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
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Andreotti et al. Nature, 457
(2009)
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2011
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Myriad of pattern forming systems
1) Finite wavenumber bifurcation
W
    A (Q  Q c )
 0
2
Lengthscale
(no room for complex
dynamics, generically )
Q
C
 0
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2011or two modes)
Amplitude equation
(one
Q
2) Zero wavenumber bifurcation
  Q  Q
2
W
4
Q
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2) Zero wavenumber bifurcation
  Q  Q
2
W
4
Q
W
Far from threshold
Q
Complex dynamics expected
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Can one say in advance if coarsening takes place ?
Yes, analytically, for a certain class of equations
and more generally …….
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What is the main idea?
Coarsening is due to phase
instability (wavelength fluctuations)
Phase modes are the relevant ones!

Eckhaus
q
stable
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unstable
General class of equations (step flow, sand ripples….)
 t u  
xx
[ B (u )  G (u )
x
[C (u )  x u ]
 t u  B (u )  G (u )  x [C (u )  x u ]
B ( u ), G ( u ), C ( u )
Arbitrary functions
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]
How can this be exploited?
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Example:Generalized Landau-Ginzburg equation
u t  u xx  B ( u )  L ( u )
(trivial solution is supposed unstable)
Example of LG eq.:
B  u  u
  1 q
u  exp( iqx   t )
Unstable if
3
q  qc  1
or
2
   c  2

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q
u t  u xx  B ( u )  L ( u )
u0 ( x)
steady solution
u0x
u 0 xx  B ( u 0 )  0
2
 V  Cte  E
2
Patricle subjected to a force B
V ( u )    duB ( u )
Example
V 
u
2
2

u
4
4
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Coarsening
V 
u
2
2
U=-1

u
4
4
U=1
Kink-Antikink anihilation
time
+1
-1
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W avelength

U=0
U nstable
Lam bda c
D 0
U=0
S table
A m plitude
A
A
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
Stability vs phase fluctuations?
 ( x, t )
Local wavenumber:
 ( X ,T )
X  x
: Fast phase
q  x 
X
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:slow phase
T  t
2

 

Full branch unstable vs phase fluctuations
 ( x, t )
Local wavenumber:
 ( X ,T )
X  x
: Fast phase
q  x 
 x  q    X
X
:slow phase
T  t
2

 

 t   T      X
2
u  u 0   u1  
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Full branch unstable vs phase fluctuations
 ( x, t )
Local wavenumber:
 ( X ,T )
X  x
: Fast phase
q  x 
 x  q    X
X
:slow phase
T  t
2

 

 t   T      X
2
u  u 0   u1  
Sovability condition:
 T  D
XX
Derivation possible for any nonlinear equation
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Full branch unstable vs phase fluctuations
 ( x, t )
 ( X ,T )
X  x
: Fast phase
q  x 
Local wavenumber:
 x  q    X
:slow phase
T  t
2

 
X
 t   T      X
2
u  u 0   u1  
Sovability condition:
D 
 q q (u 0 )
(u 0 )

 T  D
2
XX
2
...  ( 2 )
2
1
 d  ...
0
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D 
 q q (u 0 )
(u 0 )
2
2
q   u 0  B ( u 0 )  0
2
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D 
 q q (u 0 )
(u 0 )
2
2
q   u 0  F ( u 0 )  0
2
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D 
 q q (u 0 )
(u 0 )
2
2
q   u 0  B ( u 0 )  0
2
Particle with mass unity in time    / q
Subject to a force
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B
D 
 q q (u 0 )
(u 0 )
2
q   u 0  B ( u 0 )  0
2
2
Particle with mass unity in time    / q
Subject to a force
B
2 / q
q (u 0 )
2
 ( 2 )
1
1
 d  ( u 0 )  ( 2 ) J
2
0
J is the action
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D 
 q q (u 0 )
(u 0 )
2
q   u 0  F ( u 0 )  0
2
2
Particle with mass unity in time    / q
Subject to a force
F
2 / q
q (u 0 )
2
 ( 2 )
1
1
 d  ( u 0 )  ( 2 ) J
2
0
J is the action
But remind that
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 
J
E
E :energy
D 
 q q (u 0 )
(u 0 )
2
q   u 0  F ( u 0 )  0
2
2
Particle with mass unity in time    / q
Subject to a force
F
2 / q
q (u 0 )
2
 ( 2 )
1
1
 d  ( u 0 )  ( 2 ) J
2
0
J is the action
 q q (u 0 )
 
But remind that
2
 ( 2 )
1
J
q
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J


3
4
2
E :energy
E
(

E
)
1
D
has sign of

A: amplitude

A

: wavelength
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wavelength
u
c
No coarsening
amplitude
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wavelength
u
c
u
c
No coarsening
amplitude
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coarsening
wavelength
u
c
u
c
No coarsening
amplitude
u
c
Interrupted
coarsening
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coarsening
wavelength
P. Politi, C.M., Phys. Rev. Lett. (2004)
u
c
u
No coarsening
c
Coarsening
amplitude
u
c
Interrupted
coarsening
u
Coarsening
c
C.M., O. Pierre-Louis, Y. Saito, ReviewErrachidia
of Modern
2011 Physics (sous press)
General class of equations (step flow, sand ripples….)
 t u  
xx
[ B (u )  G (u )
x
[C (u )  x u ]
 t u  B (u )  G (u )  x [C (u )  x u ]
B ( u ), G ( u ), C ( u )
Arbitrary functions
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]
Sand Ripples, Csahok, Misbah, Rioual,Valance EPJE (1999).
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Example: meandering of steps
on vicinal surfaces
Wavelength
u
frozen

c
branch stops

amplitude
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)
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Andreotti et al. Nature, 457
(2009)
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2011
Dunes (Andreotti et al. Nature, 457 (2009))
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Can one say something about coarsening exponent?
D ( ) 

2
t
P. Politi, C.M., Phys. Rev. E (2006)
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Coarsening exponent
 t

LG
D ( ) 

2
D ( ) 
 q q (u 0 )
(u 0 )
t
2
2
  2 / q
GL and CH in 1d
Other types of equations
  ln( t )
 t

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Some illustrations
 t u   xx [ B ( u )  u xx ]
If non conserved: remove
 B ( A)
  xx
2
D ( )  
I A 
If non conserved
Use of
I 
I  J 
u
2
dx
0
 (  x u 0 ) dx
D ( ) 
2

t
2
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V (u ) 
 B ( u ) du
time
Coarsening
V 
u
2

2
u
4
4
 B ( A)
2
 

dx 
D ( )  
U=1
U=-1
J A 
A
A
 du /(  x u ) 
 du / V ( u )   ln( 1  A )
0
0
B  A  A  2e
3
D ( )   e
2
2 
J 

  /t
2
 (  x u 0 ) dx
2
Finite (order 1)
  ln( t )
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Ae

Remark: what really matters is the behaviour of V close
to maximum; if it is quadratic, then ln(t)
V  V (1)  a u  1
1
 

Q
Q0

 Q0

 Q0
1  / 2
D ( )  
J  1, I   ,  A   Q 0
 t
 B ( A)
2
dQ

Q0  1  A
Q ( x)  1  u ( x)
Conserved:
 /2
n
n
n
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J A 
, B  Q0
 2
3  2
 2
4  4
 1
Other scenarios (which arise in MBE)
B(u) (the force) vanishes at infinity only
 t
n
n
Conserved
B (u ) 
1
4
Non conserved
n
Benlahsen, Guedda
(Univ. Picardie, Amiens)
n
1
 2
,
2

3  2
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,
 2
u
(1  u )
2

General class of equations (step flow, sand ripples….)
 t u  
xx
[ B (u )  G (u )
x
[C (u )  x u ]
 t u  B (u )  G (u )  x [C (u )  x u ]
B ( u ), G ( u ), C ( u )
Arbitrary functions
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]
Transition from coarsening to selection of a length scale
Golovin et al. Phys. Rev. Lett. 86, 1550 (2001).
u t  
 0
 
[  xx u  u  u ]   uu x
3
xx
Cahn-Hilliard equation coarsening
Kuramoto-Sivashinsky After rescaling
u  u /
no coarsening
For a critical   0 . 47
Fold singularity of the steady branch
Wavelength
  0 . 47
  0 . 47
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Amplitude
New class of eqs: new criterion ; P. Politi and C.M., PRE (2007)
u t   u xx  u xxxx   uu x    x G ( u x )
If
 0
 0
KS equation
Steady-state periodic solutions exist only if G is odd
If not stability depends on sign of
v  velocity
periodic
v
of the interface
steady solutions
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for
Extension to higher dimension possible
C.M., and P. Politi, Phys. Rev. E (2009)
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
D ( ) 

2
t
allows extraction of coarsening exponent
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Summary
1) Phase diffusion eq. provides the key for coarsening,
D is a function of steady-state solutions
(e.g. fluctuations-dissipation theorem).
2) D has sign of

A

for a certain class of eqs
3) Which type of criterion holds for other classes of equations?
But D can be computed in any case.
4) Coarsening exponent can be extracted for any equation and at
any dimension from steady considerations, using
D ( ) 

t
2
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