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: racz@general.elte.huHomepage: 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

qq

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