coarsening versus selection of a lenghtscale
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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. - PowerPoint PPT PresentationTRANSCRIPT
Errachidia 2011
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
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2 general classes of evolution
1) Length scale selection
Time
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2 general classes of evolution
1) Length scale selection
Time
2) Coarsening
Time
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Questions
• Can one say if coarsening takes place in advance?• What is the main idea?• How can this be exploited?• Can one say something about coarsening exponent?• Is this possible beyond one dimension?• How general are the results?
A. Bray, Adv. Phys. 1994: necessity for vartiaional eqs.
Non variational eqs. are the rule in nonequilibrium systems P. Politi et C.M. PRL (2004), PRE(2006,2007,2009)
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Some examples of coarsening
Errachidia 2011Andreotti et al. Nature, 457 (2009)
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That’s not me!
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Myriad of pattern forming systems
1) Finite wavenumber bifurcation
2)( cQQA Q C Q
W
Lengthscale(no room for complex dynamics, generically )
Amplitude equation (one or two modes)
0
0
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Q
W2) Zero wavenumber bifurcation
42 QQ
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Q
W2) Zero wavenumber bifurcation
42 QQ
W
Q
Far from threshold
Complex dynamics expected
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Can one say in advance if coarsening takes place ?
Yes, analytically, for a certain class of equationsand more generally …….
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Coarsening is due to phaseinstability (wavelength fluctuations)
Phase modes are the relevant ones!
Eckhaus
What is the main idea?
q
stableunstable
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][ ])([)()( uuCuGuBu xxxxt
])([)()( uuCuGuBu xxt
)(),(),( uCuGuB
General class of equations (step flow, sand ripples….)
Arbitrary functions
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How can this be exploited?
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Example:Generalized Landau-Ginzburg equation
)()( uLuBuu xxt
(trivial solution is supposed unstable)
uB 3u)exp( tiqxu 21 q
1 cqqUnstable if 2 cor
Example of LG eq.:
q
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)()( uLuBuu xxt
)(0 xu steady solution
0)( 00 uBu xx
Patricle subjected to a force B
)()( uduBuV
Example42
42 uuV
ECteVu x 2
20
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Coarsening
42
42 uuV
U=-1 U=1
time
-1
+1
Kink-Antikink anihilation
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Amplitude
Wavelength
Stable
Unstable
U=0
U=0
Lambda c 0D
A
A
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Stability vs phase fluctuations?
),( tx : Fast phase ),( TX :slow phase
Xxq Local wavenumber:
xX tT 2
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Full branch unstable vs phase fluctuations
),( tx : Fast phase ),( TX :slow phase
Xxq Local wavenumber:
xX tT 2
Xx q XTt 2
10 uuu
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Full branch unstable vs phase fluctuations
),( tx : Fast phase ),( TX :slow phase
Xxq Local wavenumber:
xX tT 2
Xx q XTt 2
10 uuu
Sovability condition: XXT D
Derivation possible for any nonlinear equation
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Full branch unstable vs phase fluctuations
),( tx : Fast phase ),( TX :slow phase
Xxq Local wavenumber:
xX tT 2
Xx q XTt 2
10 uuu
Sovability condition: XXT D
20
20
)(
)(
u
uqD q
...)2(...2
0
1
d
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20
20
)(
)(
u
uqD q
0)( 002 uBuq
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20
20
)(
)(
u
uqD q
0)( 002 uFuq
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20
20
)(
)(
u
uqD q
0)( 002 uBuq
Particle with mass unity in time q/ Subject to a force B
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20
20
)(
)(
u
uqD q
0)( 002 uBuq
Particle with mass unity in time q/ Subject to a force B
Juduqq
1/2
0
20
120 )2()()2()(
J is the action
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20
20
)(
)(
u
uqD q
0)( 002 uFuq
Particle with mass unity in time q/ Subject to a force F
Juduqq
1/2
0
20
120 )2()()2()(
J is the action But remind that EJ
E:energy
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20
20
)(
)(
u
uqD q
0)( 002 uFuq
Particle with mass unity in time q/ Subject to a force F
Juduqq
1/2
0
20
120 )2()()2()(
J is the action But remind that EJ
E:energy
12
312
0 )(4
)2()(
Eq
Juqq
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Dhas sign of
A
A: amplitude : wavelength
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wavelength
amplitude
No coarseningcu
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wavelength
amplitude
No coarseningcoarseningc
c uu
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wavelength
amplitude
No coarseningcoarsening
Interruptedcoarsening
cc
c
uu
u
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wavelength
amplitude
No coarseningCoarsening
CoarseningInterruptedcoarsening
cc
c c
uu
u u
C.M., O. Pierre-Louis, Y. Saito, Review of Modern Physics (sous press)
P. Politi, C.M., Phys. Rev. Lett. (2004)
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][ ])([)()( uuCuGuBu xxxxt
])([)()( uuCuGuBu xxt
)(),(),( uCuGuB
General class of equations (step flow, sand ripples….)
Arbitrary functions
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Sand Ripples, Csahok, Misbah, Rioual,Valance EPJE (1999).
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Wavelength
amplitude
cu
frozen Example: meandering of stepson vicinal surfaces
branch stops
O. Pierre-Louis et al. Phys. Rev. Lett. 80, 4221 (1998) and many other examples , See :C.M., O. Pierre-Louis, Y. Saito, Review of Modern Physics (sous press)
Errachidia 2011Andreotti et al. Nature, 457 (2009)
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Dunes (Andreotti et al. Nature, 457 (2009))
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tD
2
)(
Can one say something about coarsening exponent?
P. Politi, C.M., Phys. Rev. E (2006)
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Coarsening exponent
t
20
20
)(
)()(
u
uqD q
t
D2
)(
LG
GL and CH in 1d )ln(t
Other types of equations t
q/2
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Some illustrations])([ xxxxt uuBu
If non conserved: remove xx
AIABD
)()(2
dxuI2
0
If non conserved dxuJI x2
0 )(
tD
2
)( Use of
duuBuV )()(
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Coarsening
42
42 uuV
U=-1 U=1
time
)1ln()(/)/(00
AuVduududxAA
x
eAAB 23
AJABD
)()(2
dxuJ x2
0 )( Finite (order 1)
eA
teD /)( 222 )ln(t
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Remark: what really matters is the behaviour of V closeto maximum; if it is quadratic, then ln(t)
1)1( uaVV )(1)( xuxQ AQ 10
2/10
1
00
Q
dQ
Q
10
2/0A , , ,1 QBQIJ
AJABD
)()(2
Conserved:
Nonconservednt
232
n
442
n
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Other scenarios (which arise in MBE)
B(u) (the force) vanishes at infinity only )1()( 2u
uuB
nt 41
nConserved
Non conserved2 ,
21
n
2 ,23
n
Benlahsen, Guedda(Univ. Picardie, Amiens)
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][ ])([)()( uuCuGuBu xxxxt
])([)()( uuCuGuBu xxt
)(),(),( uCuGuB
General class of equations (step flow, sand ripples….)
Arbitrary functions
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Transition from coarsening to selection of a length scale
xxxxxt uuuuuu ][ 3Golovin et al. Phys. Rev. Lett. 86, 1550 (2001).
0 Cahn-Hilliard equation
Kuramoto-Sivashinsky /uu After rescaling
coarsening
no coarseningFor a critical 47.0 Fold singularity of the steady branch
Amplitude
Wavelength47.0
47.0
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)( xxxxxxxxxt uGuuuuu
If 0 KS equation
If not stability depends on sign of v
New class of eqs: new criterion ; P. Politi and C.M., PRE (2007)
solutionssteady periodicfor interface theofvelocity v
0 Steady-state periodic solutions exist only if G is odd
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Extension to higher dimension possible
Analogy with mechanics is not possible
Phase diffusion equation can be derived
A link between sign of D and slope of a certain quantity (not the amplitude itself like in 1D)
The exploitation of
tD
2
)(
allows extraction of coarsening exponent
C.M., and P. Politi, Phys. Rev. E (2009)
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Summary
4) Coarsening exponent can be extracted for any equation and at any dimension from steady considerations, using
1)
3) Which type of criterion holds for other classes of equations? But D can be computed in any case.
Phase diffusion eq. provides the key for coarsening, D is a function of steady-state solutions (e.g. fluctuations-dissipation theorem).
tD
2
)(
2) D has sign of for a certain class of eqs
A