Extreme and smooth gradient percolation

Bernard SapovalEcole polytechnique

Agnès Desolneux (MAP 5, Université Paris 5)Andrea Baldassarri (Università La Sapienza, Roma)

Subtitle:

Fractal exponents (like 7/4) without fractals and

without SLE

and critical fluctuations without critical parameter

Diffuse distribution of particles on a lattice: gradient percolation

Particlessource

Particlessink

L columns

Lg

Lg+1lines

Smooth gradient percolation:

At each occupied site a continuous gaussian distribution with a variance s2 is attached.

F(x,y) =(1/2s2) occup. sites i exp-{[(x-xi)2 +(y-yi)2] / 2s2}

Then the contributions of the occupied sites are summed:

500 x 500

s = 2

The contributions of the occupied sites are summed:

Find the constant level lines:

Whatever the level,

one observes similar

fluctuations.Critical

fluctuations but around an

arbitrary level

Physical problems at the origin of gradient percolation:

• Diffusion fronts: geometry of diffuse contacts and soldering.

• Structure of fuzzy images.

• Corrosion fronts in the etching of random solids.

• Erosion fronts: sea-coasts geometry.

DIFFUSE STRUCTURE:

x f

The front is characterized by

its position xf ,

its width f and its length Nf.

2Lg

A source of particles is kept at a constant concentration =1

At time t > 0, the particle concentration at a distance x is given by the complementary error function:

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p(x) =1−2

π1

2e

0

xlD∫

−u2

du

where

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lD

= 2(Dt)1

2 is the diffusion length at time t

If nf(x) is the mean number per unit horizontal length of points of the front:

Diffusion fronts: geometry of

diffuse contacts and soldering.

What is gradient percolation?

Random distribution of particles

with a gradient of concentration.

The gradient percolation front is the frontier of the infinite cluster

≈ Lg4/7

Mathematical aspects:P. Nolin

Numerical observations

The average front position is such that, for large Lg, p(xf) is close to pc

pc = 0.59280 ± 10-5

Rosso, Gouyet, BS(1985)

pc = 0.592745 ± 2. 10-6

Ziff, BS (1986)

pc = 0.5927460 ± 5. 10-7

Ziff and Stell (1988)

First measurements of the front fractal dimension…

(1984)

Df = 1.76 ± 0.02.

The front width f followed a power law

f Lg with ≈ 0.57…

The front length Nf followed a power law

Nf Lg with ≈ 0.42…

1984

An island is a finite cluster because it is situated at a position x where p(x) < pc

€

ξ(x) = ξ0

p(x) − pc

−ν

€

f

= K.ξ (xf+σ

f)

€

f∝ L

g

α with α =

ν1+ν

=47

€

f

= Kξ0

p(xf+σ

f) − p

c

−ν

= Kξ0.σ

f

dpdx

(xf)

−ν

= Kξ0

σf

Lg

⎛

⎝ ⎜

⎞

⎠ ⎟

−ν

= 0.5714…

= 4/3 (den Nijs, 1983)

Front length Nf:

x f

The front width being a correlation length:

Nf is of order (L / ).Df

Nf ≈ L. (Df-1) ≈ L. Lg

= (Df -1)/(1 + )

≈ 0.426

2Lg

The fact that is the horizontal correlation length

has been shown recently by Pierre Nolin (arXiv:math/0610682)

Df / (1+ ) ≈ 1

Df = (1+ )/

but one had

+ ≈ 1

(1984)

/(1 + ) + (Df -1)/(1 + ) = 1

conjecture

Df =

7/4 ???Percolation cluster hull

The number of particles in a box of lateral width is :

(Nf /L). ≈ Df ≈ Lg Df . / (1+ ) ≈ Lg

• But (Nf /L). is the number of particles in a box of size where

is the statistical width of the frontier.

This width is defined independently of the fractal character of the

frontier.

But now we know (Duplantier, 1987, Smirnov, 2001) that for sure Df = 7/4:

Fractality does not appear in this statement

For diffusion, it means that the correlated surface is of the

order of the diffusion length at time t:

lD(t) = 2(Dt)1/2

What if the gradient is so large that the frontier is no more fractal?

The same power laws are observed

Small Lg values?

???

Is there a conservation law?

B. Sapoval And M. Rosso, Fractals, 3, 23-31 (1995)Df = 7/4Df = 4/3e, f: smoothing and filteringc, d: gradient percolationa: original objectb: fuzzy photograph

Other situation with gradient percolation: fuzzy image

Statistique des fractures: empirique ou théorique?

Attaque corrosif d’une couche mince   d'aluminium plongé dans une solution

Expériences: L.Balazs (1996) PMC Ecole Polytechnique.

Aluminium

CCD Camera

Lumière

Verre

Solution

Other situation with gradient percolation: pit corrosion of an aluminum film.

L. Balasz (Ecole polytechnique,1997)

Time evolution of the corrosion picture:

• The first circular pit grows with time.

• It roughens progressively and slows down.

• It finally stops on a fractal frontier with dimension 4/3.

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The corrosion model:•Andrea Baldassari•Andrea Gabrielli

Width of the front as a function of the gradient but where is the gradient?

Rocky coast-line erosion has marine and atmospheric causes which act on random ‘lithologic’units: random rocks

Random means that the ‘mechano-chemical’ properties of the rocks (due to structure and composition) are unknown and exhibit some dispersion.

• EROSION OF ROCKY COASTS:

Sea-coasts could be fractal because their geometry damps the sea-waves (and currents …) in such manner that, for a given ‘sea power ’, the erosion is minimized.

In that sense it is not only the coast which is eroded but the effective erosion force of the sea which is diminished by the geometry of the coast.

And this is why one observes fractal sea-

coasts ????

Phys. Rev. Lett. 2004.http://www.nature.com/nsu/031124/031124-4.html

The sea power and erosion force is a decreasing function of the coast perimeter, for example:

f (t) =1

1+Lp(t)

Lp(t=0)g

Empirical breakwater construction recipe:

x1 x2 x3 x4 x5

x6 x7 x8 x9 x10

x11 x12 x13 x14 x15

The ‘resisting’ earth is represented by a square lattice where each site presents a

random lithology or resistance to the sea.

Lp = 5 , f(t=0)

x1 x2 x3 x4 x5

x6 x7 x8 x9 x10

x11 x12 x13 x14 x15

Lp = 5 , f(t=1) = f(t=0)

x1 x3 x5

x6 x7 x8 x9 x10

x11 x12 x13 x14 x15

Lp = 7 , f(t=2) < f(t=0)

x1 x3 x5

x6 x7 x8 x10

x11 x12 x13 x14 x15

The weaker sites are

eroded and, at the same time:1- new sites (strong or weak) are uncovered.2- the coastal length is modified.

Model representation of the erosion process

Time evolution: fractal morphology is a statistical

geometrical attractor

North Sardinia real and numerical

Df, num. = 4/3Df, geo. = 1.33

Box-counting determination of the fractal dimension

Df = 4/3

The value 4/3 is the dimension of the

percolation cluster accessible perimeter.

(Grossman, Aharony, 1985)

(Duplantier et al. 1999)

Scaling behavior of the coast width

4/7 is related to gradient percolation

Final coast morphologies

The erosion model in itself is more general.

It could apply to rough but non-fractal coastline:

What if the gradient is so large that the frontier is no more fractal?

The same power laws are observed

Small Lg values?

Extreme gradient percolation or fractal exponents without fractals

A. DESOLNEUX, B. SAPOVAL, and A. BALDASSARRI, Self-Organised Percolation Power Laws with and

without Fractal Geometry in the Etching of Random Solids,

in “Fractal Geometry and Applications” Proceedings of Symposia in Pure Mathematics (PSPUM).

American Mathematical SocietyIn print. See cond-mat/0302072.

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Note that if percolation had been defined in the first place through gradient

percolation :…. pc = 1/2

Extreme gradients: is f 7/4

proportionnal to Lg ?

Lg = 2

Remark: for small Lg

One expects a law of the form 7/4 = a(Lg + b)

And Nf7/3

= c(Lg + d)

Is b = -1?

Numerical test of the exponent:

Is 7/4 the best exponent?

1- Choose arbitrary 1.6 < 1.9.

2- Find the best fit values of the law

a(Lg + b).

3- Then for each compute the distance:

€

d(α )2 =147

σf(L

g)α − a

α(L

g+b

α)[ ]

2

lg = 4

50

∑

Result:

d(

For very extreme gradients, one can determine exact values:

DISCUSSION:

Comparison between the exact values and the values extrapolated from the numerics

for Lg between 4 and 50.

Discussion:

Comparison between the exact values and the values extrapolated from the numerics

for Lg = 4 and Lg = 5 for which one “knows” standard deviations

Extrapolation from small to

large

Concluding remarks

• Exact values for Lg = 2 and 3 enters the confidence interval deduced from Lg = 4 and 5.

• There exists a unique power law relating the width and length of the gradient percolation front to the gradient length for Lg between 2 and infinity. It does not apply for Lg= 1.

• The same type of results are obtained for the triangular lattice.

• There exists an implicit relation between width and length of the type 7/4 Nf

7/3. You may measure the interface properties and find if it was created, in the past, by a gradient percolation mechanism.

• Here, 7/4 is not related to conformal invariance or SLE nor to fractality …

• Its origin is combinatory …

• There is a conservation law in diffusion on a lattice: The flux per column is proportional to the correlated surface…

• However, nature does not live on a lattice and the question remains of how to apply those results to rough coatslines for example…

Smooth gradient percolation in 2D:

Other broadening functions creates the same typeof fluctuations:

GaussianExponential

Yukawa potential…

Smooth gradient percolation in 2D: Gaussian broadening

s = 2 s = 3

threshold lines = 0.1; 0.3; 0.5; 0.7; 0.9

Ordinary percolation on a lattice is defined by the occupation or not

of sites or bonds with probability p.

In continuous percolation a Poisson distribution of points is occupied by circles.

The macroscopic manifestations of these elementary events appear

only in a narrow region near pc.

In smooth percolation, critical fluctuations are found around an arbitrary threshold.

Invariance of the geometry with respect to the arbitrary threshold

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threshold

Gaussian width

apparent fractal

dimension

Exponential broadening = exp(-r/) (1s wave functions)

= 2 = 3

levels lines = 0.1; 0.3; 0.5; 0.7; 0.9

Yukawa potential or screened Coulomb potential = [(1/(r+1)] exp(-r/)

=5

is theDebye-Huckel length

Yukawa potential or screened Coulomb potential = [(1/(r+1)] exp(-r/)

=10

Examples of Yukawa potentials

1- Distribution of radio frequencies antennas: lines of constant electric field shouldcritical geometrical fluctuations.

2- Equipotential lines due to a charge concentration distribution of ions in an electrolyte. Electrochemical reaction paths?

1D Gradient percolation is simple

but not trivial

1D Percolation

dc= distance for the first empty site

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€

dc = Σk= 0k=∞kpk (1− p) =

p

1− p

€

pc =1

€

=1

Gradient percolation in 1DStarting at the origin,

where do we find the first empty site?

The size of the fluctuation zone is Lg - 2E(T) of order Lg

Example: Lg =100; s=3

Characteristic function C(x)= 1 if

F(x) ≥ with =0.6

Average of Fs,(x)

Smooth gradient percolation in 1D

At each occupied site a continuous gaussian distribution

with a variance s2 is attached.Then the contributions of the occupied sites are summed:

€

F(x) =1

s 2π occupied kΣ e−(x−k)2/2s2

Example: Lg =1000; s=3

Characteristic function C(x)= 1 if

F(x) ≥ with =0.6