Sivashinsky

1
????????-Sivashinsky???????????
The 13th Symposium on Condensed Matter
(Non-Equilibrium Statistical ) Physics
-Winter School in Tsukuba 2004-
Jan. 20, 2005

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Cond-mat/0410429
2
1. Introduction
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Systems with many degrees of freedom
Lattice spacing
? Quantum field theory
Correlation length
? Critical phenomena
? Turbulence ?
L integral length
? Kolmogolov dissipation length
Coarse-graining
Renormalization-group (RG)
Zoom out
Step1 Coarse-graining
Step2 Rescaling
rescaling
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Summary of the key definitions
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Ballistic deposition (BD)
Mean hight
Interface width
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Scaling exponents
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Roughness exponent
Growth exponent
Scaling relation
Dynamic exponent
crossover time
Scaling law
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Yakhots conjecture (1981)
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Kuramoto-Sivashinsky (KS) equation
? chemical turbulence
? flame-front propagation
? dynamics of liquid films subject to gravity
Kardar-Parisi-Zhang (KPZ) equation
? model for interface growth
Gaussian white noise
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Properties of the KPZ equation
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Surface tension term
Nonlinear term
? Galilean invariance
? Fluctuation-dissipation (FD) theorem
Fokker-Planck equation
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Numerical simulation by Sneppen (1992)
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Crossover
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Coarse-graining method by Zaleski (1989) and
Hayot (1993)
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Noisy Burgers equation
FD theorem
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2. The noisy Kuramoto-Sivashinsky (nKS) equation
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nonconserved noise conserved noise
Cutoff
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RG-step1Coarse-graining
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Slow modes
Fast modes
11
Self-energy at one-loop order
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Bare propagator
pole
Renormalized propagator
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Intrinsic noise at one-loop order
11
13
Vertex correction at one-loop order
12
(Galilean invariance)




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Scale transformation
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Self-affinity
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RG-step2 Rescaling
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Fourier space
Real space
Coarse-graining
Rescaling
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RG flow equations
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coarse-graining
rescaling
Dimensionless parameters
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Fixed point of RG flow equations
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The values of (F, G, H) are universal in the
sense that they do not depend on the initial
values.
Scaling exponents
KPZ
(Galilean invariance)
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RG flows in the parameter space (F,G,H) for
various D
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KS eq ?
Initial values
The RG flow for (F(l), G(l), H(l)) rapidly
approaches the KPZ fixed point with incresing
the strength of D
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Fluctuation-dissipation theorem
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Correlation function
Power spectrum
Equilibriumlike equipartition law
Undoing the rescaling
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3. Numerical analysis
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Periodic boundary condition
Initial condition
KS eq
nKS eq
RG flow
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Modeling by the KPZ equation
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Dashed lines
KS
nKS
KPZ
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Estimate of the effective parameters
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numerical simulation
RG
Dashed lines
nKS
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KS vs nKS
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3DNSE vs RFNSE
A. Sain R. Pandit (1998)
A. Karma C. Misbah (1993)
?three-dimensional Navier-Stokes equation
forced at large spatial scales (3DNSE)
? randomly forced Navier-Stokes equation
(RFNSE)
The stochastic forcing seems to destroy the
well-defined filaments observed in the
3DNSE without changing the multiscaling exponents
ratios. Therefore, the existence of vorticity
filaments is not crucial for obtaining these
exponents.
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4. Conclusion
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? The RG analysis for the KS equation with
conserved and nonconserved noises in 11
dimensions is performed. The noisy KS equation is
in the same universality class as the KPZ
equation in the sense that
(i) the values of the scaling exponents obtained
in the one-loop approximation are the same
as those at the KPZ fixed point,
(ii) the fluctuation-dissipation theorem is also
satisfied in the noisy KS equation in 11
dimensions.
? The KPZ scaling can be easily observed even in
moderate-size numerical simulations of the KS
equation under stochastic noises, due to the
increase of the effective noise strength.