### Coarsening exponent

```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
Errachidia 2011
2 general classes of evolution
1) Length scale selection
Time
Errachidia 2011
2 general classes of evolution
1) Length scale selection
Time
2) Coarsening
Time
Errachidia 2011
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)
Errachidia 2011
Some examples of coarsening
Errachidia 2011
Andreotti et al. Nature, 457
(2009)
Errachidia
2011
Errachidia 2011
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
Errachidia
2011or two modes)
Amplitude equation
(one
Q
2) Zero wavenumber bifurcation
  Q  Q
2
W
4
Q
Errachidia 2011
2) Zero wavenumber bifurcation
  Q  Q
2
W
4
Q
W
Far from threshold
Q
Complex dynamics expected
Errachidia 2011
Can one say in advance if coarsening takes place ?
Yes, analytically, for a certain class of equations
and more generally …….
Errachidia 2011
What is the main idea?
Coarsening is due to phase
instability (wavelength fluctuations)
Phase modes are the relevant ones!

Eckhaus
q
stable
Errachidia 2011
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
Errachidia 2011
]
How can this be exploited?
Errachidia 2011
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

Errachidia 2011
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
Errachidia 2011
Coarsening
V 
u
2
2
U=-1

u
4
4
U=1
Kink-Antikink anihilation
time
+1
-1
Errachidia 2011
W avelength

U=0
U nstable
Lam bda c
D 0
U=0
S table
A m plitude
A
A
Errachidia 2011

Stability vs phase fluctuations?
 ( x, t )
Local wavenumber:
 ( X ,T )
X  x
: Fast phase
q  x 
X
Errachidia 2011
: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  
Errachidia 2011
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
Errachidia 2011
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
Errachidia 2011
D 
 q q (u 0 )
(u 0 )
2
2
q   u 0  B ( u 0 )  0
2
Errachidia 2011
D 
 q q (u 0 )
(u 0 )
2
2
q   u 0  F ( u 0 )  0
2
Errachidia 2011
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
Errachidia 2011
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
Errachidia 2011
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
Errachidia 2011
 
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
Errachidia 2011
J


3
4
2
E :energy
E
(

E
)
1
D
has sign of

A: amplitude

A

: wavelength
Errachidia 2011
wavelength
u
c
No coarsening
amplitude
Errachidia 2011
wavelength
u
c
u
c
No coarsening
amplitude
Errachidia 2011
coarsening
wavelength
u
c
u
c
No coarsening
amplitude
u
c
Interrupted
coarsening
Errachidia 2011
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
Errachidia 2011
]
Sand Ripples, Csahok, Misbah, Rioual,Valance EPJE (1999).
Errachidia 2011
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)
Errachidia 2011
Andreotti et al. Nature, 457
(2009)
Errachidia
2011
Dunes (Andreotti et al. Nature, 457 (2009))
Errachidia 2011
Can one say something about coarsening exponent?
D ( ) 

2
t
P. Politi, C.M., Phys. Rev. E (2006)
Errachidia 2011
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

Errachidia 2011
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
Errachidia 2011
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 )
Errachidia 2011
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
NonconservedErrachidia 2011
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
Errachidia 2011
,
 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
Errachidia 2011
]
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
Errachidia 2011
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
Errachidia 2011
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
Errachidia 2011
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
Errachidia 2011
```