Growth and Pattern Formation in the KPZ equation Hans Fogedby Aarhus University and Niels Bohr Institute Denmark
Outline • • • • • • • • • • • • • • • •
NON EQUILIBRIUM GROWTH SIMULATIONS SCALING KPZ EQUATION KPZ SCALING WEAK NOISE OSCILLATOR GENERAL WEAK NOISE WEAK NOISE KPZ GROWTH MODES PATTERN FORMATION SCALING UPPER CRITICAL DIMENSION OTHER TOPICS CONCLUSION Paris 2007
Four-monopole weak noise configuration
KPZ equation
2
NON EQUILIBRIUM • • • •
Fundamental issue in statistical physics No ensemble available Non equilibrium defined by dynamics Models:
•
Issues:
simulations discrete growth models continuum growth models scaling phase transitions growth modes pattern formation stationary states distributions correlations etc
Paris 2007
KPZ equation
3
NON EQUILIBRIUM Equilibrium
Pn0 ∝ exp(− En / kT )
Boltzmann factor
Non equilibrium
dPn = ∑ ( Pm wm→n − Pn wn→m ) dt m
Master equation
dxn 1 = − Fn + ηn dt 2
Langevin equation
Issues Pn (t )
Distribution
Pn0 = lim Pn (t )
Stationary distribution
< xn (t ) xm (t ') >
Correlations etc
t →∞
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KPZ equation
4
GROWTH Growing interfaces
Molecular beam epitaxi
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Bacterial growth
KPZ equation
Deposition of snow
5
Propagating flame front old setup
newer setup
digitized fronts
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KPZ equation
6
SIMULATIONS Random deposition
time
Random deposition with relaxation
time
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Ballistic deposition
time
KPZ equation
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SCALING Dynamical scaling hypothesis
w( L,t ) ∝ Lζ F (t / Lz ) • • • • • •
Height profile: h Saturation width: w System size: L Roughness exponent: ς Dynamic exponent: z Scaling function: F
w ~Lζ
w0
t
t 0~Lz
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KPZ equation
8
Fractal properties
Growing interface is self-affine fractal h( x) ∝ b −ζ h(bx) b scale parameter Fractal dimension dF = 2 − ζ
ζ = 1/ 2, random walk d F = 3 / 2 ( > 1) Paris 2007
KPZ equation
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KPZ EQUATION • • • • •
Continuum growth model for interface Field theoretical Langevin equation Intrinsic non equilibrium model Scaling properties KPZ maps to:
Directed polymers (disorder) Burgers equation (turbulence)
• KPZ studied by:
Perturbative DRG Strong coupling DRG Mode coupling Weak noise (present approach)
Paris 2007
KPZ equation
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Edwards-Wilkinson (EW) equation G ∂h(r , t ) G G 2 = υ∇ h ( r , t ) − F + η ( r , t ) ∂t Diffusion
Drift
G d G < ηη > (r , t ) = Δδ (r )δ (t )
Noise
Noise strength
White noise
Local correlations
• • • • • • • •
Equation for height profile: h(r,t) Damping coefficient: ν Constant drift: F Noise representing environment: η Noise strength: Δ Roughness exponent ς=(2-d)/2, d=1, ς=1/2 Dynamic exponent z=2 FD theorem - describes equilibrium interface
Paris 2007
KPZ equation
Flattening effect
11
Kardar-Parisi-Zhang (KPZ) equation ∂h λG G G G 2 = υ∇ h + ∇h ⋅ ∇h − F + η , < ηη > ( r , t ) = Δδ d ( r )δ (t ) 2 ∂t Diffusion Growth
• • • • • •
Drift Noise
Strength
Growing interface
Height profile of interface: h(r,t) Damping coefficient: ν Growth parameter: λ Constant drift: F Noise representing environment: η Noise strength: Δ Growth term
Lateral growth
δ h = ((vδ t ) 2 + (vδ t∇h) 2 )1/ 2 ≈ vδ t (1 + (∇h) 2 )1/ 2 gradient expansion δh v ≈ v + (∇h) 2 + .. δt 2 Paris 2007
KPZ equation
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Noisy Burgers equation G G G G u ( r , t ) = ∇h( r , t ), local slope G G G G G G G G G ∂u ( r , t ) 2G G = υ∇ u ( r , t ) + λ (u ( r , t ) ⋅ ∇)u ( r , t ) + ∇η ( r , t ) ∂t Diffusion
Convection
• Burgers equation describes slope of interface • Burgers equation used to model irrotational hydrodynamics: rot u=0 • Burgers equation toy model for turbulence (inverse cascade, shocks) Paris 2007
KPZ equation
Conserved noise
Slope of growing interface
13
Cole-Hopf equation Cole-Hopf transformation
⎛ 2υ ⎞ ⎛ λ ⎞ = h= ⎜ log w , w exp ⎟ ⎜ h⎟ ⎝ λ ⎠ ⎝ 2υ ⎠ maps the KPZ equation • •
∂h λG G 2 = υ∇ h + ∇h ⋅ ∇h − F + η 2 ∂t
Linear diffusion equation Driven by multiplicative noise
to the Cole-Hopf equation
∂w ⎛ λ ⎞ ⎛ λ ⎞ 2 = υ∇ w − ⎜ ⎟ wF + ⎜ ⎟ wη ∂t ⎝ 2υ ⎠ ⎝ 2υ ⎠
Paris 2007
KPZ equation
Multiplicative noise
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Directed polymers Cole Hopf equation ∂w ⎛ λ ⎞ ⎛ λ ⎞ = υ∇2 w − ⎜ ⎟ wF + ⎜ ⎟ wη ∂t ⎝ 2υ ⎠ ⎝ 2υ ⎠
Functional integral solution of Cole Hopf equation Partition function for directed polymer in quenched random potential
line tension
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Directed polymers along diagonals of square lattice with random bonds
random potential
KPZ equation
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KPZ SCALING • • • • • • • • •
Dynamical Renormalization group calculation (DRG) d=2 lower critical dimension Expansion in d-2 Strong coupling fixed point in d=1, z=3/2 Kinetic phase transition for d>2 zL Lässig (operator expansion) zWK Wolf-Kertesz (numerical) zKK Kim-Kosterlitz (numerical) d=4 upper critical dimension
DRG phase diagram Δλ 2
υ3
Δλ 2
υ3
d DRG equation
g= Paris 2007
KPZ equation
Δλ 2
υ3
,
dg = (2 − d ) g + cst. g 2 dl 16
WEAK NOISE Langevin equation
dx 1 = − F ( x) + η (t ) dt 2 Drift
Noise
• •
Noise η drives x into stationary stochastic state Noise strength Δ singular parameter Δ=0, relaxational behavior Δ~0, stationary behavior Crossover time diverges as Δ→0
Noise correlations
• • •
= Δδ (t)
Working hypothesis:
Noise strength
Paris 2007
Weak noise limit Δ→0 captures interesting physics KPZ equation
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Langevin equation
dx 1 = − F ( x ) + η (t ) 2 dt Fokker-Planck equation
∂P 1 ∂ 2 P 1 ∂ = Δ 2 + ( FP ) ∂t 2 ∂x 2 ∂x Diffusion
Drift
• Δ is effective Planck constant • Δd/dx is the momentum operator • Distribution P is the wave function • Weak noise corresponds to classical limit (ћ→0)
WKB ansatz
Schrödinger equation in imaginary time
⎛ S⎞ P ∝ exp ⎜ − ⎟ ⎝ Δ⎠
∂P 1 2 ∂ P Δ ∂ Δ = Δ + ( FP ) 2 ∂t 2 ∂x 2 ∂x 2
Time Kinetic evolution energy
Paris 2007
Potential energy (velocity dependent)
KPZ equation
Classical action
Noise strength
18
Weak noise limit Δ→0
Weak noise recipe
Hamiltonian H=
•
1 2 1 ∂S ∂S p − Fp, p = , H =− 2 2 ∂x ∂t
Equations of motion
dx 1 = − F ( x) + p dt 2 dp 1 dF ( x) = p dt 2 dx
Noise replaced by momentum Equation for momentum
Action
Comments •
x ,T
S ( x, xi , T ) =
Solve equations of motion for orbit from initial xi to final x in time T • Momentum p is a ”slaved” variable • Evaluate action S(xi,x,T) for orbit • Evaluate transition probability according to WKB ansatz P(xi,x,T)≈exp[- S(xi,x,T)/Δ]
⎛ dx ⎞ dt ∫ ⎜⎝ p dt − H ⎟⎠ x ,0
•
Stochastic problem replaced by dynamical problem in weak noise limit Dynamical action is generic weight function (compare with Boltzmann factor exp(-E/kT))
i
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KPZ equation
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Phase space description • • • • • • •
Stochastic Langevin equation replaced by coupled deterministic Hamilton equations of motion Noise replaced by canonical momentum p Solutions of equations of motion determine orbits in canonical phase space spanned by x and p Energy is conserved – orbits lie on energy surfaces Long time orbits through saddle point (SP) Orbits on zero-energy surface yield stationary state Action evaluated for orbit from x0 to x in time t yields transition probability from x0 to x in time t Paris 2007
KPZ equation
Canonical Phase Space
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Stochastic – Quantum analogue • • • • • •
Distribution corresponds to wave function Fokker-Planck equation corresponds to Schrödingers equation Noise yields kinetic energy Drift yields (v-dependent) potential energy Small noise strength corresponds to small Planck constant Weak noise corresponds to small quantum fluctuations, i.e., the correspondence limit Paris 2007
• • • • • •
KPZ equation
WKB approximation yields in both cases a ”classical” action Principle of least action is ”operational” Classical orbits and phase space discussion Structure of energy manifolds different Stationary zero-energy state corresponds to saddle point in stochastic case Method goes back to Onsager 21
OSCILLATOR Langevin equation for overdamped oscillator
dx = −γ x + η (t ) , η (t ) dt = Δδ (t )
elementary step
Example
white noise
Mean square displacement
∝
Suspended Brownian particle of size R in viscous medium with viscosity η
Δ 2γ
γ
Fokker-Planck equation for overdamped oscillator
∂P ( x, t ) 1 ∂ ⎡ ∂⎤ γ = + Δ 2 x P ( x, t ) ∂t 2 ∂x ⎢⎣ ∂x ⎥⎦
dx = − kx + ξ (t ) dt
< ξ (t )ξ (0) >= 2Tk Bγδ (t )
Stokes law
γ = 6π Rη
Time-dependent distribution
⎛ γ [ x − x0 exp( −γ t )]2 ⎞ P ( x, t ) ∝ exp ⎜ − ⎟ Δ − − [1 exp( 2 γ t )] ⎝ ⎠ Stationary distribution
⎛ γ x2 ⎞ Pstat ( x) ∝ exp ⎜ − ⎟ ⎝ Δ ⎠ Paris 2007
KPZ equation
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Weak noise limit Δ→0 Langevin equation
Action
dx = −γ x + η , F = 2γ x dt
γ [ x − xi exp( −γ T )]2 S ( x, xi , T ) = 2 1 − exp( −2γ T )
Hamiltonian
H=
1 2 p − γ xp 2
Distribution
⎛ γ [ x − xi exp( −γ T )]2 ⎞ P ( x, xi , T ) ∝ exp ⎜ − ⎟ Δ − − 2 1 exp( 2 γ T ) ⎝ ⎠
Equations of motion
dx = −γ x + p dt dp =γ p dt Paris 2007
Text book result
Stationary distribution
⎛ γ x2 ⎞ Pstat ( x ) ∝ exp ⎜ − ⎟ 2 Δ ⎝ ⎠ KPZ equation
Gaussian distribution
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Phase space discussion Hamiltonian
H=
1 2 1 p − γ xp = p ( p − 2γ x) 2 2
Canonical phase space
Stationary manifold
Finite time orbit Infinite time orbit
Transient manifold
Saddle point (long time orbits pass close to saddle point)
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KPZ equation
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GENERAL WEAK NOISE Generic Langevin equation for stochastic field wn(t) (Stratonovich formulation):
1 Δ d = − Fp − ∑ Gmn G pn + ∑ G pnηn , p = 1...N 2 2 mn dt dwm n
dwp
< ηn (t )ηm (0) >= Δδ nmδ (t ), Fp ({wq }), Gmn ({wq }) Generic Fokker-Planck equation for distribution P({wn},t):
dP 1 d = ∑ dt 2 n dwn
⎡ ⎤ d K mn ⎥ P ⎢ Fn + Δ ∑ m dwm ⎣ ⎦
K pm ({wq })= ∑ G pn ({wq }) Gmn ({wq }) n
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KPZ equation
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Weak noise scheme WKB ansatz
Equations of motion
⎛ S⎞ P ≅ exp ⎜ − ⎟ ⎝ Δ⎠
dwn 1 = − Fn + ∑ K nm pm dt 2 m dpn 1 dF 1 d = ∑ pm m − ∑ pm pq K mq dt dwn 2 mq dwn 2 m
Hamilton-Jacobi equation
Action
∂S +H =0 , ∂t
∂S p= ∂w
w,T
S=
∫
dt ∑ pn n
dwn − HT dt
Hamiltonian
Reduced action
1 1 H = K nm pn pm − pn Fn 2 2
1 S= 2
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w,T
∫ dt ∑ K
KPZ equation
nm
pn pm
nm
26
Generic canonical phase space
Stationary manifold (zero energy, H=0)
Finite time orbit (on H≠0 manifold Infinite time orbit (on H=0 manifolds) Saddle point (Markov behavior)
Transient manifold (zero energy, H=0)
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KPZ equation
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WEAK NOISE KPZ The KPZ equation ∂h λG G 2 = υ∇ h + ∇h ⋅ ∇h − F + η 2 ∂t Diffusion
Growth
Drift
Noise
Noise correlations
G d G < ηη > ( r , t ) = Δδ ( r )δ (t )
Height field
Noise strength
Paris 2007
White noise
•
The Kardar-Parisi-Zhang (KPZ) equation is a generic continuum nonequilibrium growth model • The KPZ equation is a nonlinear Langevin equation driven by locally correlated white noise • The KPZ equation is at a critical point and has strong coupling scaling properties • The KPZ equation is amenable to the weak noise analysis
KPZ equation
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Weak noise scheme Nonlinear Cole-Hopf transformation
⎛ 2υ ⎞ ⎛ λ ⎞ h= ⎜ ⎟ log w, w = exp ⎜ h ⎟ ⎝ λ ⎠ ⎝ 2υ ⎠ maps the KPZ equation
∂h λG G 2 = υ∇ h + ∇h ⋅ ∇h − F + η 2 ∂t to the Cole-Hopf equation
•
Langevin equation with multiplicative noise • Weak noise scheme also works in the case of multiplicative noise
∂w ⎛ λ ⎞ ⎛ λ ⎞ = υ∇ 2 w − ⎜ ⎟ wF + ⎜ ⎟ wη ∂t ⎝ 2υ ⎠ ⎝ 2υ ⎠
Paris 2007
KPZ equation
Multiplicative noise
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Weak noise scheme Cole-Hopf Hamiltonian
1 ⎡ ⎤ H = ∫ d x ⎢ −υ∇w ⋅ ∇p − υ k 2 wp + k02 w2 p 2 ⎥ 2 ⎣ ⎦ d
Parameters
λF λ , k = , F drift in KPZ equation 0 2 2υ 2υ Field equations k2 =
∂w = +υ∇2 w − υ k 2 w + k02 w2 p ∂t ∂p = −υ∇2 p + υ k 2 p − k02 p 2 w ∂t Paris 2007
KPZ equation
Action w ,T
1 S = k02 ∫ d d xdt (wp ) 2 2 wi ,0 Transition probability
P exp [ − S / Δ ]
Phase space
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GROWTH MODES Program: • • •
Seek localized solutions to static field equations Boost static solutions to propagating growth modes Construct dynamical network of dynamical growth modes Static field equations
υ∇2 w − υ k 2 w + k02 w2 p = 0
Linear diffusion equation 2 2
∇ w = k w, p = 0
υ∇2 p − υ k 2 p + k02 p 2 w = 0
Nonlinear Schrödinger equation
consistent for p=0 (or w=0) identical for p≈w
∇2 w = k 2 w − k02 w3 , p ∝ w
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KPZ equation
Nonlinear
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Diffusion equation
Diffusion equation
∇2 w = k 2 w, p = 0 d 2 w ( d − 1) dw 2 + = k w 2 dr r dr
•
Cone
•
Radial Cole-Hopf, height and slope fields
•
w+ ( r ) ∝ r (1−d ) / 2 exp( kr )
•
h+ ( r ) =(1/ k0 ) log[ w+ ( r )] ∝ ( k / k0 ) r G G G u+ ( r ) =∇h+ ( r ) ∝ ( k / k0 )( r / r )
•
Cole-Hopf field exponentially growing Height field linearly growing cone (pit) Slope field constant amplitude monopole field - positive charge k continuous charge, k2≈F, F drift in KPZ equation Drift F fixes charge k
Positive monopole Radial vector field
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KPZ equation
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Non linear Schrödinger equation NLSE bound state
Nonlinear Schrödinger equation (NLSE)
∇2 w = k 2 w − k02 w3 , p = υ w d 2 w ( d − 1) dw 2 2 3 + = k w − k 0w 2 dr r dr Radial Cole-Hopf, height and slope fields
w− ( r ) ∝ r
(1− d ) / 2
• •
Inverted cone
• •
exp( − kr )
h− ( r ) =(1/ k0 ) log[ w− ( r )] ∝ −(k / k0 ) r G G G u− ( r ) =∇h+ ( r ) ∝ −(k / k0 )( r / r )
•
Cole-Hopf field exponentially damped Height field linearly growing inverted cone (tip) Slope field fixed amplitude monopole field - negative charge k continuous charge, k2≈F, F drift in KPZ equation Drift F fixes charge k
Negative monopole Radial vector field
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KPZ equation
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Bound state solution for the NLSE Solution of radial NLSE by Runge-Kutta (matlab) In d=1
Bound states (numerical)
w− ( x) = 2(k / k0 ) cosh −1 (kx) h− ( x) = −(1/ k0 ) ln cosh(kx) u− ( x) = −(k / k0 ) tanh(kx)
w_(r)
1D domain wall
In higher d bound state narrows, amplitude increases In d≥4 bound state disappears! Paris 2007
KPZ equation
r 34
Galilei transformation KPZ - Burgers - Cole-Hopf scheme admits dynamical symmetry Galilei boost
G G G0 r → r − λu t G0 G h → h +u ⋅r G G G u → u + u0
Growth modes in 2D Moving frame Constant slope Constant shift
Tilted height cone
Propagating growth modes
G G G G G h± ( r , t ) ≈ ± ( k / k0 ) | r − ri (t ) | + u 0 ⋅ r G G G G r − ri (t ) G u± ( r , t ) ≈ ± ( k / k 0 ) G G + u0 | r − ri (t ) | G G G ri (t ) = r i0 − λ u 0t Paris 2007
Shifted vector slope field
KPZ equation
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PATTERN FORMATION Dynamical network Global solution built from localized modes Low density ’instanton’ scheme • • • • • • • • • • •
Localized modes as building blocks Boost modes by means of Galilei transformation Treat amplitudes k as charges, k2≈F Use charge language – positive and negative charges Growing mode, positive charge Decaying mode, negative charge Impose flat interface at infinity – zero slope Form self-consistent dynamical network from nodes Convenient to implement for slope field Flat interface at infinity corresponds to charge neutrality Evolving network corresponds to growing interface
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KPZ equation
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Dynamical network Construct network of static modes in terms of vector slope field Assign velocities to modes according to Galilean invariance and matching conditions
G G0 r − ri G G u (r ) = (1/ k0 )∑ ki G G 0 , | r − ri | i G G ri 0 − rl 0 G vi = −2υ ∑ kl G 0 G 0 | ri − rl | l ≠i
∑k i
t
Boost modes to assigned velocities
G G G0 ri (t ) = ∫ vi (t ')dt '+ ri 0
Construct self-consistent dynamical network in terms of slope field and height field
G G r − r (t ) G G u (r , t ) = (1/ k0 )∑ ki G Gi | r − ri (t ) | i G G G h(r , t ) = (1/ k0 )∑ ki | r − ri (t ) | i
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KPZ equation
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i
=0
Pattern formation in 1D Positive charge Right hand domain wall u
Negative charge Left hand domain wall u
u+ u--
u--
u
u3
u4
u1
u+ x
h
;; ;;;
Growing interface
u5
x
x
x h
u2
p
x x
x
Two-soliton excitation – quasi particle h
x
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KPZ equation
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Pattern formation in 2D Dipole mode in 2D
Propagating height mode
Propagating slope mode
Velocity
Monopoles
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KPZ equation
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Four-monopole height profile in 2D
Asymptotically flat interface with height offset Height field
As monopoles propagate subject to periodic boundary conditions interface grows
x-y plane
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KPZ equation
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SCALING Dipole mode
Action
Distribution
S ∝ Tk 4− d , k charge
L4− d S ∝ 3− d , P ( L, T ) ∝ exp(− S / Δ ) T P ( L, T ) ∝ exp(−cst.L4− d / T 3− d )
Pair velocity
v∝k Distance in time T
L = vT Hurst exponent
3− d H= 4−d
Displacement
< δ L2 >∝ T 2 H = T 2 / z Dipole random walk
Hurst exponent vs d
Dynamic exponent
zdip
4−d = 3− d
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KPZ equation
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Scaling in 1D Dipole mode
Dynamics
Dispersion law
Action: S ∝ υλ u T 3
Energy: E ∝ −υλ u
3
Momentum: Π ∝ υ u 2
E∝−
λ 3/ 2 | Π | υ 1/ 2
Spectral representation
< uu > ( x, t ) = ∫ d ΠF (Π ) exp(− Et / Δ + iΠ x / Δ ) Dynamical scaling
< uu > (r , t ) ∝ F (t / r z ) E ∝ Πz z = 3/ 2 Paris 2007
• • • • • •
Stochastic interpretration Spectrum of dipole mode Gapless dispersion Spectral representation Comparison with dynamical scaling ansatz Dispersion law exponent yields dynamical exponent z
Dynamical scaling exponent
z = 3/ 2 KPZ equation
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KPZ SCALING • • • • • • • • •
Dynamical Renormalization group calculation (DRG) d=2 lower critical dimension Expansion in d-2 Strong coupling fixed point in d=1, z=3/2 Kinetic phase transition for d>2 zL Lässig (operator expansion) zWK Wolf-Kertesz (numerical) zKK Kim-Kosterlitz (numerical) d=4 upper critical dimension
DRG phase diagram Δλ 2
υ3
Δλ 2
υ3
d DRG equation
g= Paris 2007
KPZ equation
Δλ 2
υ3
,
dg = (2 − d ) g + cst. g 2 dl 43
UPPER CRITICAL DIMENSION General remarks • • • • • •
•
Upper critical dimension usually considered in scaling context Mode coupling gives d=4; above d=4 maybe glassy, complex behavior (Moore et al.) DRG shows singular behavior in d=4 (Wiese) Numerics inconclusive! Issue of upper critical dimension unclear and controversial In present context we interprete upper critical dimension as dimension beyond which growth modes cease to exist Numerical computation of bound state shows
DRG phase diagram
Δλ 2
υ3
d
d=4
Paris 2007
KPZ equation
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Derrick’s theorem Scale transformation:
NLSE from variational principle yields IDENTITY 1 Scale transformation yields IDENTITY 2 IDENTITY 2 involves dimension d Demanding finite norm of bound state implies d 0 and I > 0 it follows that
d (t ) = Δδ (t )δ nm dt 2 dΦ Fn = , Pstat ( xn ) ∝ exp(−Φ / Δ ), FD theorem dxn Fn ≠
∂P = HP, HPstat = 0 ∂t zero energy eigenvalue
Δ
dΦ , Pstat ( xn ) = ?, no FD theorem dxn
Weak noise method
Phase space (one degree of freedom)
One degree of freedom 1 2 1 1 p − Fp = p ( p − F ) 2 2 2 zero energy manifolds: p = 0 and p = F dΦ F= FD theorem, Φ free energy dx dx S = ∫ dt ( p − H ) → ∫ Fdx = Φ dt Pstat ∝ exp( −Φ / Δ ) Boltzmann H=
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KPZ equation
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Stationary state
1D Noisy Burgers case Phase space (1D Burgers)
Hamiltonian manifolds
Hamiltonian: H = E = ∫ dx hdensity ( x ) Hamiltonian density: hdensity ( x ) hdensity ( x ) = p[υ∇ 2u + λ u∇u − (1/ 2)∇ 2 p ] Zero energy manifolds, H = 0 for p = 0, transient manifold p = 2υ u, stationary manifold (hdensity = (2 / 3)υλ∇(u 3 ) = total differential)
Stationary state Action du ⎡ du ⎤ S = ∫ dtdx ⎢ p − hdensity ⎥ → 2υ ∫ dtdxu dt ⎣ dt ⎦ → υ ∫ dxu = υ ∫ dx (∇h ) 2
2
Stationary state ⎛ υ ⎞ Pstat ∝ exp ⎜ − ∫ dx (∇h )2 ⎟ ⎝ Δ ⎠ FD theorem in 1D
Paris 2007
Comments:
• Stationary state given by zero-energy manifold • Zero-energy manifold not identified in D>1 • Hamiltonian density has form: G G G G G G G 2G hdensity = p[υ∇ u + λ (u ⋅ ∇)u − (1/ 2)∇(∇ ⋅ p )] KPZ equation
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Symmetric exclusion process 1D interacting lattice model with exclusion Defined by Master equation Particle density ρ, 0< ρ ( x, t ) = Δδ ( x)δ (t ) Paris 2007
KPZ equation
Langevin equation Noise correlations
49
Asymmetric exclusion process 1D interacting lattice model with exclusion Defined by Master equation Particle density ρ, 0< ρ >1, hopping rates p≠q Bulk-driven non equilibrium state
Hydrodynamical limit
∂ρ υ 2 = ∇ ρ − λ∇( ρ (1 − ρ )) + ∇ ( ρ (1 − ρ )η ), λ ∝ p − q ∂t 2 < ηη > ( x, t ) = Δδ ( x)δ (t )
Paris 2007
KPZ equation
Langevin equation Noise correlations
50
Scaling limit of SEP and ASEP • • • • • • • • •
Map SEP and ASEP to spin ½ model, Pauli spins Spin up - occupied site, spin down - empty site Employ quantum language Represent configurations by basis states Represent distribution by wave function Map Master equation to Schroedinger equation Identify Hamiltonian driving the system Ground state corresponds to stationary distribution Spectrum yields relaxation rates
Paris 2007
KPZ equation
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Details of scheme
Quantum scheme
Master equation
n → σ , σ = ±1
dPn = ∑ ( wm→n Pm − wn→m Pn ) dt m
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| Ψ (t ) >= ∑ P (t )| σ > σ
d | Ψ (t ) > = − H | Ψ (t ) > dt P (σ → σ ', t ) =< σ | exp(− Ht ) | σ ' > KPZ equation
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Hamiltonians H SEP
JJG JG = −∑ [σ i ⋅ σ i +1 − 1] i
SEP mapped to isotropic Heisenberg spin ½ chain Hamiltonian is hermitian Stationary state corresponds to aligned FM state Excitations: Spin waves, quadratic dispersion Dynamics is diffusive
H ASEP
JJG JG JJJGJG JG = −∑ [σ i ⋅ σ i +1 − 1 + iε e ⋅ (σ i × σ i +1 )], ε = ( p − q ) /( p + q ) i
ASEP mapped to Heisenberg spin ½ chain with spin wave interaction Hamiltonian is non-hermitian (signature of non equilibrium) Stationary state corresponds to aligned FM state Excitations: domain walls with superimposed spin waves Dynamics is ballistic with superimposed diffusion Paris 2007
KPZ equation
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Excitations Align chain in x direction – half occupancy Construct Kadanoff block spins, spin length s Oscillator representation of spins about FM state Scaling/continuum limit a→0 (a lattice distance)
< 0 | σ x | 0 >= s, < 0 | σ z | 0 >= 0, < 0 | σ y | 0 >= 0 1 2 [ ϕˆi , uˆi ] = iδ ik , ϕˆi position, uˆi momentum
σ x = s − (uˆi2 + ϕˆi2 ), σ y = s1/ 2ϕˆi , σ y = s1/ 2uˆi
uˆ ( x) = a −1/ 2uˆi , ϕˆ ( x) = a −1/ 2ϕˆi [uˆ ( x), ϕˆ ( x ')] = iδ ( x − x ') Expand HASEP
spin waves
Paris 2007
spin wave interaction
KPZ equation
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Equations of motion
Classical limit Permanent profile solutions Domain walls (kinks, solitons) u
u u+
u--
u-u+ x
x
Velocity
v = −ε [u+ + u− ]
Width
w∝ J /v
Paris 2007
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Dispersion Energy 2 2 ⎧ J ⎡⎛ ∂u ⎞ ⎛ ∂ϕ ⎞ ⎤ 2 ∂ϕ ⎫ E ∝ ∫ dx ⎨ ⎢⎜ ⎟ + ⎜ ⎬ ⎟ ⎥ + iε u ∂x ⎭ ⎩ 2 ⎢⎣⎝ ∂x ⎠ ⎝ ∂x ⎠ ⎥⎦
Momentum P ∝ ∫ dxu
∂ϕ ∂x
Pair of kinks Elementary excitation
E ∝ v3 / ε 2 P ∝ v2 / ε 2 E ∝ ε P 3/ 2
Quantization E → ω, P → k
ω ∝ εkz, z = 3/ 2
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CONCLUSION • • • • • • • • • •
Nonperturbative asymptotic weak noise approach Equivalent to WKB approximation in QM Stochastic problem mapped to dynamical problem Stochastic equation replaced by dynamical equations Canonical phase space representation Scheme captures strong coupling features For KPZ equation localized propagating growth modes Dynamical network represents strong coupling aspects Method yields a picture of stochastic pattern formation Only NLSE localized mode below d=4, upper critical dimension for KPZ equation !
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FIN
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