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### Modeling of “Anomalous” Behaviors of Soft Matter

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 & mapping”, “earth sciences & engineering”, “engineering mechanics”)
- 1 department-scale institute (geotechnic institute)
- Around 3000 undergraduate students + 1000 graduate students

Wen Chen （陈文）

Institute of Soft Matter Mechanics

Hohai University, Nanjing, China

3 September 2007

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.

Typical soft matters

- Granular materials
- Colloids, liquid crystals, emulsions, foams,
- Polymers, textiles, rubber, glass
- Rock layers, sediments, oil, soil, DNA
- Multiphase fluids
- Biopolymers and biological materials

highly deformable, porous, thermal fluctuations play major role, highly unstable

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

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.

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.

Challenges

- Mostly phenomenological and empirical models, inexplicit physical mechanisms, often many parameters without clear physical significance;
- Computationally very expensive;
- Few cross-disciplinary research, less emphasis on common framework and problems.

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.

More features (courtesy to N. Pan)

- Very slow internal dynamics
- Highly unstable system equilibrium
- Nonlinearity and friction
- Entropy significant

a jammed colloid system, a pile of sand,

a polymer gel, or a folding protein.

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.

Typical “anomalous” (complex) behaviors

- “Anomalous” diffusion（heat conduction, seepage, electron transport, diffusion, etc.）
- Frequency-dependent dissipation of vibration, acoustics, electromagnetic wave propagation.

- Basic postulates of mechanics.

conservation of mass, momentum and energy

- Basic concepts of mechanics

stress, strain, energy and entropic elasticity

- Constitutive relations and initial–boundary-value problems.

Outline:

- Part I: Progresses and problems – a personal view
- Part II: Our works in recent five years

What in Part I?

- Field and experimental observations
- Statistical descriptions
- Mathematical physics modelings

The absorption of many materials and tissues

obeys a frequency-dependent power law

Courtesy of Prof. Thomas Szabo

Anomalous diffusion

- ＝1, Normal (Brownian) diffusion
- 1, Anomalous ( >1 superdiffusion， <1 subdiffusion)

Levy self-similar random walks

A characteristic Levy walk

Stretched Gaussian diffusion:

Gaussian diffusion:

Tsallis distribution (nonequlibrium system)

Tsallis non-extensive entropy

Max s

Boltzmann-Gibbs entropy

Tsallis distribution

A comparison of diverse distributions

A. Komnik, J. Harting, H.J. Herrmann

Progresses in statistical descriptions

- Continuous time random walk, fractional Brownian motion, Levy walk, Levy flight;
- Levy distribution, stretched Gaussian, Tsallis distribution.

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?

Physics behind “normal diffusion”

- Darcy’s law (granular flow)
- Fourier heat conduction law
- Fick’s law
- Ohlm law

- Fick diffusion:

Nonlinear Modelings

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:

Anomalous diffusion equation in Fractional calculus

Master equation (phenomenological)

Physical significances

- Histroy dependency (memory, non-Markovian)corresponding to fractional Brownian motion.
- Singular Volterra integral equation.
- Numerical truncation is risky!

Constitutive relationships

- Hookian law in ideal solids:
- Ideal Newtonian fluids:
- Newtonian 2nd law for rigid solids:
- One model of soft matter:

Numerical fractional time derivative

- Volterra integral equation；
- Finite difference formulation：Grunwald-Letnikov definition；
- “Short memory” approach (truncation and stability)
- Something new？

I. Podlubny, Fractional Differential equation, Academic Press, 1999

Numerical fractional space derivative

- Full numerical discretization matrix；
- Boundary condition treatments；
- Fast algorithm (e.g., fast multipole method).

Progresses in PDE modeling

- Fractional time derivative, fractional Laplacian;
- Hausdorff derivative;
- Growing PDE models in various areas.

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 modelings

Fractional vs. Nonlinear systems

- History dependency
- Global interaction
- Fewer physical parameters (simple= beautiful)
- Competition or complementary

Summary

- New definition of fractional Laplacian;
- Introduction of positive fractional time derivative, and modified Szabo dissipative wave equations;
- Mathematical physics explanation of [0,2] frequency power dependency via Levy statistics;
- Fractal time-space transforms underlying “anomalous” physical 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.

Definitions based on Fourier transform

Fractional derivative：

Positive fractional derivative：

Positivity requirments in modeling of dissipation

Traditional definition in space

0< <1

Samko et al. 1993. Fractional Integrals and Derivives: Theory and Applications

Our definition

0< <1

- Merits：
- Weak vs. strong singularity，
- Accurate vs. approximate，
- Finite domain with boundary conditions vs. infinite domain

Journal of Acoustic Society of America, 115(4), 1424-1430, 2004

Fractional derivative modelings of frequency-dependent dissipative medical ultrasonic wave propagation

Imaging Comparisons

Courtesy of Prof. Thomas Szabo

Medical ultrasound

Imaging (sonography) and ablating the objects inside human body for medical diagnosis and therapy.

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 fitting.

Time-space wave equations of integer-order partial derivative only exist for y=0, 2

Thermoviscous wave equation (y=2):

Damped wave equation (y=0):

Linear fractional Laplacian wave equation for arbitrary frequency dependency

Dr. Richter’s New clinical approach

- stabilize each deformable breast between two plates,
- detect breast cancer via speed change & attenuation.

“Anomalous” diffusion equation for frequency dependent dissipation

Phenomenological master equation

Fourier transform of probability density function of Lévy -stable distribution is the characteristic function of solution of “anomalous” diffusion equation：

To satisfy the positive probability density function, the Lévy stable index must obey

1) In terms of Lévy statistics, the media having >2 power law attenuation are not statistically stable in nature;

2) =0 is simply an ideal approximation.

Fractal time-space transforms, Hausdorff derivative, fractional quantum and phonon

Perplexing issues in anomalous diffusion

- Levy stable process and fractional Brownian motion
- The mean square displacement dependence on time

Fractional (fractal) time-space transforms

Special relativity transforms:

Two hypotheses for “anomalous” physical processes

- The hypothesis of fractal invariance: the laws of physics are invariant regardless of the fractal metric spacetime.
- The hypothesis of fractal equivalence: the influence of anomalous environmental fluctuations on physical behaviors equals that of the fractal time-space transforms.

Fractional quantum in complex fluids

Fractional quantum relationships between energy and frequency, momentum and wavenumber（fractional Schrodinger equation）

Statistical and Reynolds equation modelings of turbulence via fractional derivative

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)

Richardson superdiffusion

Richardson diffusion consistent with Kolmogorov scaling

Fractional Laplace

statistical equation:

Fractional derivative Reynolds equation

Navier-Stokes equation

Reynolds decomposition

Reynolds equation

Three order of magnitude: turbulence vs. molecule viscousity

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.

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

Thinking future?

- Phenomenological models & physics mechanisms of soft matter；
- Time-space mesostructures and statistical models；
- Numerical solution of fractional calculus equations；
- Verification and validation of models and engineering applications.

A Journal Proposal

- Title: Journal of Power Laws and Fractional Dynamics
- Publisher: Springer
- Proposers: W. Chen, J. A. T. Machado, Y. Chen

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.

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)

Scattered 3D geologicial data reconstruction

471,031 scattered data made by U.S. Geological Survey

Difficulties in simulation of high-dimensional, high wavenumber and frequency

Wavenumber:

- 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

200 millions DOFs matrix for 2D FEMengineering precision; 30000 full matrix & 13.5Gb storage for the standard BEM

2D case

SLangdon, Lecture notes on “Finite element methods for acoustic scattering”, July 11, 2005

S. Chandler-Wilde & S. Langdon, Lecture notes on “Boundary element methods for acoustics”, July 19, 2005

Advantages

- 1/100 storage，1/1000 computing cost of the BEM;
- High accuracy, simple program, non numerical integration, meshfree, suitable for inverse problems；
- Irregularly-shaped boundary, high-dimensional problems, symmetric matrix.

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 problems （killer applications)

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