A. Helal & M.A. Fardin 07/2/2010 - MITweb.mit.edu › nnf › education › Summer2010 › ahelal_mafardin.pdf07/2/2010 Scaling with Fractional Calculus. Normal vs anomalous relaxation

A. Helal & M.A. Fardin

07/2/2010

Scaling with Fractional Calculus

Normal vs anomalous relaxation

Fractional relaxation processes and fractional rheological models for the description of a class ofviscoelastic materials, Ralf Metzler & Theo Nonnenmacher, International Journal of Plasticity, 2003

2

1 ( , )d

K tdt

x

21( , )

K k tt e k

1 2

0 ( , )td

K D x tdt

0 1

0

1 ( , )( ( , ))

( )

t

t

W tD W t dt

t t

xx

2( , )t E K k t k

0

( )

(1 )

n

n

zE z

z

2

2 exp(1 )

K k tE K k t

1

2 2 (1 )E K k t K k t

Maxwell-Debye relaxation Anomalous relaxation Fractional integration & derivation

1

0 0( ( , )) [ ( ( , ))]t td

D W t D W tdt

x x.( , ) ( , )it e t k xx k

Mittag-Leffler function

At short times,

At long times,

Inverse power-law / Nutting

Stretched exponential / KWW

Mittag-Leffler

Nutting

KWW

Scaling and fractional calculusA. Helal & M.A. Fardin

2

( )

1/ 2

0 ( )tC

D Ct

1/2 1/2

0 ( ) 0tD t

0 tD

0 tD

0 0 0 0 0 0( ( )) ( ( )) ( ( )) ( ( ))t t t t t tD f t D f t D D f t D D f t

Short Introduction to fractional calculus, Mauro Bologna

0 []tD

In general, NOT commutative

Explanation :

Let be the space of constant functions for the operator of order

Example :

Constant functions don’t belong to (1/ 2)

(1/ 2)1/ 2t belongs to

Practical rule

( )f tLet have a power series expansion

We have

if ( )f t doesn’t contain any function which is constant for the operators &


3

Commutativity of the operator

( )m e eG G G

0 0 0( ) ( ) ( ) ( ) ( )q q q q q q

t e m t e tt D t G G D t G D t

0 ,1 ,1( ) ( ( ) ) ( ) ( ( ) )q q

q e qG t G E t G t E t

.0

( )

( )

n

n

zE z

z

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

q q

eG t G tG tq q

0 m eG G G

, (0,1)q

mG

eG

mG

q

Generalized Cole-Cole behavior and its rheological relevance, Chr. Friedrich and H. Braun. Rheo. Acta, 1992

Standard Zener model

Fractional Zener Model

Stress relaxation

Generalized Mittag-Lefflerfunction

Asymptotic behaviour

(Adapted from original article because of typos)


4

Fractional rheological models: Zener model

Generalized Cole-Cole behavior and its rheological relevance, Chr. Friedrich and H. Braun. Rheo. Acta, 1992

1 2

0

cos( ( ) / 2) cos( ( 2)( )

1 2 cos( / 2)

q q

e

q q

e

qG Gg

G G q

2 2

0

sin( ( ) / 2) sin( ( 2)( )

1 2 cos( / 2)

q q

q q

e

qGg

G G q

Oscillatory testing

Experimental fitting for Carbon filled natural rubber

Carbon filled naturalrubber (0, 30, 60 phrfrom bottom to top)


5

Experimental results & Cole-Cole plots

0

0

( )

( )

G s sds

G s ds

( ) ( ) ( )

t

t G t t t dt

0

0

( )

( )

G s sds

G s ds

0( )t

G t G e

0( )t

G t G


6

Time scales & Deborah number I

obs

DeT

Maxwell-Debye relaxation

Linear Viscoelasticity

Anomalous relaxation

?De

0( )t

G t G e

0( )t

G t G

1

logd G dt constant

1

log /d G dt t


7

Time scales & Deborah number II

1

log ( )( )

log

obsobs

obs

d G TDe T

d T

( )obsobs

De TT

( )obsobs

De TT

1( )obsDe T

Differential definition of the average relaxation time

1

logd G dt

Application

Dynamic Deborah number

Maxwell-Debye

KWW

Nutting

On physically similar systems; Illustrations of the use of dimensional equations, E. Buckingham, 1914

[ ] [ ] [ ] [ ]a b cK M L T

3

, ,K a b c

0 2 1

1[ ] [ ] [ ] [ ]K M L T 2 /m s

2 /m s

3

, ,K a b c

0 2[ ] [ ] [ ] [ ]K M L T


8

Scaling & Buckingham-Pi Theorem

Relevant parameters space

Normal situation

is a vector space over

Anomalous situation

is a vector space over

Example


What are the relevant parameters?

Do they depend on rescaling (theory) /resampling (experiment) ?


What are the relevant parameters?

Do they depend on rescaling (theory) /resampling (experiment) ?

Examples:

Average size of a cluster?

Density?

Diffusion coefficient?

Distribution in X, Y...

t0

t0 x 5 t

0 x 10 t

0 x 50 t

0 x 100

Note:

Even for a “linear system” finite size and

time introduce non trivial coarse-

graining10


Avoiding the question: The tradition of “Linear” (and infinite) systems

Calculus (fr:Analyse): Taylor exp., approx. for “smooth” functions, solving equations...

Mechanics: Model systems, spring, dashpot...

Physics: Free systems, mode decomposition, superposition principle, input/output...

Statistics: LLN, Central limit theorem, independent probability, Ergodicity...

Philosophy: Traditional causality, traditional reductionism, Occam's razor...

Lingo: Linear [Forces] = Quadratic [Energies] = Gaussian [Partition f.]

Introduction to Nonlinear Physics, By Lui Lam (2003)

Why do we love linear systems?

11


Rescaling a Gaussian system: introduction to coarse-graining

Polymer in theta solvent / random walk...

Scaling concepts in polymer physics, PGG (1979)

a

(1) a' = a g1/2

® r'=rg-1/2

N'=N/g (g=10)

R~aN1/2

(1(

®

r(s)

The parameter a→ (monomer or mean free path) keeps its “bare” value →The coarse-grained and original statistics are the same

Integer fractal dimension – pseudo classical space

Hausdorff dimension: Lℤ

F'=F

12

( ) Dr r


Rescaling a non Gaussian system: non trivial coarse-graining

(more) realistic polymer chain / self avoiding random walk


a(1(

The parameter a→ and v are renormalized. They don't keep their bare value→The coarse-grained and original statistics are the same only if F is renormalized

Non-integer fractal dimension – pure fractal space

Hausdorff dimension: D= = 1.56...

...

...

(2v(

a

v'


Renormalization: a powerful method to find relevant parameters


Lectures on phase transitions and the renormalization group, N. Goldenfeld (1993)

Renormalization group theory: its basis and formulation in statistical physics, Rev. Mod. Phys. 70, 653 (1998)

→Quantities at fixed points have Levy stable distributions→Fixed-point=Self-similar, it maps to itself upon coarse-graining=”universality class” The difficulties are in computing the flow, i.e. finding how F is renormalized→(diagrams)→The method is well established only when F exists, i.e. at equilibrium (no T) →Extremely relevant in an empirical approach: measures involve coarse-graining

www.wolframalpha.com

Dependence on d

14


Geometrical interpretation of anomalous (spatial) dimensions

(...Geometric self-similarity and fractal (Chris—Fractal gels→→Link between fractal and fractional calculus (1 to 1 correspondence still debated)→Finite (incomplete) fractals and asymptotic fixed points (“phases”)→Some psychophysics roots too...

Lℤ | Lℝ | LℤFu-Yao Ren et al., Phys. Lett. A 219 (1996), 59-68.R.R. Nigmatullin, Theoret. and Math. Phys. 90, No 3 (1992), 242-251.

R.S. Rutman, Theoret. and Math. Phys. 105, No 3 (1995), 1509-1519. 15

+ + + =

Time correlation → Tℝ


How to interpret fractional time?

(...Some out-of-equilibrium renormalization (surface growth process→→Anomalous diffusion (Jason—diffusion on fractal network, trap models...) →Fractal representation of fractional elements →Link with temporal correlations

M.Marsili, Rev. Mod. Phys. 68 (1996), 963.

C. Friedrich et al. in Advances in the Flow and Rheology of Non-Newtonian Fluids (1999)

+ + +...

=

No time correlation

→ Tℤ

Tℤ on Lℝ → Tℝ

...

Levy stable

16


Non Newtonian time measure

Geometric and physical interpretation of fractional integration and fractional differentiation, Igor Podlubny (2002)

M. Balland et al. PRE (2006) 021911 ,74

Indeed, how do we measure time intervals? Only by observing some processes, which we consider

as regularly repeated. G. Clemence wrote:

“The measurement of time is essentially a process of counting. Any recurring phenomenon

whatever, the occurrences of which can be counted, is in fact a measure of time.”

CG

CG regular continuum Tℤ

fractional continuum Tℝ

17


Non Newtonian time measure: another insight to experiments

Geometric and physical interpretation of fractional integration and fractional differentiation, Igor Podlubny (2002)

M. Balland et al. PRE (2006) 021911 ,74 18


Fundamental question of Non Newtonian/Riemannian time measure

uniform

measure

non uniform

measure

uniform measure Riemann measure Kolmogorov extension

Gelfand-Naimark-Segal

construction

non uniform

measure

Lebesgue measure

Hausdorff measure

Frame transformations?? (Arezoo)

There is a very famous example of frame transformation with non uniform measure...19


20

Back to the Deborah number

1

log ( )( )

log

obsobs

obs

d G TDe T

d T

( )obsobs

De TT

( )obsobs

De TT

1( )obsDe T

Differential/local definition of the average relaxation time

1

logd G dt

Dynamic Deborah number

Maxwell-Debye

KWW

Nutting

Local definition of the Deborah number, measures the stretching of time


Keep the balance between empiricism and rationalism

Superposition principle

Ergodicity

Reductionism

theta solvent

random walk

relevant parameters

anomalous diffusion

Anomalous dimension

→Don't get lost in lingo

→Watch for sloppiness

J.P. Sethna et al. , Sloppy-Model, PRL (2006) 150601 97

http://www.lassp.cornell.edu/sethna/Sloppy/WhatAreSloppyModels.html

In a variety of contexts, physicists study complex, nonlinear models

with many unknown or tunable parameters to explain experimental

data. We explain why such systems so often are sloppy: the system

behavior depends only on a few ‘‘stiff’’ combinations of the

parameters and is unchanged as other ‘‘sloppy’’ parameter

combinations vary by orders of magnitude.

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A. Helal & M.A. Fardin 07/2/2010 - MITweb.mit.edu › nnf › education › Summer2010 › ahelal_mafardin.pdf07/2/2010 Scaling with Fractional Calculus. Normal vs anomalous relaxation