.
Ô(t|r) conditional probability density of a particle transition
duration given the excursion length.
λ shorthand notation of 1 + α (see above).
μ shift parameter in Levy’s stable distributions.
η anomalous diffusion exponent.
Ó(h) autocorrelation function at a separation of (h).
σ scale (spread) parameter in Levy’s stable distributions.
ν exponent relating particle walk size and velocity.
ψ(k) characteristic function of a random variable X (i.e.,
E[eikX]).
Ò Pareto density lower cutoff.
τ mean particle transition duration (T).
ω order of fractional time derivative.
-
4
ξ separation distance in autocovariance functions.
aL longitudinal dispersivity (L).
A rate of change of a particle’s 1st moment (L T—ω).
B rate of change of a particle’s αth moment (Lα T—ω).
Dqa+ positive--direction fractional derivative of order q with
lower limit (a).
Dq+ positive--direction fractional derivative of order q with
lower limit (--").
Dqa− negative--direction fractional derivative of order q with
upper limit (a).
Dq− negative--direction fractional derivative of order q with
upper limit (").
D diffusion or dispersion tensor (Lα T—ω).
E() expectation of a random variable.
ERF(z) the error function.
F(f(x)) Fourier transform of function f(x).
Iqa+ fractional integral of order q integrating from a in the
positive direction.
k Fourier variable.
L(f(x)) Laplace transform of function f(x).
N the Gaussian Normal distribution.
Rvv(h) autocovariance of v at a separation of h (L2T--2).
s Laplace variable.
sign(k) sign of the variable (k) times unity (i.e., sign(k) =
--1 for k < 0, and 1 otherwise).
SERFα the α--stable error function.
t time.
v velocity (LT--1).
VAR() variance of a random variable.
expectation of a random variable.
d= equal in distribution (as in random variables).
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5
CHAPTER 2
REVIEW OF CLASSICAL THEORY
The difference between landscape and landscape is small; but
there is a great differ-ence between the beholders.
-- Ralph Waldo Emerson, Nature
2.1 Advection--Dispersion Equation
Nearly all current descriptions of solute transport make use of
the Advection--Dispersion Equation (ADE):
∂∂xi− viC + Dij ∂C∂xj = ∂C∂t (2.1)
where C is solute concentration, v and D are the velocity and
dispersion tensors (respectively), x is the spatialdomain and t is
time. The ADE is based on the classical definition of the
divergence of a vector field. Thedivergence is defined as the the
ratio of total flux through a closed surface to the volume enclosed
by thesurface when the volume shrinks toward zero (c.f., Schey
[1992]):
∇ ⋅ J ≡ limV→0
1V
S
J ⋅ ndS (2.2)
where J is a vector field, V is an arbitrary volume enclosed by
surface S, and n is a unit normal. Implicitin this equation is that
the limit of the integral exists, i.e. the vector function J exists
and is smooth as V !0. This is well suited to atomic force vectors
such as Maxwell’s equations of electromagnetic fields, sincethe
flux is indeed a “point” vector quantity. Conversely, solute
dispersion is primarily due to velocityfluctuations that arise only
as an observation space grows larger. The ADE is an implementation
of Gauss’Divergence Theorem using solute flux as the vector
function. In the ADE, J is replaced by the solute flux,so as an
arbitrary control volume shrinks, the ratio of total surface flux
to volume must converge to a singlevalue. The solute flux (J) is
due to the combined effects of mean velocity (advection) and
velocityfluctuations or variance (dispersion). The dispersive
fluxes for a given volume are averaged in some fashion(volumetric,
statistical) and usually approximated by a process using Fick’s
first Law, i.e. J = vC -- DoC.Since velocity itself is a variable
function of space, as a control volume shrinks, the velocity
fluctuations
disappear and the dispersive flux shrinks to zero. Thus, if one
uses the definition of divergence in (2.2), theflux cannot contain
a dispersive term (except perhaps for molecular diffusion, which is
generally negligible).In mathematical terms, the classical
divergence of solute flux reduces to:
-
6
∇ ⋅ (vC − D∇C) ≡ limV→0
1V
S
(vC − D∇C) ⋅ n dS =
limV→0
1V
S
(vC) ⋅ n dS = ∇ ⋅ (vC)
(2.3)
In this setting, the classical divergence theorem is of little
use in subsurface hydrology since the boundaryvalue problem for
o¡(vC) = --#C/#t is infinitely complex. Because of this complexity,
a de facto definitionof divergence has long been used to quantify
advection and dispersion. The divergence is associated witha finite
volume and is given by the first derivative of total surface flux
to volume (Figure 2.1). The dispersioncoefficient tensor does not
grow (scale) if the ratio of surface flux to volume is constant
over some range ofvolume (solid lines in Figure 2.1). An example is
a column of uniform glass beads. At the pore scale, theratio is
non--constant and no constant dispersion parameter can be assigned.
At some larger scale, the ratio
VOLUME
S
(vC + v′C′) ⋅ ndS
0
slope = div(vC--v′C′)
0
div(vC)
div(vC -- DaoC)
1VS
(vC + v′C′) ⋅ ndS
aa
Figure 2.1 Illustration of the definition of the divergence of
solute flux over many scales.The solid lines denote assumption of
local homogeneity and multiscale, integer--order(classical)
divergence. Dashed lines denote continuum--heterogeneity and the
resultingnoninteger--order divergence. To reconcile the growth in
the integer divergence (usingcurrent theories) from scale a to b,
the first order fluctuations v!C! are approximated byDoC with
increasing, spatially local D.
div(vC -- DboC)
bb
SUR
FAC
EFL
UX
VO
LU
ME
TR
ICa) b)
NOTE: SURFACE FLUX =NOTE: VOLUMETRICSURFACE FLUX =
first
VOLUME
derivative
SUR
FAC
EFL
UX
localhomogeneity
-
7
of total surface flux to volume is constant over a large range
of arbitrary volumes and the first derivative (thede facto
divergence) is relatively constant. Solute flux within
heterogeneous aquifers violates this principlebecause increases in
an arbitrary volume result in a growing amount of dispersive flux.
The first derivativeof surface flux to arbitrary volume is not
constant when a travelling solute plume samples more of the
velocityvariations.
When an integer--order divergence is assumed, the ratio of
surface flux to volume is forced to take on aconstant value over
some volumetric range. This action approximates the monotonically
increasing ratio ofsurface flux to volume by a step function
(Figure 2.1b). An effective parameter (D) with scaling propertiesis
used to account for the fact that the de facto divergence is
ill--defined in continually evolving heterogeneity.The parameter D
is intimately tied to a specific volume, and the ADE is no longer
self--contained with aclosed--form solution for all scales.
Estimation techniques include small perturbation solution of a
linearizedstochastic ADE and substitution of a local effective
parameter D into the ADE for a specific plume size(Gelhar and
Axness [1983], Dagan [1984]). More recent suggestions include
simple power lawmultiplication of D (Su [1995]), but this leads to
an equation that is not dimensionally correct. Thesesolutions
suffer primarily from using a special case (integer--order
divergence) for a more general problem.The first derivative of the
surface flux to volume (Figure 2.1a) is not constant, i.e. the
first derivative doesnot account for continuous growth of the
surface flux to volume ratio as a plume grows in
heterogeneousmedia. A more robust description of the volumetric
surface flux growth will be constant over a greater range,perhaps
even the entire range expected for a plume’s lifetime. Not only is
the dispersion coefficient tied tothe smallest control volume
scale, but the scale of measurement (due to concentration
averaging) as well.The scale of measurement must be much larger
than the scale of heterogeneity in order for the relative sizeof
the control volume to approach zero. This is why the Fickian
approximation works well in certaininstances. An approach that
integrates the measurement scale would also be desirable. For these
reasons,the description of solute transport is better suited for
fractional derivatives.
Similar arguments also apply to purely statistical treatments of
the dispersive fluxes, in which the CentralLimit Theorem (CLT)
suggests Gaussian (Fickian) dispersion when the scale of
measurement includes a largenumber of independent, finite--variance
velocities (c.f., Bhattacharya and Gupta [1990]). The
Gaussianvelocity distribution suggests that the probability
distribution of travel times can be modelled by a Markovprocess
with standard Brownian motion. This yields (through the Kolmogorov
forward equation) aFokker--Planck equation of the solute particle
probability and therefore concentration. The assumption ofa large
scale compared to the velocity fluctuations fulfills the CLT and
gives a dispersion tensor that isasymptotically fixed and Fickian
dispersion ensues.
As an interesting aside, the average velocity is considered to
(roughly) change from an arithmetic to ageometric mean as the
amount of heterogeneity encountered increases. This also is a scale
effect that ariseswhen hydraulic conductivity appears as a
parameter in the Poisson Equation (for an overview
ofscale--dependent mean flow, see Gelhar [1993]). The derivatives
of velocity are not smooth and are tied anirreducible finite size,
or “representative elementary volume” (REV). A fractional
derivative approach to thegroundwater flow problem is suggested for
further study.
The classic mode of operation with the integer--order derivative
formulation (2.1) is to estimate the valuesof the parameters within
the ADE at any particular point in a plume’s history. The
parameters (D and to alesser extent v) are thought to change as the
plume size changes. This is due to the fact that the
autocorrelationlengths within the velocity field are large compared
to the scale of measurement. In order to have a predictive
-
8
tool for plume behavior, several theories have arisen to
estimate values of “effective” parameters, so that theADE recreates
the observed plume moments. These include:
S Volume and statistical averaging (e.g., Gray [1975]; Cushman
[1984]);
S Techniques based on a small--perturbation, stochastic
differential equation (Gelhar andAxness [1983]; Dagan [1984];
Neuman and Zhang [1990]);
S Power--Law Dispersivity growth via empirical ADE (e.g., Su
[1995]) in which Deff = xsD,with s = empirical constant;
S Purely statistical formulation using Kolmogorov forward
equation with simple velocityfunction (Bhattacharya and Gupta
[1990]).
These methods suffer from various drawbacks due to their
inherent assumptions. The first method, volumeaveraging, invokes a
hierarchy of distinct scales wherein the averaging length scale is
much larger than thescale of perturbation. In other words, the
averaged quantity is composed of a relatively homogeneouscollection
of smaller (perturbed) quantities. An example is the laboratory
scale being much larger than thepore scale. These methods are not
valid at the “in--between” scales or in smoothly--varying
heterogeneity.
The spectral methods rely on a linearized stochastic ADE that
cannot predict spreading when the velocitycontrasts are large.
Typically ln(v) is given by a Gaussian normal with a variance less
than unity. Thiscondition limits the application of this technique
to relatively homogeneous aquifers. The power--lawdispersivity
growth (bullet #3 above) does not yield a parameter that is
dimensionally correct, thus theparameter does not have a sound
physical basis. Moreover, this represents an unjustified, empirical
additionof another parameter into the “governing” equation.
Berkowitz and Scher (1995) demonstrate that atime--dependent
dispersion tensor is also unsound. Purely statistical methods make
broad assumptions aboutthe functional basis of a velocity field.
Clearly, the work dedicated to evaluating an “effective”
parameterhas lost sight of where the actual scaling occurs within
(2.1). First, smooth integer derivatives of the fluxdo not exist in
natural porous media. Second, dispersion cannot be considered a
point flux.
If we looked at how the divergence is defined for solute flux in
porous material, we might start with a plotof the total surface
flux versus volume for an arbitrary volume at the leading edge of a
plume (Figure 2.1).As the volume goes to zero, the surface flux is
a real number, and the slope of the line at the origin is o⋅(vC).As
the size of the arbitrary volume increases, so does the total
surface flux. If the medium is homogeneousover some scale, then the
slope of the line (the ratio of surface flux to volume) is a
constant (solid lines inFigure 2.1). Within that scale, the
dispersion is Fickian and one can assign a divergence of the flux
accordingto o⋅(vC -- DoC). Within that length scale range, the
divergence is associated with an arbitrary and finitevolume. Since
the ratio of surface flux to volume is constant over the range, the
value of D appliescontinuously throughout the range. If homogeneity
is present in several distinct stages, then the dispersiveflux at
all smaller scales are averaged into the effective dispersion
coefficient at the largest measured scale.Because the slope is
constant within a distinct scale, the first derivative of the
surface flux with respect tovolume (not as the volume approaches
zero) is used as a de facto definition of divergence.
Typically, plumes at the field scale are in a pre--Fickian stage
where an increase in the size of an arbitraryvolume (or measurement
size) encloses material with different velocity. This leads to a
non--constant ratioof dispersive flux to volume (the curved, dashed
line in Figure 2.1a) and an apparent increase in the“divergence”
(dashed line in Figure 2.1b). Since many analytic solutions already
exist to the classical ADE,it has been advantageous to assume that
the non--constant volumetric surface flux can be approximated
by
-
9
a step function wherein each rise is described by a growing D.
When an effective parameter D is derivedthrough volume or
statistical averaging, it is only valid at that particular volume
(or scale). Further increasesin the slope of the dispersive flux
(increasing scale) require a new D value. This simply arises
because thefirst spatial derivative of the dispersive flux (which
defines the de facto divergence) is not constant. Ratherthan assume
a step function exists and force D to take on increasing values,
one might assume that describingthe evolving dashed curves in
Figure 2.1b would more accurately replicate plume histories and
give apredictive tool as well. The mathematical tools of fractional
calculus are better suited to describing the curvesin Figure 2.1b
than the classical (integer--order) divergence. This will be
demonstrated in Chapter 5.
The classical ADE is based on the the diffusion equation, which
is linked to an underlying physical orprobabilistic model of
particle movement. It is instructive to analyze that link before
generalizing the notionof particle movements and seeking the
governing equation of these generalized movements. The
physicalbasis of the diffusion equation is well known to be
Brownian motion.
2.2 Brownian Motion
There are several ways to construct a Brownian motion in one or
more dimensions. The first and mostintuitive way is to restrict the
motions to a regular lattice so that a particle can move in only
one directionduring each jump. The probability of moving to an
adjacent lattice location is always equally distributed.In one
dimension, the position of a particle at time t is a random
variable given by X(t). If the distance to thenext lattice point is
nx and the time spent in transit is nt, then
X(t) = Δx(X1 + + X[t∕Δt]) (2.4)
where Xi =+ 1 if the ith step is forward− 1 if the ith step is
backward
and [t/nt] is the largest integer ≤ t/nt. The probability that
Xi = +1 is equal to the probability that Xi = --1,which is 1/2 for
symmetric walks.
Denote the expectation of a random variable E[X] and the
variance VAR[X]. Since E[Xi] = 0 and VAR[Xi]= E[(Xi)2] = 1, E[X(t)]
= 0 and VAR[X(t)] = nx2(t/nt). Now the limit must be carefully
defined as nx andnt go to zero. If nx and nt simply go to zero,
then VAR[X(t)] converges to zero. If, however,
Δx∕(Δt)1∕2 = c, with (c) a positive constant, then E[X(t)] = 0
and VAR[X(t)] = c2t. By the Central LimitTheorem, as the number of
jumps becomes large (i.e. let the increments become very small),
X(t) is a Normalrandom variable with zero mean and variance
c2t.
Brownian motion is characterized by its independent increments.
Since each jump is independent of theprevious jump, for all t1 <
t2 < < tn the increments X(tn) − X(tn−1), X(tn−1) − X(tn−2),
... ,X(t2) − X(t1), and X(t1) are also independent and stationary,
since the variance of any increment dependsonly on the interval,
not on time. The density function for the random variable X(t) is
given by
ft(x) = 12Õc2t
e−x2∕2c2t (2.5)
Each increment of finite size X(t + z) -- X(t), where z is a
finite constant, is composed of infinitely manysmaller jumps. The
increment itself is therefore a Normal random variable with zero
mean and variance of
-
10
c2z. It is easily seen that a Brownian motion is an addition of
successive increments that are themselvesindependent, identically
distributed (iid) random variables. These variables also have the
important featureof finite variance. So the limiting distribution
of the sum of a large number of iid finite--variance
randomvariables is the Normal distribution. The variance of the sum
of independent variables is the sum of theindividual variances.
The term Standard Brownian motion, given the symbol B(t) refers
to a Brownian Motion with unit (c). AnyBrownian motion can be
related to the standard by B(t) = X(t)/c.
2.3 The Diffusion Equation and Brownian Motion
Several methods are used to relate the diffusion equation and
Brownian motion. Solutions to 2--variablepartial differential
equations can be facilitated by integral transform in order to
remove dependance on oneof the independent variables.
Throughout this text, the Fourier transform F and its inverse
F--1 are defined as:
F(f(x)) = f~(k) =
∞
−∞
e−ikxf(x)dx (2.1)
F−1(f~(k)) = f(x) = 1
2Õ∞
−∞
eikxf~(k)dx (2.2)
The pair of functions f(x) and f~(k) are unique. Each function
uniquely implies the other. The change of
variable from x → k implies Fourier transform throughout this
text.
The Fourier transform of the diffusion equation Ct = DCxx with
respect to the space variable is:
∞
−∞
e−ikx ∂C∂t dx = ∞
−∞
e−ikxD ∂2C∂x2
dx (2.3)
If C and its derivatives vanish as |x| ! ∞, than integration by
parts twice gives
ddt∞
−∞
e−ikxCdx = − k2DC~ (2.4)
dC~
dt= − k2DC
~ (2.5)
where the tilde indicates the Fourier transformed function.
Given a Dirac delta function initial condition:
C(t = 0, x) = δ(x − 0) (2.6)
C~(t = 0, k) = 1 (2.7)
gives the Gaussian (Normal) density for the Green’s function
solution:
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11
C~(k, t) = exp(− k2Dt) (2.6)
With inverse transform:
C(x, t) = 12ÕDt
exp(− x2∕2Dt) (2.7)
The width of the concentration profile (the distance between two
concentration percentiles) is equal to (Dt)1/2.The Green’s function
of the diffusion equation is identical to the solution for Brownian
motion where D =c2 = Δx2/Δt. Since Fourier transform pairs are
unique, the diffusion equations implies Brownian motion asan
underlying probabilistic model.
Another method of relating the diffusion equation and a Brownian
motion relies on the fact that thedifferential displacement of
particles dX(t) = X(t + dt) − X(t) is Gaussian and satisfies Ito’s
stochasticdifferential equation (Bhattacharya and Gupta [1990])
:
dX(t) = f ⋅ dt + g ⋅ dB(t) (2.8)
In one or more dimensions, f represents the drift of the
process, or the mean velocity vector. The functiong is a constant
tensor of the standard deviation of the Gaussian process X(t). This
process satisfies theFokker--Planck equation of the “flow” of
probability in time and space:
∂P∂t =
∂∂x (− f ⋅ P) +
∂2∂x2
(g ⋅ P) (2.9)
If many particles are simultaneously released and do not affect
each other, the probability and concentrationare interchanged to
give the ADE. One can simplify the problem further by describing a
mean--removedequation that follows a moving frame of reference that
travels at the mean velocity. The diffusion equationis
recovered.
If the underlying physical model described above is altered to
allow a higher probability of long--rangeparticle transitions, then
the 2nd--order diffusion equation is no longer the governing
equation of those walks.The link between Brownian motion, the
2nd--order diffusion equation and its Gaussian fundamental
solutionis generalized to “Lévy motion,” a fractional--order
equation, and fundamental solutions that are superset ofthe
Gaussian. These superset probability densities are covered in the
next Chapter.
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12
CHAPTER 3
STABLE LAWS
All things are difficult before they are easy.
-- Thomas Fuller, Gnomologia
3.1 Characteristic Functions
The properties of many probability distributions are more easily
investigated in terms of their characteristicfunction. The
characteristic function is a description of the Fourier transform
of the probability density func-tion. (Actually it is more akin to
the reverse Fourier transform, but this is merely a
re--parameterization.)Also useful is the moment generating
function, similar to the Laplace--transformed density.
The characteristic function ψ of a random variable X with a
density f(x) is given by E[eikX] where E(⋅) isthe expectation:
E(eikX) = ψ(k) = ∞
–∞
eikxf(x)dx (3.1)
The Fourier transform of the density is closely related to the
characteristic function by f^(− k) = ψ(k).
The uniqueness of Fourier transform pairs guarantees that the
characteristic function defines the density andvice--versa. Unless
noted otherwise, the Fourier transforms in this study will place
the constant 1/2Õ on theinverse transform to more closely resemble
the characteristic function.
For positive domain distribution functions, the one--sided
Laplace transform (moment generating function)is useful:
E(esX) = Ô(s) = ∞
0
esxf(x)dx (3.2)
Integration by parts gives the Laplace transform of a cumulative
distribution function F(x):
Ô(s)s =
∞
0
esxF(x)dx (3.3)
Inversion of the characteristic functions follows the inverse
Fourier transform:
f(x) = 12Õ∞
–∞
e−ikxψ(k)dk (3.4)
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13
Or in Laplace space:
f(x) = 12Õiγ+i∞
γ–i∞
e−sxÔ(s)ds (3.5)
where γ is a real number greater than the real component of any
singularities in the function Ô(s).
3.2 Stable Distributions (Stable Laws)Chapter 2 contained a
demonstration that a Brownian motion can be created by a sum of
independent, identi-cally distributed (iid) Normal random
variables. It is intuitive that a sum of iid Normal variables would
keepthe same distribution after dividing by a normalizing constant.
One might wonder if sums of random vari-ables with other
distributions maintain the distributions of the individual
summands. A large family of thesedistributions were shown to exist
by Paul Lévy in 1924. The Normal distribution is merely a member
ofLévy’s family of stable distributions. Lévy’s relevant and
oft--cited work (1924; 1937) has not been translatedinto English.
Lucid summaries and extensions are provided by Feller (1966),
Zolotarev (1986), Samorod-nitsky and Taqqu (1994), and Janicki and
Weron (1994).
Lévy’s distributions arise when describing a “stable” sum that
is distributed identically to the summands.It is easiest to use
shifted, or zero--mean random variables, so a scaled sum of (n)
zero--mean iid random vari-ables is:
Sn =X1 + X2 + + Xn
cn(3.6)
A number of assumptions about the probability functions are
omitted for clarity. See Feller (1966, Ch. XVIIand others) or
Körner (1988, Ch. 50) for more complete development.
The characteristic function of a sum of two independent
variables X1 and X2 is given by
E(eik(X1+X2)) = E(eikX1eikX2) = ψX1(k)ψX2(k)(3.7)
In a similar manner, the characteristic function of the sum of a
sequence of iid Xn is simply (ψX(k))n. We canalso calculate the
characteristic function of the expectation of a scaled and shifted
variable aY + b:
E(eik(aY+b)) = E(eikaY ⋅ eikb) = eibkψY(ak) (3.8)
Now the scaled sum cnSn in (3.6) can be related to the density
of the iid variables Xn:
ψcnSn(k) = ψSn(cnk) = (ψX(k))n (3.9)
Equating the characteristic functions ψX and ψSn and taking
logarithms:
logψ(cnk) = n logψ(k) (3.10)
This equality is fulfilled by a power law:
logψ(k) = Akα (3.11)
where A is a constant and the value of the exponent α is limited
to 0≤α≤2 (Feller [1966]).
The equality in (3.10) will only be true when cnα = n, or cn =
n1/α. With these constraints, the scaled sumand summands are
identically distributed. The result is a generalization of the
Central Limit Theorem forsum of n random variables (X) that are
iid:
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14
Sn d=X1 + X2 + + Xn
n1∕α(3.12)
An entire family of distributions that includes the Gaussian is
described when the value of the exponent αranges from 0 < α≤ 2
(Feller [1966, Ch. XVII]). The constant A can be complex
(indicating skewness),and the variable can have non--zero mean, so
the characteristic functions of these α--stable distributions
takethe general form (Samorodnitsky and Taqqu [1994]):
ψ(k) = exp(–|k|ασα 1–iβsign(k) tan(Õα∕2) + iμk) α ≠ 1 (3.13)
where the parameters σ, β andμ describes the spread, the
skewness and the location of the density, respective-ly. The
sign(k) function is --1 for k < 0 and 1 otherwise. The
characteristic function for α = 1 (the Cauchydistribution) is
slightly different and will not be listed for clarity.
When the density is symmetric, the skewness parameter (β) is
zero, and the symmetric characteristic functionis:
ψ(k) = exp(− σα|k|α+ iμk) (3.14)
A standard α--stable density function has unit “spread” and is
centered on the origin, so σ = 1 and μ = 0. Itis a simple matter to
show that forα > 1, E(X) = μ. The mean is undefined forα≤ 1. A
standard, symmetricα--stable distribution (SSαS) is characterized
by the compact formula:
ψ(k) = exp(− |k|α) (3.15)
In this form it is easy to see that the Gaussian (Normal)
density is α--stable with α = 2. Note, however, thatwhen the scale
factor of the stable law σ = 1, the standard deviation of the
Normal (α = 2) distribution (N)
is 2 :
N(k) = exp− 2σ2k2 + iμk (3.16)
The most important feature of the α--stable distributions (3.13)
is the characteristic exponent (also called theindex of stability)
α. The value of α determines how “non--Gaussian” a particular
density becomes. As thevalue of α decreases from a maximum of 2,
more of the probability density shifts toward the tails. Figure3.1
shows the standard α--stable distribution functions for α= 1.6,
1.8, 1.9, and 2. Note that the distributionsappear very Gaussian in
untransformed coordinates, and that the difference lies in the
relative weight presentin the tails. For probabilities between 1
and 99 percent, the different distributions appear
near--normal.
Non--standard (σ≠1 andμ≠ 0) stable distribution functions (F)
and densities (f) are related to their standardcounterparts by the
relations:
Fαβ(x, σ, μ) = Fαβ(x − μ)σ , 1, 0 (3.17)
fαβ(x, σ, μ) =1σ fαβ(x − μ)σ , 1, 0 (3.18)
Cauchy and Lévy sought closed--form formulas for the stable
densities (in real, not Fourier, space) for allvalues of α. They
found that direct inversion of the characteristic function ψ(k) is
only possible when α =½, 1, or 2. A number of accurate
approximations are available for other values. Sinceψ(k) is known
exactly,
-
15
99.9
90
99
70
50
30
10
1
0.1--10 --5 5 100
α = 2.0 1.9 1.8 1.6
PRO
BA
BIL
ITY
(PE
RC
EN
T)
Figure 3.1 Plots of the distribution function F(x) versus x for
several standard sym-metric α--stable distributions using a) linear
scaling and b) probability scaling. TheGaussian normal (α= 2.0)
plots as a straight line using probability scaling for the
ver-tical axis.
x
--10 --5 5 100
PRO
BA
BIL
ITY
(PE
RC
EN
T)
x
0
20
40
60
80
100
(a)
(b)
-
16
a fast numerical Fourier inversion can yield accurate densities.
The Fourier inversion formula also has manyreal--valued integral
representations that yield quick numerical solutions (c.f.,
McCulloch [1994, 1996]; Zo-lotarev [1986]). In particular,
McCulloch (1996) gives the integral representation of the standard
forms forσ =1 and μ = 0 of the cumulative probability function
(Fαβ):
Fαβ(x) = C(α, Ò) +sign(1 − α)
2
1
−Ò
exp− x* αα−1 Uα(Ô, Ò)dÔ (3.19)
where
x* = c*x
c* = 1 + β tan(Õα∕2)2−1
2α
Ò = 2Õα tan−1β tan(Õα∕2)
C(α, Ò) = 1, α > 1(1 − Ò)∕2, α< 1Uα(Ô, Ò) = sin Õ2 α(Ô+
Ò)cos Õ2Ô
α1−α
The densities are obtained by differentiating the cumulative
probabilities with respect to x. Note McCul-loch’s (1996) mistaken
standard density (fαβ) that should read:
fαβ(x) =x*
1α−1αc*
2|1 − α| 1
−Ò
Uα(Ô, Ò)exp− |x*| αα−1 U(Ô, Ò)dÔ (3.20)
Equations (3.19) and (3.20) were coded using a simple
trapezoidal rule to return values of the distribution(Figure 3.1)
and the density (Figure 3.2) for various values of α. Listings of
the FORTRAN subroutines(DFASTD.F and CFASTD.F) are given in
Appendix I.
Several series expansions of the standard α--stable densities
are listed in readily--available recent documents(c.f., Feller
[1966]; Nikias and Shao [1995]; Janicki and Weron [1994]).
Bergstrom (1952) and Feller arecredited with independently deriving
similar expansions. Feller (1966) also gives series expansions for
aslightly different parameterization, using γ to quantify the
skewness, rather than β:
fαγ(x) = 1Õx∞
k=0
Γ(kα+ 1)k! (
− x)−kα sin Õk2 (γ− α) 0 ≤ α < 1(3.21)
fαγ(x) = 1Õx∞
k=0
Γ(kα−1 + 1)k! (
− x)k sin Õk2α (γ− α) 1 < α ≤ 2(3.22)
Feller’s skewness parameter γ is obtained by equating the
canonical form (3.13) to his equivalent representa-tion of the
standard characteristic function:
-
17
10--1
10--2
10--3
10--4
10--5
10--6
10--7
100
10 100110--1
Figure 3.2 Plots of symmetric α--stable densities showing
power--law “heavy” tailedcharacter. a) linear axes, and b) log--log
axes.
1.8
1.4
1.2
1.6
2.0 (Gaussian)
--5.0 --3.0 --1.0 1.0 3.0 5.00.0
0.2
0.4
α = 2.0(Gaussian)
α = 1.4
(a)
x − μσ
σ ⋅ fαx − μσ α =
σ ⋅ fαx − μσ
(b)
-
18
ψ(k) = exp− |k|αeiÕ(sign k)γ∕2 (3.23)
Resulting in (Samorodnitsky and Taqqu [1994]):
γ =
− 2Õ arctan(β tan(Õα∕2)) 0 < α< 1
2Õ arctan(β tan(Õ(α− 2)∕2)) 1 < α < 2
(3.24)
For symmetric densities, setting γ=0 in (3.22) yields a formula
that converges with reasonably few terms evenwith large arguments.
This expansion will be used throughout this study:
fα(x) = 1Õ∞
k=0
(− 1)k(2k + 1)! Γ
2k + 1α + 1x2k 1 < α ≤ 2 (3.25)Portions of this study require
a generator of random variables that share anα--stable
distribution. Janicki andWeron (1994) gives an algorithm for
generating a standard, symmetric stable random variate X based on
auniform random variable V on (--Õ/2,Õ/2) and an exponential
variable W with unit mean:
X = sin(αV)(cos(V))1∕α
⋅ cos(V − αV)W
(1−α)∕α (3.26)3.3 Moments and Quantiles
It is interesting to note the behavior of the α--stable
densities in the large x limit. The simplest characteristicfunction
of an α--stable random variable can be approximated for small k
(large x) by ψ(k) = exp(--|k|α) ≈1 -- |k|α, with an inverse
transform f(x)≈ Cx--1--α. Figure 3.2b is a log--log plot of the
positive half of severalof the α--stable densities, clearly showing
the power--law tail behavior. The Gaussian density lacks the
pow-er--law tail, although theα--stable family represents a
continuum. Asα approaches 2, the power--lawbehavioronly becomes
evident at very large values of |x|. It has been shown that the
power--law tail behavior is presentwith any values of α, σ and β
(Samorodnitsky and Taqqu [1994]).
The moments of a distribution with density f(x) can be defined
by the integral
μr ≡ ∞
−∞
xrf(x)dx (3.27)
The first several integer moments are historically those that
are studied in physical sciences. In order for amoment to exist,
the integral (3.27) must converge. Since at least one of the tails
of any α--stable densityfollows a power law for large |x| (Feller
[1966]), we can integrate (3.27) to check for convergence. At
the
tails we have lim|x|→∞
C|x|r−α which is finite only if r < α. So the moments higher
than the real number α do
not exist for these distributions. In particular, the variance
and standard deviation are undefined for all α--stable
distributions except the Gaussian, when α = 2.
The infinite variance of α--stable laws can aid in their
detection. The calculated variance of a series of α--stable random
variables will not tend to converge using standard variance
estimators. The failure of this esti-
-
19
mator to converge will require a large population as α grows
closer to 2, since the probability of extreme val-ues is only
slightly larger than predicted by the Normal distribution.
The fact that the variance of an α--stable random variable is
infinite does not preclude measurement of the“spread” of the
density of the variable. Nikias and Shao (1995) advocate the use of
fractional moments (anyrth moment with r < α). Janicki and Weron
(1994) use quantiles of the distribution when investigating
thespread of an α--stable process. The quantiles qp are defined
here as F--1(p) with its pair F--1(1--p), where pis a desired
probability. Thus for a random variable X with distribution F(x),
the quantiles qp are the pointsx where F(x) = p and F(x) = 1--p.
The probabilities 0.159 and 0.841 are typically used for the Normal
distribu-tion since these numbers correspond to the mean one
standard deviation. These quantiles will be usedin this study for
convenience.
Another useful formula gives the value of the moments of order
less than α. Nikias and Shao (1995) showthat the fractional
lower--order moments of a symmetric SαS variable X are calculated
by the formula:
E(|X|r) =2r+1Γr+12 Γ(− r∕α)
α Õ Γ(− r∕2)σr (3.28)
In particular, the expectation of the absolute value of X (r =
1) reduces to
E(|X|) =2Γ(1 − 1∕α)
Õ σ(3.29)
Figure 3.3 is a plot of E(|X|) for 1
-
20
CHAPTER 4
PHYSICAL MODEL
The process of irregular motion which we have to conceive of as
the heat--content ofa substance will operate in such a manner that
the single molecules of a liquid willalter their positions in the
most irregular manner thinkable.
-- Einstein (1908)
The stochastic process of Brownian motion was reviewed in
Chapter 2. Brownian motion is a continuoustime random walk (CTRW)
with Gaussian increments that is also the limit process
ofuncorrelated, unit jumpson a lattice. The movement of a particle
in aquifer material clearly does not follow Brownian motion
sincegeologic material is deposited in continuous, correlated
units. A particle travelling faster than the mean atsome instant is
much more likely to still be travelling faster than the mean some
later time due to the spatialautocorrelation of aquifer hydraulic
conductivity. The same is true for particles travelling slower then
themean velocity. This suggests that particle excursions that
deviate significantly from the mean are much morelikely than
traditional Brownian motion can model. This Chapter examines
another (superset) model of par-ticle random walks that
accommodates these large deviations from the mean particle
trajectory.
Many other random physical processes are characterized by
extreme and/or persistent behavior (the Josephand Noah effects
coined by Mandelbrot and Wallis [1968]) for which Brownian motion
is an inadequate mod-el. Notable among these is the dispersion of a
passive scalar in near--turbulent (chaotic) and turbulent flow(see
Klafter et al. [1996] for a survey). In these flows, a particle
tends to spend long periods of time trappedin vortices that are
essentially stagnant with respect to mean flow. Mixing within a
vortex may in fact followBrownian motion, but a particle can
occasionally escape and travel with high velocity “jets” between
vortices(Weeks, et al. [1995]). These relatively rare, high
velocity events represent a heavier--tailed probability
dis-tribution for the particle excursions. These particle motions
are described by Lévy flights and Lévy walks,which are similar to
Brownian motion but differ in the probability distribution of the
jumps. Rather thanhaving Gaussian increments, they have Lévy’s
α--stable, or power--law (Pareto) distribution increments. Itis
instructive to analyze how these Lévy motions can arise as a limit
process of jumps on a lattice, just as wasdone with Brownian motion
in Chapter 2. This analysis leads to a more general model of random
walks ona lattice and provides a link between the memory of a
fractional derivative and the memoryless property ofrandom walks
and Markov processes.
The particle “propagator” describes the probability of finding a
particle somewhere in space at some time.Solving the equations for
the propagator start with the mathematical representation of a
single particle re-leased at the Cartesian origin. This propagator
is a surrogate for concentration if it represents a large
quantityof independent solute “particles.” Since a contaminant mass
placed in an aquifer is composed of a huge num-ber of these
particles, the propagator density is “filled in” by the solute
particles. So the task of deriving agoverning equation for the
movement of an instantaneously released slug of solute tracer is
reduced to solvingthe equations for the propagator. This
equivalence often will be used in this study.
-
21
It is instructive to follow the development of this propagator,
starting again with random jumps on a lattice.With prior knowledge
of the α--stable distributions, one might expect that a particle
will be given a higherpropensity to make longer excursions than a
particle experiencing Brownian motion. If thewalks areuncorre-lated
with respect to time, they must be spatially correlated in order to
embark on these longer walks. Brow-nian motion’s unit walks on a
lattice are uncorrelated in space (although it will be shown that a
Brownianmotion defined by Gaussian increments has some very
short--range spatial correlation). Thus the differencebetween
Brownian motion, and its superset Lévy motion, is the range of
spatial correlation. This is shownin Section 4.3. The first two
Sections (4.1 and 4.2) are primarily a review of current theories
that are applica-ble to solute transport, with minor corrections to
the originals where indicated. The final Section (4.3) alsoincludes
a new derivation of the statistical properties of a particle
undergoing Lévy walks to enable estimationof certain parameters
from aquifer characteristics.
4.1 Lévy Flights -- Discrete Time
Hughes et al. (1981) describe a random walk on a
one--dimensional infinite lattice. The probability of findingthe
particle at lattice position j after n jumps is denoted Pn(j). Let
p(m) be the probability of jumping m latticesites during a single
step. The Markov property of jump independence dictates that the
probability of findingthe walker at site j at the next step is the
sum of the transition probabilities from all other lattice sites
(j′) multi-plied by the probability of being at those sites. This
is stated mathematically in the Chapman--Kolmogorovequation:
Pn+1(j) = ∞
j′=−∞p(j − j′)Pn(j′) (4.1)
This is a convolution, so the Fourier--transformed probabilities
are used where:
P~
n(k) = ∞
j=−∞eikjPn(j) (4.2)
p~(k) = ∞
m=−∞eikmp(m) (4.3)
The Fourier transformed walk probability p~(k) is know as the
“structure function.” Montroll and Weiss(1965) solve the
convolution with a Green’s function equation, i.e. using an initial
condition that a walk start-ing at the origin has a delta function
initial probability: P0(i) = δ(i--0). The resulting probability
Pn(l) is calledthe “propagator” since it describes the n--step
spatial evolution of a single event at the origin. By inductionand
the definition of convolution, (4.1) and the initial condition
gives:
P~
n(k) = (p~(k))n (4.4)
The inverse transform gives the spatial probability density of a
walker after n steps:
Pn(j) = 12Õ2Õ
0
e−ikj(p(k))ndk (4.5)
-
22
A Brownian motion must also be described by (4.4). One way to
recover Brownian motion is to restrict par-ticle movement to 1
lattice position in either direction of the current (mth) position
using the Dirac delta func-tion distribution (Hughes et al.
[1981]):
p(m) = 12
(δ(m + 1) + δ(m − 1)) (4.6)
The transformed probability is:
p~(k) = eik + e−ik
2= cos k (4.7)
When the number of transitions becomes large (n→∞),
P~(k∕ n ) = (p~(k∕ n ))n = cos(k∕ n )n = 1 − k22n + O(1∕n)
n
≈ exp− k22 (4.8)Fourier inversion gives the Gaussian
profile:
Pn(j∕ n ) ≈ 12Õ2Õ
0
exp(− ikj) exp− k22dk = 1
2Õexp(− j2∕2) (4.9)
A change to the real variable x = j/(n1/2) results, for large n,
in the approximation
Pn(x) = 12Õn
exp(− x2∕2n) (4.10)
which is normal with zero mean and variance (n). Another more
general way to generate the Brownian motion
is to define each jump by a random variable with finite variance
m2 where ⋅ denotes expectation:
m2 = ∞
l=−∞m2p(m) (4.11)
The resulting limiting value of the transformed step probability
is Gaussian:
p~(k) ≈ 1 − m2 k22≈ exp− m2 k2
2 (4.12)
And the overall trajectory density of an individual walker with
any finite--variance transition probability is
Pn(j) = 12Õm2n
exp − j22m2n (4.13)
The distinguishing characteristic of all Brownian motions is
held in (4.6) and (4.11). If the distance that arandom walker
instantaneously travels has finite variance, then the random walk
is asymptotically Gaussian.A number of researchers have
investigated the limit process that results when each jump is 1)
random withinfinite variance, and/or 2) not instantaneous. Both of
these modifications lead to models that simply andconcisely
describe a wealth of macroscopic, non--Gaussian processes (Bouchard
[1995]). These real--worldphenomena include faster--than Fickian
dispersion (often called superdiffusion) in chaotic to turbulent
flow(Shlesinger, et al. [1987]; Weeks, at al. [1995]), structure of
DNA (Stanley, et al. [1995]), movement of
-
23
charges in semiconductors (c.f., Geisel [1995] and references
within), and quantum Hamiltonian systems(Zaslavsky [1994a]).
Consider a random walk in which each particle jump has a travel
distance probability that isα--stable. Thesewalks would favor
larger deviations from the mean because of the tail--heavy density
of the distributions.A symmetric jump probability has a purely real
Fourier transform:
p~(k) = exp(− σα|k|α) (4.14)
And the probability propagator is also an α--stable
distribution, by virtue of its characteristic function:
P~
n(k) = exp(− nσα|k|α) ⇔ P~
n(k∕n1∕α) = exp(− σα|k|α) (4.15)
These random motions are named Lévy flights since the particles
are instantaneously moved from point topoint. These random
diffusion paths have infinite variance. Figure 4.1 shows a series
of points in two dimen-sions that follow a Lévy flight. The
distance of each flight is an α--stable random variable and the
directionis a uniform random [0,2Õ] variable. (In this case, the
direction changes were limited to multiples of Õ/2 toshow simulated
movement on an orthogonal lattice). The left plot (Figure 4.1)
shows a Lévy flight with acharacteristic exponent of 1.9, which is
not too dissimilar to a Brownian motion (α=2.0). The rightmost
plotuses a more tail--heavy distribution with α=1.7. Note that
these flights are characterized by clusters that areseparated by
relatively infrequent, long--distance jumps. The set of turning
points is a random fractal, andmagnification of the clusters shows
the same clustered behavior on increasingly smaller scales. Figure
4.2shows the cumulative displacement of a one--dimensional walker
undergoing Lévy flights with exponentsof α=1.7 and 1.9. The
variance of both processes are infinite, but the “spread” of these
processes can be ana-lyzed using quantiles. Note that within the
definition of the Levy flights, a “particle” does not visit the
pointsin space between turning points, i.e. the connecting lines in
Figures 4.1 and 4.2. Rather, a particle is instantlymoved from
turning point to turning point.
--50 0 50
α = 1.9
--50
--50--100
0 50 100 150
α = 1.7
Figure 4.1 Lévy flights in two dimensions.
50
0
100
150
200
-
24
--50
50
150
100
0
20001000500 1500
Figure 4.2 Numerical approximation of one--dimensional,
continuous--time, random Lévy walks.
α=1.7α=1.9
X(t)
t
The symmetricα--stable random variables used to create random
walks were generated using equation (3.26)and identical seeds for
the random number generator, hence the similarity of the traces
using value of α of1.7 and 1.9 (Figures 4.1 and 4.2). A listing of
the FORTRAN code (SIMSAS.F) used to generate the Levyflights is
given in Appendix I.
It was shown (4.6) -- (4.13) that a single--lattice jump becomes
a Brownian motion (i.e. a Bernoulli processbecomes a Gaussian
process) as the number of jumps or trials becomes large. Hughes et
al. (1981) describea simple clustering symmetric walk that
approximates a Lévy process in the same way. They use jumps ona
lattice in which the probability of travelling a step of length x
is
p(x) = a − 12a∞n=0
a−n(δ(x − Δbn) + δ(x + Δbn)) (4.16)
where b2 > a > 1 and $ > 0 is the lattice spacing. If b
is an integer, then jumps of size 1, b, b2 ... bn latticepositions
are allowed. The larger jumps are less likely by a factor of a--n.
On average, a cluster of (a) jumpsof length 1 are linked by a jump
of length b. About (a) of these clusters are linked by a jump of
length b2,and so on in a classically fractal manner. The structure
function of this walk probability is (Hughes, et al.[1981]):
p~(k) = a − 1a ∞
n=0a−ncos(Δbnk) (4.17)
which is the everywhere--continuous, nowhere (integer)
differentiable, self--similar Weierstrass function.This function
has been shown to be fractionally differentiable up to order
ln(a)/ln(b) (Kolwankar and Gangal[1996]). A quick check also shows
that this density has an infinite variance.
Equation 4.16 follows the scaling relationship
-
25
p~(k) = 1a p~(bk) + a − 1a cos(Δk)
(4.18)
Hughes et al. (1981) show that the asymptotic (k → 0) behavior
is satisfied by an exponential function:
p~(k) ≈ exp(− C|k|α) where α= ln(a)ln(b) and C = −(Δα∕τ) Õ2
Γ(α) sin(Õα∕2)(4.19)
In fact, for wave numbers smaller than that of the lattice
spacing, the exponential function that describes anα--stable
transition density matches quite well (Figure 4.3). By the
Tauberian Theorem (Feller [1966]), thetails of the densities (4.17)
and (4.19) are identical. Note that Δα/τ is a constant, analogous
to the constant(Δx)2/t defined for Brownian motion.
The probability propagator is also asymptotically (in the domain
of attraction of) an α--stable density, sinceits Fourier transform
follows:
P~
n(k∕n1∕α) = p~(k∕n1∕α)n≈ exp(− C|k|α) (4.20)
which indicates that the propagator is asymptotically invariant
after scaling by n1/α.
0.0 5.0 10.0--0.5
0.0
0.5
1.0
wave number (k)
p(k)
Weierstrass Function (4.17)
Exponential (4.19)
Figure 4.3. Graphs of the Fourier transform of the particle jump
probability (thestructure function) of a “clustered” walk on a
discrete lattice. For this graph, the latticespacing (Δ) was set to
unit length. Note the good approximation of the completeWeierstrass
function by the exponential function for wave numbers smaller than
theinverse of the lattice spacing (i.e. k
-
26
Levy Flights -- Continuous TimeTo generalize this random walk to
non--integer jump sizes and real--valued time, first allow jumps of
non--in-teger size, i.e. b∈9 > 1. An integral replaces the
summation for the n--step propagator:
Pn+1(x) = ∞
−∞
p(x − x′)Pn(x′)dx′ (4.21)
If each jump takes an equal amount of time (τ) to complete, then
the change in probability over a single jumpis
Pn+1(x) − Pn(x)τ =
∞
−∞
1τ (p(x − x′) − δ(x − x′))Pn(x′)dx′
(4.22)
Noting that the limit as τ→ 0 is the time derivative of the
propagator, then the motion can be generalizedto a continuous time
function. If we denote the time of the nth step as t, we have
∂P(x, t)∂t =
∞
−∞
limτ→0
1τ (p(x − x′) − δ(x − x′))P(x′, t)dx′
(4.23)
The integral now requires that the step sizes dx′ are
infinitesimally small, which requires that the lattice spac-ingΔ→
0. The rate at which the spacing shrinks must depend on how τ→ 0.
In Fourier space, the last resultbecomes
∂P~ (k, t)∂t = limτ,Δ→0
1τ (p
~(k) − 1))P~ (k, t) (4.24)
Two options arise for the transition density, finite or infinite
variance. The finite variance case should recoverBrownian motion
(i.e. solve the diffusion equation). This requires taking the
limitsΔ and τ so that the expres-sion is not trivially 0 or
infinity, i.e. Δ2/τ is a constant. The finite--variance structure
function where eachjump is $j in size is now
p~(k) ≈ 1 − Δ2j2 k22
(4.25)
leading to the evolution of the propagator:
∂P~ (k, t)∂t = limτ,Δ→0
− Δ2j22τP~ (k, t) ≡ − DP~(k, t) (4.26)
which is the Fourier transform of the integer--order diffusion
equation with a diffusion coefficient definedby
D = limτ,Δ→0
Δ2j22τ
(4.27)
To generalize the walk, Hughes et al. (1981) use the structure
function for infinite variance walks. The limitsof Δ and τ are once
again taken so that zero or infinity do not result, i.e. $α/τ =
constant. Now the structurefunction is approximated by 1 -- C|$k|α,
so
-
27
limτ,Δ→0
− Cτ |Δk|
αP(k, t) = limτ,Δ→0
− CΔα
τ |k|αP(k, t) (4.28)
And the probability propagator follows the equation
∂P~ (k, t)∂t = − Dα|k|
αP~(k, t) (4.29)
and is therefore α--stable:
P~(k, t) = exp(− Dαt|k|α) (4.30)
Hughes at al. (1981) use the full structure function equation
(not listed here for simplicity) to derive the lastequation with a
complete description of the diffusion coefficient for Lévy
flights:
Dα = limΔ,τ→0
Δατ ⋅
Õ2Γ(α) sin(αÕ∕2)
(4.31)
The velocity of a particle undergoing the continuous time random
walks (including Brownian motion) have
infinite velocity (v) whenα>1, since v = limΔ,τ→0
Δτ = limΔ,τ→0
Δα ⋅ Δ1−ατ = limΔ→0