ノイズのある蔵本 -Sivashinsky 方程式のくりこみ群解析

1. The 13th Symposium on Condensed Matter (Non-Equilibrium Statistical ) Physics -Winter School in Tsukuba 2004- Jan. 20, 2005 ノイズのある蔵本-Sivashinsky方程式のくりこみ群解析 上之 和人（名大工）、　坂口 英継（九大総合理工）、　岡村 誠（九大応研） Cond-mat/0410429

2. 1 Systems with many degrees of freedom 1. Introduction Lattice spacing ● Quantum field theory Correlation length ● Critical phenomena ● Turbulence ? L: integral length η: Kolmogolov dissipation length Coarse-graining Renormalization-group (RG) Zoom out Step1: Coarse-graining Step2: Rescaling rescaling

3. 2 Summary of the key definitions Ballistic deposition (BD) Mean hight Interface width

4. 3 Roughness exponent Growth exponent Scaling exponents Scaling relation Dynamic exponent crossover time Scaling law

5. 4 Kuramoto-Sivashinsky (KS) equation Yakhot’s conjecture (1981) ● chemical turbulence ● flame-front propagation ● dynamics of liquid films subject to gravity Kardar-Parisi-Zhang (KPZ) equation ●model for interface growth Gaussian white noise

6. 5 Surface tension term Nonlinear term Properties of the KPZ equation ●Galilean invariance ●Fluctuation-dissipation (FD) theorem Fokker-Planck equation

7. 6 Numerical simulation by Sneppen (1992) Crossover

8. 7 Coarse-graining method by Zaleski (1989) and Hayot (1993) Noisy Burgers equation FD theorem

9. 8 2. The noisy Kuramoto-Sivashinsky (nKS) equation nonconserved noise + conserved noise Cutoff

10. 9 Slow modes RG-step1:Coarse-graining Fast modes

11. + = + 10 Self-energy at one-loop order Bare propagator pole Renormalized propagator

12. = + 11 Intrinsic noise at one-loop order

13. 12 Vertex correction at one-loop order (Galilean invariance) = + + +

14. 13 Scale transformation Self-affinity

15. 14 Fourier space Real space RG-step2: Rescaling Coarse-graining Rescaling

16. 15 coarse-graining rescaling RG flow equations Dimensionless parameters

17. 16 Fixed point of RG flow equations 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)

18. 17 RG flows in the parameter space (F,G,H) for various D 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

19. 18 Fluctuation-dissipation theorem Correlation function Power spectrum Equilibriumlike equipartition law Undoing the rescaling

20. 19 3. Numerical analysis Periodic boundary condition Initial condition KS eq nKS eq RG flow

21. 20 Modeling by the KPZ equation Dashed lines KPZ KS nKS

22. 21 Estimate of the effective parameters : numerical simulation + RG Dashed lines nKS

23. 23 3DNSE vs RFNSE A. Sain & R. Pandit (1998) KS vs nKS 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.

24. 22 4. Conclusion ●The RG analysis for the KS equation with conserved and nonconserved noises in 1+1 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 1+1 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.