1/121 南京市 nanjing city. 2/121 河海大学 -hohai university
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1/121南京市 Nanjing City
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河海大学 -Hohai University
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College of Civil Engineering in Hohai University
Basic facts of the College (largest in our university):• Close to 200 staffs• 4 departments (“civil engineering”, “survey & ma
pping”, “earth sciences & engineering”, “engineering mechanics”)
• 1 department-scale institute (geotechnic institute)• Around 3000 undergraduate students + 1000 grad
uate students
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Modeling of “Anomalous” Behaviors
of Soft Matter
Wen Chen （陈文）Institute of Soft Matter Mechanics
Hohai University, Nanjing, China
3 September 2007
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Soft matter ？• Soft matters, also known as complex fluids,
behave unlike ideal solids and fluids.
• Mesoscopic macromolecule rather than microscopic elementary particles play a more important role.
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Typical soft matters
Granular materialsColloids, liquid crystals, emulsions, foams, Polymers, textiles, rubber, glassRock layers, sediments, oil, soil, DNA Multiphase fluidsBiopolymers and biological materials
highly deformable, porous, thermal fluctuations play major role, highly unstable
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Lattice in ideal solids
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Polymer macromolecules: fractal mesostructures
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Fractured microstructures
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Soft Matter Physics
Pierre-Gilles de Gennes proposed the term in
his Nobel acceptance speech in 1991.
widely viewed as the beginning of the soft matter science.
_ P. G. De Gennes
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Why soft matter?
• Universal in nature, living beings, daily life, industries.
• Research is emerging and growing fast, and some journals focusing on soft matter, and reports in Nature & Science.
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Engineering applications
• Acoustic wave propagation in soft matter, anti-seismic damper in building ， geophysics ， vibration and noise in express train ；
• Biomechanics ， heat and diffusion in textiles, mechanics of colloids, emulsions, foams, polymers, glass, etc ；
• Energy absorption of soft matter in structural safety involving explosion and impact ；
• Constitutive relationships of soil, layered rocks, etc.
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Challenges
• Mostly phenomenological and empirical models, inexplicit physical mechanisms, often many parameters without clear physical significance;
• Computationally very expensive;• Few cross-disciplinary research, less emph
asis on common framework and problems.
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Characteristic behaviors of soft matter
• “Gradient laws” cease to work, e.g., elastic Hookean law, Fickian diffusion, Fourier heat conduction, Newtonian viscoustiy, Ohlm law;
• Power law phenomena, entropy effect;• Non-Gaussian non-white noise, non-Markovian
process;• In essence, history- and path-dependency, long-
range correlation.
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More features (courtesy to N. Pan)
Very slow internal dynamicsHighly unstable system equilibriumNonlinearity and frictionEntropy significant
a jammed colloid system, a pile of sand,
a polymer gel, or a folding protein.
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Major modeling approaches
• Fractal (multifractal), fractional calculus, Hausdorff derivative, (nonlinear model?);
• Levy statistics, stretched Gaussian, fractional Brownian motion, Continuous time random walk;
• Nonextensive Tsallis entropy, Tsallis distribution.
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Typical “anomalous” (complex) behaviors
• “Anomalous” diffusion （ heat conduction, seepage, electron transport, diffusion, etc. ）
• Frequency-dependent dissipation of vibration, acoustics, electromagnetic wave propagation.
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• Basic postulates of mechanics.conservation of mass, momentum and energy
• Basic concepts of mechanicsstress, strain, energy and entropic elasticity
• Constitutive relations and initial–boundary-value problems.
Mechanics of Soft Matter
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Outline:
• Part I: Progresses and problems – a personal view
• Part II: Our works in recent five years
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What in Part I?• Field and experimental observations
• Statistical descriptions
• Mathematical physics modelings
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Field and experimental observations
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Anomalous electronic transport
Non-dissipation Normal dissipation
Anormalous dissipation
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The absorption of many materials and tissues obeys a frequency-dependent power law
0
yf
Courtesy of Prof. Thomas Szabo
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Statistical descriptions
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Anomalous diffusion
＝ 1, Normal (Brownian) diffusion 1, Anomalous ( >1 superdiffusion ，
<1 subdiffusion)
tx 2
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Random walks
Left: Brownian motion; Right: Levy flight
with the same number (7000) of steps.
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Levy self-similar random walks
A characteristic Levy walk
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Stretched Gaussian Distribution
2
4
4),(
2
d
Dtx
Dt
etxP
)1
exp(),(2
2
xa
xtxP
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Measured probability density of changes of the wind speed over 4 sec
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Stretched Gaussian diffusion:
Gaussian diffusion:
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Levy stable distribution
11
ˆ1
,t
xp
ttxp
dkeep kiq
2
1ˆ tx
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Two cases of Levy distributions
Gaussian (=2)
Cauchy distribution (=1)
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Tsallis distribution (nonequlibrium system)
qxqZ
xp 1
1
111
)(
N
i
qiq p
q
ks
1
)1(1
N
iii pps
1
ln
Tsallis non-extensive entropy
Boltzmann-Gibbs entropy
Tsallis distribution
1q
Max s
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Tsallis distribution cases
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A. Komnik, J. Harting, H.J. Herrmann
A comparison of diverse distributions
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Progresses in statistical descriptions
• Continuous time random walk, fractional Brownian motion, Levy walk, Levy flight;
• Levy distribution, stretched Gaussian, Tsallis distribution.
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Problems in statistical descriptions
• Relationship and difference between Levy distribution, stretched Gaussian, and Tsallis distribution?
• Calculus corresponding to stretched Gaussian and Tsallis distrbiution?
• Infinite moment of Levy distribution?
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Partial differential equation modeling
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Anomalous diffusion equation in fractional derivatives
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Physics behind “normal diffusion”
• Darcy’s law (granular flow)• Fourier heat conduction law• Fick’s law• Ohlm law
ukJ
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ukt
u
• Continuity:
• Fick diffusion:
Jt
u
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Nonlinear Modelings
1 numk u
k
Multirelaxation models, nonlinear models; varied models for different media with quite a few parameters having no explicit physical significance. For instance, nonlinear power law fluids:
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Anomalous diffusion equation in Fractional calculus
0 ptp
10 10
Master equation (phenomenological)
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Fractional time derivative in Fourier domain
PipDFT
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Physical significances
• Histroy dependency (memory, non-Markovian) corresponding to fractional Brownian motion.
• Singular Volterra integral equation.• Numerical truncation is risky!
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Operation case ： 2121
21 2t
dt
td
Ap
pdt
tpd
0
021
21
terfcAetp tSolution:
Equation case ：
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Weierstrass function (differentiability order 0.5)
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Constitutive relationships
• Hookian law in ideal solids:
• Ideal Newtonian fluids:
• Newtonian 2nd law for rigid solids:
• One model of soft matter:
kxF
2
2
dt
xdmF
y
uF
t
xF
20
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Numerical fractional time derivative
• Volterra integral equation ；• Finite difference formulation ： Grunwald-Let
nikov definition ；• “Short memory” approach (truncation and st
ability)• Something new ？
I. Podlubny, Fractional Differential equation, Academic Press, 1999
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Numerical fractional space derivative
• Full numerical discretization matrix ；• Boundary condition treatments ；• Fast algorithm (e.g., fast multipole method).
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Progresses in PDE modeling
• Fractional time derivative, fractional Laplacian;
• Hausdorff derivative;
• Growing PDE models in various areas.
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Problems in PDE modelings
• Relationship and difference between fractional calculus and Hausdorff derivative?
• Fractional time and space modelings?• Computing cost• Nonlinear vs. fractional modeling;• Physical foundation of phenomenological modelin
gs
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Fractional vs. Nonlinear systems
• History dependency • Global interaction• Fewer physical parameters (simple= beautiful)
• Competition or complementary
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Part II: Our works
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Summary• New definition of fractional Laplacian; • Introduction of positive fractional time derivative, and mo
dified Szabo dissipative wave equations; • Mathematical physics explanation of [0,2] frequency powe
r dependency via Levy statistics;• Fractal time-space transforms underlying “anomalous” phy
sical behaviors, and two hypotheses concerning the effect of fractal time-space fabric on physical behaviors,
• Introduction of Hausdorff fractal derivative;• Fractional derivative modeling of turbulence.
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Positive fractional time derivative
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Definitions based on Fourier transform
Positive fractional derivative ： PpDFT
Fractional derivative ： PipDFT
Positivity requirments in modeling of dissipation
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Definitions in time domain
,21,1
1
,10,1
0 1
2
0
1
dt
pD
q
dt
pD
qpD
t
t
21cos12
q
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New definition of fractional Laplacian
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Definitions via Fourier transform
PkpF 2
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Traditional definition in space
Samko et al. 1993. Fractional Integrals and Derivives: Theory and Applications
d
x
pxp
dp d 2
1
0< <1
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Our definition
dSx
N
xnDhxpxp
S dd
2222*
1
0< <1
Journal of Acoustic Society of America,Journal of Acoustic Society of America, 115(4), 1424-1430, 2004 115(4), 1424-1430, 2004
MeritsMerits ：：• Weak vs. strong singularityWeak vs. strong singularity ，，• Accurate vs. approximateAccurate vs. approximate ，，• Finite domain with boundary conditions vs. infinite domainFinite domain with boundary conditions vs. infinite domain
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Fractional derivative modelings of frequency-dependent dissipative medical ultra
sonic wave propagation
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Imaging Comparisons
Type Mechanism Safety Speed Access Cost
X-Ray
oA5.15.
Ion. Rad.
still 2 sides Through
$
CT oA5.15.
Ion. Rad.
still 360 deg Through
$$$
MRI RF Precess In H,1mm
Excel- lent
fast 360 deg Through
$$$$
Ultra-sound
mm5.11. Mechanical
Very Good
very fast
Small footprintpulse- echo
$$
Courtesy of Prof. Thomas Szabo
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Medical ultrasoundImaging (sonography) and ablating the objects inside human body for medical diagnosis and therapy.
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xexSxxS
y 0
2,0 0
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Conventional nonlinear and multirelaxation models ：• Material-dependent models ；• Quite a few artifical (non-physical) parameters, in
essence, empirical and semi-empirical models.
Our fractional calculus models ：• Few phyiscally explicit parameters ，• Parameters available from experimental data f
itting.
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Time-space wave equations of integer-order partial derivative only exist for y=0, 2
Thermoviscous wave equation (y=2):
pt
ct
p
cp
002
2
20
21
t
p
ct
p
cp
0
02
2
20
21
Damped wave equation (y=0):
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Linear positive time fractional derivative wave equation
pDct
p
cp y 1
0
02
2
20
21
pDpDppD 220 ,
pDpDDpDDpD yyyy 1111
where
Note
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Linear fractional Laplacian wave equation for arbitrary frequency dependency
ptct
p
cp
y
2
10
02
2
20
21
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脂肪 肌肉 肝脏水
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Dr. Richter’s New clinical approach
• stabilize each deformable breast between two plates,
• detect breast cancer via speed change & attenuation.
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Courtesy of Prof. Thomas Szabo
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3D configuration
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Levy Statistical explanation of
frequency dependent power 2,0
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“Anomalous” diffusion equation for frequency dependent dissipation
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Phenomenological master equation
02 ptp
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Fourier transform of probability density function of Lévy -stable distribution is the characteristic function of s
olution of “anomalous” diffusion equation ：
)exp( kkW
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To satisfy the positive probability density function, the Lévy stable index must obey
20 1) In terms of Lévy statisticsLévy statistics, the media having >2 power law attenuation are not statistically stable in nature;
2) =0 is simply an ideal approximation.
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Fractal time-space transforms, Hausdorff derivative, fractional quantum
and phonon
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Perplexing issues in anomalous diffusion
• Levy stable process and fractional Brownian motion• The mean square displacement dependence on time
tx 22
2
1mv
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Fractional (fractal) time-space transforms
.
,
tt
xx tx 2
.
,1
tvt
xv
x
Special relativity transforms:
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Two hypotheses for “anomalous” physical processes• The hypothesis of fractal invariance: the laws of physic
s are invariant regardless of the fractal metric spacetime.
• The hypothesis of fractal equivalence: the influence of anomalous environmental fluctuations on physical behav
iors equals that of the fractal time-space transforms.
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Fractional quantum in complex fluids
Fractional quantum relationships between energy and frequency, momentum and wavenumber （ fractional Schrodinger equation ）
hE ˆ
xV
m
h
thei
2
ˆˆ
22
khp ˆ
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Fractional phonon and vibrational absorption energy spectrum?
hE ˆ
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Hausdorff derivative under fractal
t
tg
tt
tgtglim
t
tgtt ˆ
ˆ
dt
dx
td
xdv
ˆˆ
ˆGeneralized velocity:
uDt
u Hausdorff derivative diHausdorff derivative di
ffusion equationffusion equation :
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Statistical and Reynolds equation modelings of turbulence via fractional derivative
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Kolmogorov -5/3 scaling of turbulence
• Validation in sufficiently high Reynolds number tubulence• Narrow spectrum of -5/3 scaling in finite Reynolds
number turbulence, i.e., intermittency (non-Gaussian distribution)
3532 kCkE
tu 2 322 ru
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k
E(k
)
100 101 102
10-7
10-6
10-5
10-4
10-3
10-2
10-1
100
k-5/3
k-3
y+=501
Energy spectrum (obtained from Kim ， Moin’s DNS database)
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Turbulence distribution ： Gauss vs. Levy
Nature, 409, 1017–1019, 2001 Levy distribution
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Plasma turbulenceOak Ridge National Laboratory
Power law decay of Levy distribution
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Richardson superdiffusion
031
Pt
P Fractional Laplace Fractional Laplace
statistical equationstatistical equation:
Richardson diffusion consistent with KoRichardson diffusion consistent with Kolmogorov scalinglmogorov scaling
22 tCr 32
31)2( C
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Fractional derivative Reynolds equation
jij
ij
ij
i uux
upx
uu
t
u ~~1
uuuu
p
t
1Navier-Stokes equation
Reynolds decomposition uuu ppp
Reynolds equation
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ijij
uuux
31~~
Fractional Reynolds equation ：
uuuuu 31
31Re
ˆ
Re
1
pEtt
St
Three order of magnitude: turbulence vs. molecule viscousity
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Relevant Publications• W. Chen, S. Holm, Modified Szabo’s wave equation models for lossy
media obeying frequency power law, J. Acoustic Society of America, 2570-2574, 114(5), 2003.
• W. Chen, S. Holm, Fractional Laplacian time-space models for linear and nonlinear lossy media exhibiting arbitrary frequency dependency, J. Acoustic Society of America, 115(4), 1424-1430, 2004.
• W. Chen, Lévy stable distribution and [0,2] power law dependence of acoustic absorption on frequency in various lossy media, Chinese Physics Letters ， 22(10) ， 2601-03, 2005.
• W. Chen, Time-space fabric underlying anomalous diffusion, Soliton, Fractal, & Chaos, 28(4), 923-929, 2006. .
• W. Chen,. A speculative study of 2/3-order fractional Laplacian modeling of turbulence: Some thoughts and conjectures, Chaos, (in press), 2006.
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Keywords:
• Geophysics, bioinformatics, soft matter, porous media
• frequency dependency, power law
• Fractal, microstructures, self-similarity,
• Fractional calculus (Abel integral equation; Volterra integral equation)
• Entropy & irreversibility
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Thinking future?
• Phenomenological models & physics mechanisms of soft matter ；
• Time-space mesostructures and statistical models ；• Numerical solution of fractional calculus equation
s ；• Verification and validation of models and enginee
ring applications.
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A Journal Proposal
• Title: Journal of Power Laws and Fractional Dynamics
• Publisher: Springer
• Proposers: W. Chen, J. A. T. Machado, Y. Chen
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Research issues covered in this journal
• Empirical and theoretical models of a variety of “anomalous” behaviors characteristic in power law such as history-dependent process, frequency-dependent dissipation etc.;
• Novel physical concepts, mathematical modeling approaches and their applications such as fractional calculus, Levy statistics, fractional Brownian motion, 1/f noise, non-extensive Tsallis entropy, continuous time random walk, dissipative particle dynamics, etc.;
• Numerical algorithms to solve the relevant modeling equations, which often involve non-local time-space integro-differential operators;
• Real-world applications in all engineering and scientific branches such as mechanics, electricity, chemistry, biology, economics, control, robotic, image and signal processing.
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Something else
• Non-stationary data processing
• Large-scale multivariate scattered data processing (radial basis functions)
• Meshfree computing and software (e.g., high wavenumber acoustics and vibration)
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Scattered 3D geologicial data reconstruction
471,031 scattered data made by U.S. Geological Survey
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Difficulties in simulation of high-dimensional, high wavenumber and frequency
• Ultrasonics(1-100MHz),microwaves 0.1GHz-100GHz ，seismics ；
• High wavenumber for 2D problems N>100 ， 3D problems N>20 ；
• FEM requires at least 12 points in each wavenumber
Wavenumber:Wavenumber:c
LLN
)O((20N)d
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2D Helmholtz (k=802D Helmholtz (k=80 ，， N=160) problemN=160) problem
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Multiple-edged outdoor noise barrier design
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2D case200 millions DOFs matrix for 2D FEM engineering precision; 30000 full matrix & 13.5Gb st
orage for the standard BEM
SS Langdon, Lecture notes on “Finite element methods for acoustic scattering”, July 11, 2005Langdon, Lecture notes on “Finite element methods for acoustic scattering”, July 11, 2005S. Chandler-Wilde & S. Langdon, Lecture notes on “Boundary element methods for acoustics”, July 19, 2005S. Chandler-Wilde & S. Langdon, Lecture notes on “Boundary element methods for acoustics”, July 19, 2005
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Advantages
• 1/100 storage ， 1/1000 computing cost of the BEM;• High accuracy, simple program, non numerical int
egration, meshfree, suitable for inverse problems ；• Irregularly-shaped boundary, high-dimensional pr
oblems, symmetric matrix.
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Drawbacks
• For extremely high-wavenumber （ high frequency and/or large domain), full and ill-conditioned matrix; fast algorithm is desirable, e.g., fast multipole method.
• Exterior problems.• Nonlinear? Software package for real-world p
roblems （ killer applications)
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“There is nothing so practical as a sound scientific theory.”
