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Coarsening versus selection of a lenghtscale. Chaouqi Misbah , LIPHy (Laboratoire Interdisciplinaire de Physique) Univ . J. Fourier, Grenoble and CNRS, France. with P. Politi , Florence , Italy. 2 general classes of evolution. 1) Length scale selection. Time.

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slide1

Coarsening versus selection of a lenghtscale

ChaouqiMisbah,

LIPHy (Laboratoire Interdisciplinaire de Physique)

Univ. J. Fourier, Grenoble and CNRS, France

with P. Politi, Florence, Italy

Errachidia 2011

slide2

2 general classes of evolution

1) Length scale selection

Time

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slide3

2 general classes of evolution

1) Length scale selection

Time

2) Coarsening

Time

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slide4

Questions

  • Can one say if coarseningtakes place in advance?
  • Whatis the main idea?
  • How canthisbeexploited?
  • Can one saysomething about coarseningexponent?
  • Is this possible beyond one dimension?
  • How general are the results?

Bray, Adv. Phys. 1994: necessity for vartiaionaleqs.

Non variationaleqs. are the rule in nonequilibriumsystems

P. Politi et C.M. PRL (2004), PRE(2006,2007,2009)

Errachidia 2011

slide7

That’s not me!

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Myriadof pattern formingsystems

1) Finite wavenumber bifurcation

Lengthscale

(no room for complex

dynamics, generically )

Amplitude equation (one or two modes)

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2) Zero wavenumber bifurcation

Far from threshold

Complex dynamics expected

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Can one say in advance if coarseningtakes place ?

Yes, analytically, for a certain class of equations

and more generally …….

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Whatis the main idea?

Coarseningis due to phase

instability (wavelength fluctuations)

Phase modes are the relevant ones!

Eckhaus

stable

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unstable

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Example:GeneralizedLandau-Ginzburgequation

(trivial solution is supposed unstable)

Example of LG eq.:

or

Unstable if

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steady solution

Patricle subjected to a force B

Example

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Coarsening

U=1

U=-1

Kink-Antikink anihilation

time

+1

-1

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Stability vs phase fluctuations?

:slow phase

: Fast phase

Local wavenumber:

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Full branch unstable vs phase fluctuations

:slow phase

: Fast phase

Local wavenumber:

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Full branch unstable vs phase fluctuations

:slow phase

: Fast phase

Local wavenumber:

Sovability condition:

Derivation possible for anynonlinearequation

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Full branch unstable vs phase fluctuations

:slow phase

: Fast phase

Local wavenumber:

Sovability condition:

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Particle with mass unity in time

Subject to a force

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Particle with mass unity in time

Subject to a force

is the action

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Particle with mass unity in time

Subject to a force

is the action

But remind that

:energy

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Particle with mass unity in time

Subject to a force

is the action

But remind that

:energy

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has sign of

A: amplitude

: wavelength

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wavelength

No coarsening

amplitude

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wavelength

No coarsening

coarsening

amplitude

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wavelength

No coarsening

coarsening

amplitude

Interrupted

coarsening

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wavelength

P. Politi, C.M., Phys. Rev. Lett. (2004)

No coarsening

Coarsening

amplitude

Interrupted

coarsening

Coarsening

C.M., O. Pierre-Louis, Y. Saito, Review of Modern Physics(sous press)

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slide36

Example: meandering of steps

on vicinal surfaces

Wavelength

frozen

branch stops

amplitude

O. Pierre-Louis et al. Phys. Rev. Lett. 80, 4221 (1998) and manyother

examples ,

See :C.M., O. Pierre-Louis, Y. Saito, Review of Modern Physics(sous press)

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Can one saysomething about coarseningexponent?

P. Politi, C.M., Phys. Rev. E (2006)

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Coarsening exponent

LG

GL and CH in 1d

Other types of equations

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some illustrations
Some illustrations

If non conserved: remove

If non conserved

Use of

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Coarsening

time

U=1

U=-1

Finite (order 1)

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Remark: what really matters is the behaviour of V close

to maximum; if it is quadratic, then ln(t)

Conserved:

Nonconserved

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Other scenarios (which arise in MBE)

B(u) (the force) vanishes at infinity only

Conserved

Non conserved

Benlahsen, Guedda

(Univ. Picardie, Amiens)

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Transition from coarsening to selection of a length scale

Golovin et al. Phys. Rev. Lett. 86, 1550 (2001).

Cahn-Hilliard equation

coarsening

Kuramoto-Sivashinsky

After rescaling

no coarsening

For a critical

Fold singularity of the steady branch

Wavelength

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Amplitude

slide47

New class of eqs: new criterion ; P. Politi and C.M., PRE (2007)

KS equation

If

Steady-state periodic solutions exist only if G is odd

If not stability depends on sign of

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slide48

Extension to higherdimension possible

C.M., and P. Politi, Phys. Rev. E (2009)

Analogywithmechanicsisnot possible

Phase diffusion equationcanbederived

A linkbetweensignof D and slope of a certain quantity

(not the amplitude itselflike in 1D)

The exploitation of

allows extraction of coarsening exponent

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slide49

Summary

Phase diffusion eq. providesthe key for coarsening,

D is a function of steady-state solutions

(e.g. fluctuations-dissipation theorem).

1)

2) D has sign of for a certain class of eqs

3) Which type of criterionholds for other classes of equations?

But D canbecomputed in any case.

4)Coarseningexponentcanbeextracted for anyequation and at

any dimension fromsteadyconsiderations, using

Errachidia 2011