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Ageing of the 2+1 dimensional Kardar-Parisi Zhang model

Ageing of the 2+1 dimensional Kardar-Parisi Zhang model Géza Ódor , Budapest (MTA-TTK-MFA) Jeffrey Kelling, S. Gemming Dresden ( H ZD R ) ,. MECO39 Coventry 08 / 04 /201 4. www.mfa.kfki.hu/ ~ odor. The K ardar- P arisi- Z hang ( KPZ ) equation.

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Ageing of the 2+1 dimensional Kardar-Parisi Zhang model

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  1. Ageing of the 2+1 dimensional Kardar-Parisi Zhang model Géza Ódor, Budapest(MTA-TTK-MFA)Jeffrey Kelling, S. GemmingDresden (HZDR), MECO39 Coventry08/04/2014 www.mfa.kfki.hu/~odor

  2. The Kardar-Parisi-Zhang (KPZ) equation th(x,t) = 2h(x,t) +λ ( h(x,t))2 + (x,t)‏ σ: (smoothing) surface tension coefficient λ :local growth velocity, up-down anisotropy η : roughens the surface by a zero-average, Gaussian noise field with correlator:<(x,t) (x',t')> = 2 D d (x-x')(t-t')‏ Fundamental model of non-equilibrium surface physicsRecent interest : Solvability in 1+1 dim, experimental realizations in 2+1 d Simple scaling of the surface growth:Interface Width: Exhibits simple power-laws:

  3. Mapping of KPZ onto ASEP in 1d • Attachment (with probability p) and • Detachment (with probability q) • Corresponds to anisotropic diffusion of particles (bullets) along the 1d base space (Plischke & Rácz 1987) The simple ASEP (Ligget '95)is an exactly solved 1d lattice gas Many known features: response to disorder, different boundary conditions ... are known. Widespread application in biology Kawasaki' exchange of particles

  4. Mapping of KPZ growth in 2+1 d • Generalized Kawasaki update: • Octahedron modelDriven diffusive gas of pairs (dimers) • G. Ódor, B. Liedke and K.-H. Heinig, PRE79, 021125 (2009) • G. Ódor, B. Liedke and K.-H. Heinig, PRE79, 031112 (2010) • Surface pattern formation via dimer model • G. Ódor, B. Liedke and K.-H. Heinig, PRE79, 051114 (2010)

  5. CUDA code for 2d KPZ • Each 32-bit word storesthe slopes of 4 x 4 sites • Speedup230 x (Fermi)with respect a CPUcore of 2.8 GHz up to:131072 x 131972 size J. Kelling and G. Ódor Phys. Rev. E 84 (2011) 061150

  6. Physical ageing in systems without detailed balance Known & practically used since prehistoric times (metals, glasses) systematically studied in physics since ~1970 Discovery : ageing effects are reproducible & universal !They occur in different systems: structural glasses, spin glasses, polymers, simple magnets, . . . Dynamical scaling, growing length scale: L(t) ~ t1/z Broken time-translation-invariance

  7. Two-time aging observables Time-dependent order-parameter field: (t; r) t : observation time, s : start time Scaling regime: Two-time correlator: Two-time response: a) System at equilibrium : fluctuation-dissipation theorem b) Far from equilibrium : C and R are independent !C, R, a, b can be independent

  8. Ageing in 1+1 d KPZ (Henkel, Noh & Pleimling 2012) Fluctuation-dissipation for: t>>s Different from equilibrium:

  9. Two-dimensional KPZ ageing simulations • Two-time integrated response for : • Sample A with pi = p0 = 0.98 deposition prob. for all times • Sample B with pi = p0 up to time s , and pi = p0 later

  10. Simulation results for the auto-correlation • Method is confirmed by restricting the communication to 1d • CPU and GPU results agree, but saturation for the latter for t/s large ageing exponent: b = -2 = -0.483(2) C /z = 1.21(1) + oscillations due to kinematic vawes simulation by Kerch (1997) : C ~ (t/s) -1.7 •  marginally supports Kallabis & Krug hypothesis: C = d,

  11. Universality (in permission with Timothy Halpin Healy) Completely new RSOS, KPZ Euler, and Directed Polymer in Random Medium (DPRM) simulations: 2014 EPL 105 50001 Full agreement

  12. Auto-response results Fast oscillating decay, Low signal/noise ratio, Very slow convergence GPU and CPU results agree and provide a = 0.3, R/z= 1.25(1) Fluctuation – Dissipation is broken weakly

  13. Conclusions & outlook • Fast parallel simulations due to mapping onto stochastic cellular automata (lattice gases) Extremely large scale (215 x 215) simulations on GPUs and CPUs GPU speedup ~230 with respect to a single CPU core Ageing exponents of 2+1 d KPZ are determined numerically This also describes the behavior of driven lattice gas of dimers Lack of fluctuation-dissipation is shown explicitly Generalization to higher dimensions is straightforward Local Scale Invariance hypothesis can be tested • Acknowledgements: DAAD-MÖB, OTKA, OSIRIS FP7, NVIDIA Publications:H. Schulz, G. Ódor, G. Ódor, M. F. Nagy, Computer Physics Communications 182 (2011) 1467. J. Kelling and G. Ódor, Phys. Rev. E 84, 061150 (2011), G. Ódor, B. Liedke, K.-H. Heinig J. Kelling, Appl. Surf. Sci. 258 (2012) 4186R. Juhász, G. Ódor, J. Stat. Mech. (2012) P08004J. Kelling, G. Ódor, M. F. Nagy, H. Schulz and K. -H. Heinig, EPJST 210 (2012) 175-187 G.Ódor, J. Kelling, S. Gemming, Phys. Rev. E 89, 032146 (2014)

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