PatternPatterns and Frontss and Fronts
Zoltán RáczZoltán Rácz
IntroductionIntroduction
(1) Why is there something instead of nothing?
Homogeneous vs. inhomogeneous systems Deterministic vs. probabilistic description
Instabilities and symmetry breakings in homogeneous systems
(2) Can we hope to describe the myriads of patterns?
Notion of universality near a critical instability. Common features of emerging patterns. Example: Benard instability and visual hallucinations. Notion of effective long-range interactions far from equilibrium. Scale-invariant structures.
(3) Should we use macroscopic or microscopic equations?
Relevant and irrelevant fields -- effects of noise. Arguments for the macroscopic. Example: Snowflakes and their growth. Remanence of the microscopic: Anisotropy and singular perturbations.
Institute for Theoretical PhysicsEötvös UniversityE-mail: [email protected]: cgl.elte.hu/~racz
Patterns from stability analysisPatterns from stability analysis
(1) Local and global approaches.
Problem of relative stability in far from equilibrium systems.
(2) Linear stability analysis.
Stationary (fixed) points of differential equations. Behavior of solutions near fixed points: stability matrix and eigenvalues. Example: Two dimensional phase space structures Lotka-Volterra equations, story of tuberculosis Breaking of time-translational symmetry: hard-mode instabilities Example: Hopf bifurcation: Van der Pole oscillator Soft-mode instabilities: Emergence of spatial structures Example: Chemical reactions - Brusselator.
(3) Critical slowing down and amplitude equations for the slow modes.
Landau-Ginzburg equation with real coefficients. Symmetry considerations and linear combination of slow modes. Boundary conditions - pattern selection by ramp.
(4) Weakly nonlinear analysis of the dynamics of patterns.
Secondary instabilities of spatial structures. Eckhaus and zig-zag instability, time dependent structures.
(5) Complex Landau-Ginzburg equation Convective and absolute instabilities of patterns. Benjamin-Feir instability - spatio-temporal chaos. One-dimensional coherent structures, noise sustained structures.
PatternsPatterns fromfrom moving frontsmoving fronts
(1) Importance of moving fronts: Patterns are manufactured in them.
Examples: Crystal growth, DLA, reaction fronts. Dynamics of interfaces separating phases of different stability. Classification of fronts: pushed and pulled.
(2) Invasion of an unstable state.
Velocity selection. Example: Population dynamics. Stationary point analysis of the Fisher-Kolmogorov equation. Wavelength selection. Example: Cahn-Hilliard equation and coarsening waves.
(3) Diffusive fronts.
Liesegang phenomena (precipitation patterns in the wake of diffusive reaction fronts - a problem of distinguishing the general and particular).
LiteratureLiterature
M. C. Cross and P. C. Hohenberg, Pattern Formation Outside of Equilibrium, Rev. Mod. Phys. 65, 851 (1993).
J. D. Murray, Mathematical Biology, (Springer, 1993; ISBN-0387-57204).
W. van Saarloos, Front propagation into unstable state, Physics Reports, 386 29-222 (2003)
Why is there Something instead of Nothing? (Leibniz)
Homogeneous (amorphous) vs. inhomogeneous (structured)
Actorsand
spectators (N. Bohr)
Deterministic vs. probabilistic aspects I.
Bishop to Newton:
Now that you discovered the laws governing the motion of the planets, can you also explain the regularity of their distances from the Sun?
Newton to Bishop:
I have nothing to do with this problem. The initial conditions were set by God.
The question of the origins of order:
Titius-Bode law
(Cornell Universty)
?
Deterministic vs. probabilistic aspects II.
Mechanics, electrodynamics, quantum mechanics: initial conditions
Thermodynamics, statistical mechanics: S=max (equilibrium is independent of initial conditions) (at given constraints)
Stability
Disorder wins?
The question of the origins of order:
Order in equilirium
T>Tc
(E>Ec)
Mg
g
NN
Instability - symmetry breaking - critical slowing down
T<Tc
(E<Ec)
Instabilities and Symmetry Breakings
Basic approach: Understand more complex through studies of (symmetry breaking) instabilities of less complex
(Elmer Co.)
Rayleigh-Bénard:temperature field
velocity field
M.Schatz (shadowgraph images of convection patterns):
The wonderful world of stripes
Clouds
Precipitation patternsin gels
CuCl2 +NaOH
CuO + ...
P. Hantz
NASA
Sand dunes
Characteristic length:
~10 m
~10 m
2
-4
~10 - 10 m-1 4
Visual Hallucinations and the BVisual Hallucinations and the Bénard Instabilitynard Instability
BBénardnard experiments (G. Ahlers et al.)experiments (G. Ahlers et al.) Visual hallucinations (H. Kluver)Visual hallucinations (H. Kluver)
(lattice, network, grating honeycomb)
Caleidoscope
tunnel funnel spiral
cobweb
Visual hallucinations: retina visual cortex mapping
Visualcortex
Eye Retina
r
rlog~
Retina Visual cortex
J. D. Cowan
Scale Invariant Structures
MgO2 in Limestone C.-H. Lam
DLA (diffusion limited aggregation)
Oak tree
1 million particles N=100 million
DNR /1
(H. Kaufman)
Level of description: Microscopic or macroscopic?
(1) No two snowflakes are alike
(2) All six branches are alike
Parameters determining growth fluctuateon lengthscales larger than 1mm.
(3) Sixfold symmetry
Microscopic structure is relevanton macroscopic scale.
(4) Twelvefold symmetry (not very often)
Initial conditions may be remembered.
Fluctuations and Noise
disorder instability order
Rayleigh-Benard near but below the convection instability:
G. Ahlers et al.G. Ahlers et al.
Power spectrum
)(1 xeikx
xVk
2k ||~)( kS
homogeneous large fluctuations
Emergence of spatial structures: Soft mode instabilities
),...,,,( 2 nnnfn xx
0),,(
0),,(
212
211
nnf
nnf
),()( txntnn
Spatial mixing: convection, diffusion, … Reaction-diffusion systems
),,(
),,(
212222
211111
nnfnDn
nnfnDn
Chemical reactions in gels
(model example: Brussellator)
Stability analysis:
(1) Stationary homogeneous solutions:Stab
Local and global approaches
dxdU
x
Stability (absolute and relative stability)
x
U(x)
xmin
Umin
?
Regions of attraction of stationary points
0dxdU
Stationary points
The problem of non-potential systems
xxmin xminxmin(1) (3)(2)
Linear stability analysis
Assumption: The system is desribed by autonomous differential equations
),,(
),,(
2122
2111
nnfn
nnfn
fields control parameter
0),,(
0),,(
212
211
nnf
nnf
Stationary (fixed) points of the equations:21 ,nn
Behavior of solutions near fixed points
)()(
222
111
tnnntnnn
2
1
2
1
n
n
n
n
A
Linearization
21 ,nn
),,(
),,(
2122
2111
nnfn
nnfn
Stability matrix
Diagonalization
2
1
0
0~A
EigenvaluesSolution
eett
ccn
n
21
2212
1 ee1
Eigenvectors
ee11ee22
Lotka-Volterra systems: Hare-lynx problem
Lotka 1925 (osc. chem. react.)Volterra 1926 (fish pop.)
2212
2111
nnnn
nnnn
2n - concentration of hare (rabbits)
- conc. of lynx (foxes)
1n
2n21nn
1n
- rabbits eat grass and multiply
1( sets the time-scale)
- foxes perish without eating
- rabbits perish when meeting foxes 1( sets the -scale)1n
21nn - foxes multiply when meeting rabbits 1( sets the -scale)2n
21n - finite grass supply
Fixed points: 0,0 21 nn 1, 21 nn
Hare-lynx problem: Fixed point structure
2212
2111
nnnn
nnnn
0,0 21 nn 1, 21 nn
DoomDoom CoexistenceCoexistence
21 1 i2,1
01
0
0
01A
1n
2n
Problems of fluctuations and discretenessProblems of fluctuations and discreteness
Fixed point structures
221 ,,, ee1),,(
),,(
2122
2111
nnfn
nnfn
i ,0, Re Im
Re Re
Im Re Im 0Re
Node I Saddle
Spiral Centre
Re Node II
Re Star
1n
2n 1e
2e
Fixed point structures and the problem of tuberculosisFixed point structures and the problem of tuberculosis
Cause: Koch bacillus; Treatment: antibiotics; Immunity by vaccination
Characteristics: periodicity in the course of illness
antibodies
Koch bacilli
Breaking of time-translational symmetry: Limit cycles
Example: Skier on a wavy slope
Example: Van der Pole oscillator
xxxx )1( 2
xcxbaF )cos(
)~,~( UxIx
),( xx
Van der Pole oscillator
L
C
R
M
I
Ia
U
Ia
U0
Io
30 3
Ug
sUIIa
t
a IdC
IRdtdI
MdtdI
L0
01
xCI
LCRCMs
tLCt xxxx )1( 2
URCMs
Mgx
Emergence of spatial structures: Soft mode instabilities
),...,,,( 2 nnnfn xx
0),,(
0),,(
212
211
nnf
nnf
),()( txntnn
Spatial mixing: convection, diffusion, … Reaction-diffusion systems
),,(
),,(
212222
211111
nnfnDn
nnfnDn
Chemical reactions in gels
(model example: Brussellator)
Stability analysis:
(1) Stationary homogeneous solutions:Stab
Brusselator - oscillations and spatial patterns in chemical reactions
(Prigogin and Lefever, 1968)
A B + UU
3U
V + D
2U + V U E
1k
4k3k
2k
Concentrations: A, B, U(x,t), V(x,t) A, B
D, Egel
VUkBUkVDV
UkVUkBUkAkUDU
v
u
232
42
321
Brusselator - rescalings and canonical form of the equations
A B + UU
3U
V + D
2U + V U E VUkBUkVDV
UkVUkBUkAkUDU
v
u
232
42
321
41 / kAkccvVcuU
4/ kDxx u4/ ktt time space
concentrations
vubuvdv
vuubuu2
2
)1(1
uv DDd /
42 / kBkb
34
221
33 / kAkk
parameters
control parameterequations
Brusselator II - rescalings and canonical form of the equations
A B + UU
3U
V + D
2U + V U E VUkBUkVDV
UkVUkBUkAkUDU
v
u
232
42
321
vcVcuU xx tt
time space concentrations
vucckBcukvcD
vc
cukvucckBcukAkucD
uc
v
u
22322
422
3212
equations
Brusselator III - rescalings and canonical form of the equations
A B + UU
3U
V + D
2U + V U E VUkBUkVDV
UkVUkBUkAkUDU
v
u
232
42
321
vcVcuU xx tt time space concentrations
vuckucBck
vD
v
ukvucckuBkcAk
uD
u
v
u
223
22
42
321
2
uv DDd /
42 / kBkbuD/2 4/1 k
4
1
kAk
cc b
1
34
23
21
kAkk
1 1
Brusselator IV - perturbations at the homogeneous fixed point
vubuvdv
vuubuu2
2
)1(1
fixed point
/ ,1 ** bvu
Linearization
ikxk
ikxk
evvv
euuu
*
*
kkk
kkk
vdkbuv
vukbu
)(
)1(2
2
tkk
tkk
k
k
evv
euu
)0(
)0(
0 ,
,12
2
dkb
kb
k
k
),,,( dbkfk
Brusselator V- eigenvalue analysis
k2
0))(1(])1(1[ 2222 dkkbbkdb kk
k
kkkk 2
2kk
k
kstable
unstable
soft mode instability
hard mode instability
0k unstable
0 ,0 kk
0 ,0 kk stable
2kk stable or unstable focus
0k 0k
Brusselator VI- hard mode instability
k2
0))(1(])1(1[ 2222 dkkbbkdb kk
kkkkk 2
2kk
k
kstable
unstable
hard mode instability
0 ,0 kk
(1) k=0 instability
kk Re
2/)1( b
(2) instability point 1bc(3) exists only for 1b 00 k
k
Brusselator VII- soft mode instability
k2
0))(1(])1(1[ 2222 dkkbbkdb kk
kkkkk 2
2kk
k
kstable
unstable
soft mode instability
0 ,0 kk
instability point
42)( dkkdbdk
02 k
k0k
2)1( bdc
kmink
Soft mode instabtility first if
hardc
softc 1
11
bb
d uv DD
Emergence of spatial structures: Stability analysis
ikx
k
ikxk
ennn
ennn
222
111
k
k
k
k
n
nk
n
n
2
1
2
1 )(
A
Linearization
21 ,nn
Stability matrix
Diagonalization
)(0
0)(~
2
1
k
k
A
EigenvaluesSolution
eetk
kk
tk
kkk
k ccn
n )(
22
)(
12
1 21
ee1
Eigenvectors
e1ke2k
),,(
),,(
212222
211111
nnfnDn
nnfnDn
Critical slowing down and classification of instabilities
)()(2,1 kk
- with the largest real part
)(Re k
Instability: 0)(Re k
c
0
0 )(Im ckc
0
0 ckk
c
ck
c
c
Possibilities:
soft
hard
Classification of instabilities - emerging structures
)(Re k 0ck
k
)(Re k 0ck
k
c
c
c
c
0)(Im ckc 0)(Im ckc
spatially homogeneous
stationary limit cycle
t
spatially structured
stationary
t
time- and space-dependent
Stationary structures emerging in d=2 homogeneous systems
)(Re k 0ck
k
c
c
0)(Im ckc
d=2isotropy
tkxkikk ean )(~
)()( kk
+++
Swinney et al. 1991Turing patterns
gel
Beyond the instability: Amplitude equation for slow modes
)(Re kck
k
c
c
c
k k
c Band of unstable modes
0)(Im ckc c
c c
2)()( cc kkak
)(Re k smooth function of and
What is the steady state?
ck c
k k
k
control parameter from now on
k
Variation of the amplitude of the periodic structure
on lengthscale and on timescale .
Amplitude equation: Characteristic lengths and times
)(Re k
kk k
Band of unstable modes
tkikxk endkntxn )(* ~ ),(
k
k
tkxkkik
xik cc endke )()( ~
2)()( ckkak
00ck
0
~~~
),(0 txAe xikc
/1~ /1~ /1~
Amplitude equation
)(Re k
kk k
Band of unstable modes
2)()( ckkak
00ck
0
),(),(),( 0 txAetxAentxn xikxik cc
/1~ /1~
Plug it in the original equation and expand.
AAxA
AtA 2
2
2
Amplitude equation:
Amplitude eq.: Derivation from the Swift-Hohenberg equation
)(Re k0k
k
0
0
0
k k
0)(Im 00 k
0
00
3220 )( uukuut
),( txh
__
hhtxhtxu ),(),(
near instability:
kkt ukku ])([ 220
2 linearization:
)(k
Detailed derivation in separate pdf file
Amplitude equation: Simple solutions
0 ),(),(),( 0 txAetxAentxn xikxik cc
AAxA
AtA 2
2
2
small
zvz
z-component of the velocity
)cos( v xkA cz A const.
A
AAxA
AtA 2
2
2
general solution
),(0 txAA
Amplitude equation: Why is it so general?
0 ),(),(),( 0 txAetxAentxn xikxik cc
/1~
AAxA
AtA 2
2
2
small
zvz
z-component of the velocity
Slow, large-scale motion around the stationarystructure should not depend on the position of the underlying structure
),(),(
),(),( )(
txBetxAee
txAetxAexiklikxik
lxikxik
ccc
cc
lowest order in spatial derivatives preserving symmetry
Linear stability changes with changing sign
Amplitude equation: What can we get out of it?
0),(v txAe xikz
c
/1~
AAxA
AtA 2
2
2
)(xA0A
0v zxik
zcev
AAxdAd
A2
2
2
0
boundary conditions
stationarystate
xA
2th
Scale of A and x are determined
Amplitude equation: Fixing the time-scale
00),(v txAe xikz
c
AAxA
AtA 2
2
2
0
Quenching from ordered into disordered state
Amplitude equation should be still good
0 )0( A
ikxk etatxA )(),(
kk aka )( 2
2
1kk
Relaxation time
),( txA
t
Amplitude equation: Secondary instabilities I
0
0v ae xikz
c
AAxA
A2
2
2
0
iqxqeaA
0|| 22 qq aaq
Large number of possible stationary states
22|| qaq
qxqkixik
z aeAe cc )(v
Meaning: Shift in the wavelength of the pattern
q
2|| qa
Amplitude equation: Secondary instabilities II
0 AAxA
A2
2
2
0
iqxqeaA
Large number of possible stationary states
qxqkixik
z aeAe cc )(v q
2|| qa
q
final state Phase winding solutions
Amplitude equation: Secondary instabilities III
AAxA
AtA 2
2
2
qxqkixik
z aeAe cc )(v
final state (?)
Eckhaus instability line
q
Zig-zag instability
Stability analysis:
iqxq eaaA )(
),,( tyxaa
Amplitude eq.: Secondary instabilities: Phase diffusion
AAAAA xt
22
)()( qxiq eaaAAe xik
zcv
Eckhaus instability line: phase diffusion becomes unstable:
Stability analysis:
decays on timescale
does not decay
/1~a
const
Qxbsin~decays as
2/1~ QQ),( xfa
Qa
2
2
23xt q
q
3/q
Y. Pomeau, P. Manneville
Amplitude equation for A(x,y,t): Secondary instabilities
AAAki
AA yc
xt
22
2
...)(),,( 4/12/10
2/1 xtyxAu ),,(v tyxAe xik
zc
Zigzag instability:
Stability analysis:
decays on timescale decays on timescale
),( xfa
22
2
2
23
yc
xt kq
0q
)()( qxiq eaaA
),,( tyx
Dynamics of secondary instabilities: Topological defects
final state (?)
Eckhaus instability line
q
Initial state
Ae xikz
cv
Zig-zaginstab.
)()( qxiq eaaA
2
Not consistent with smooth change
time
0A
Translating structuresA.J. Simon, J. Bechofer, A. Libchaber
Isotropic-nematic transition
Solitary wave to moving to the left
Collision of two solitary waves
)(),(),(),( 0)( txktxAtxAentxn cctxki cc
Complex Landau-Ginzburg equation
t
time- and space-dependent
0)(Im ,0 cc kkc
),(),( ])(Im[ txAentxn tkxki cc
AAicxA
icAxA
vtA 2
32
2
1 )1()1(
cc kkv /)(Im
Velocity of the wave
xt31,cc are varied
H. Chate’
CLG equation: Secondary Instabilities
AAicAicAA xt
2
32
1 )1()1(
Benjamin-Feir instability
31,cc increased
Phase winding solutions
)(,
tqxiqeaA
22
2
32
1
aq
acqc
Linear stability
q
linearly stable region decreases
131 ccNo linerly stable region exists.
Newell criterion
CLG equation: Coherent structures AAicAicAA xt
2
32
1 )1()1(
Source
Front
Pulse
A
x
A
x
A
x
A
x
Sink
v 1v
2v1v
2v
Two different phase-winding solutions= ? Pattern left behind?v
CLGE - Phase diagram AAicxA
icAtA 2
32
2
1 )1()1(
3c
1c
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