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 (HZDR), Géza Ódor, Budapest (MTA-TTK-MFA) Jeffrey Kelling, S. Gemming Dresden (HZDR), MECO39 Coventry 08/04/2014

The Kardar-Parisi-Zhang (KPZ) equation  t h(x,t) =  2 h(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: = 2 D  d (x-x')(t-t')‏ Fundamental model of non-equilibrium surface physics Recent interest : Solvability in 1+1 dim, experimental realizations in 2+1 d Simple scaling of the surface growth: Interface Width: Exhibits simple power-laws:

 Attachment (with probability p) and  Detachment (with probability q)  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 The simple ASEP (Ligget '95) is an exactly solved 1d lattice gas Many known features: response to disorder, different boundary conditions... are known. Many known features: response to disorder, different boundary conditions... are known. Widespread application in biology Widespread application in biology Mapping of KPZ onto ASEP in 1d Kawasaki' exchange of particles

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

CUDA code for 2d KPZ Each 32-bit word stores the slopes of 4 x 4 sites Each 32-bit word stores the slopes of 4 x 4 sites Speedup 230 x (Fermi) with respect a CPU core of 2.8 GHz up to: x size Speedup 230 x (Fermi) with respect a CPU core of 2.8 GHz up to: x size J. Kelling and G. Ódor Phys. Rev. E 84 (2011)

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) ~ t 1/z Broken time-translation-invariance

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

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

Two-dimensional KPZ ageing simulations Two-time integrated response for : Sample A with p i = p 0 = 0.98 deposition prob. for all times Sample B with p i = p 0   up to time s, and p i = p 0 later

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  = (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,

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

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

Conclusions & outlook Fast parallel simulations due to mapping onto stochastic cellular automata (lattice gases) Fast parallel simulations due to mapping onto stochastic cellular automata (lattice gases) Extremely large scale (2 15 x 2 15 ) simulations on GPUs and CPUs Extremely large scale (2 15 x 2 15 ) simulations on GPUs and CPUs GPU speedup ~230 with respect to a single CPU core GPU speedup ~230 with respect to a single CPU core Ageing exponents of 2+1 d KPZ are determined numerically Ageing exponents of 2+1 d KPZ are determined numerically This also describes the behavior of driven lattice gas of dimers This also describes the behavior of driven lattice gas of dimers Lack of fluctuation-dissipation is shown explicitly Lack of fluctuation-dissipation is shown explicitly Generalization to higher dimensions is straightforward Generalization to higher dimensions is straightforward Local Scale Invariance hypothesis can be tested Local Scale Invariance hypothesis can be tested Acknowledgements: DAAD-MÖB, OTKA, OSIRIS FP7, NVIDIA Acknowledgements: DAAD-MÖB, OTKA, OSIRIS FP7, NVIDIA Publications: H. Schulz, G. Ódor, G. Ódor, M. F. Nagy, Computer Physics Communications 182 (2011) J. Kelling and G. Ódor, Phys. Rev. E 84, (2011), G. Ódor, B. Liedke, K.-H. Heinig J. Kelling, Appl. Surf. Sci. 258 (2012) 4186 R. Juhász, G. Ódor, J. Stat. Mech. (2012) P08004 J. Kelling, G. Ódor, M. F. Nagy, H. Schulz and K. -H. Heinig, EPJST 210 (2012) G.Ódor, J. Kelling, S. Gemming, Phys. Rev. E 89, (2014) Publications: H. Schulz, G. Ódor, G. Ódor, M. F. Nagy, Computer Physics Communications 182 (2011) J. Kelling and G. Ódor, Phys. Rev. E 84, (2011), G. Ódor, B. Liedke, K.-H. Heinig J. Kelling, Appl. Surf. Sci. 258 (2012) 4186 R. Juhász, G. Ódor, J. Stat. Mech. (2012) P08004 J. Kelling, G. Ódor, M. F. Nagy, H. Schulz and K. -H. Heinig, EPJST 210 (2012) G.Ódor, J. Kelling, S. Gemming, Phys. Rev. E 89, (2014)