Ageing of the Ageing of the 2+12+1 dimensional dimensional Kardar-Parisi Zhang modelKardar-Parisi Zhang model

Géza ÓdorGéza Ódor, , BudapestBudapest (MTA-TTK-MFA)(MTA-TTK-MFA)

Jeffrey Kelling, S. GemmingJeffrey Kelling, S. Gemming Dresden (Dresden (HHZDZDRR)),,

MECO39 Coventry 08/04/2014

www.mfa.kfki.hu/~odor

The The KKardar-ardar-PParisi-arisi-ZZhang (hang (KPZKPZ) equation) 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:

AAttachmentttachment (with probability (with probability pp) and ) and

DDetachmentetachment (with probability (with probability qq)) Corresponds Corresponds to anisotropic diffusion of to anisotropic diffusion of particles (bullets) along the particles (bullets) along the 1d 1d base space base space ( (Plischke & Plischke & RRáácz 1987)cz 1987)

The simple The simple ASEPASEP (Ligget '95) (Ligget '95) iis an s an exactly solved 1d lattice gasexactly solved 1d lattice gas

Many known features: response toMany known features: response to disorder, different boundary disorder, different boundary conditions ... are known.conditions ... are known.

Widespread application in biologyWidespread application in biology

Mapping of KPZ onto ASEP in Mapping of KPZ onto ASEP in 1d1d

Kawasaki' exchange of particles

Mapping of KPZ growth in Mapping of KPZ growth in 2+1 d2+1 d

Generalized Kawasaki update:Generalized Kawasaki update:

Octahedron modelOctahedron model

Driven diffusive gas of Driven diffusive gas of pairs (dimers) pairs (dimers)

G. Ódor, B. Liedke and K.-H. Heinig, G. Ódor, B. Liedke and K.-H. Heinig, PRE79, 021125 (2009 PRE79, 021125 (2009))

G. Ódor, B. Liedke and K.-H. Heinig, G. Ódor, B. Liedke and K.-H. Heinig, PRE79, 031112 (2010 PRE79, 031112 (2010))

Surface pattern formation via Surface pattern formation via dimer modeldimer model

G. Ódor, B. Liedke and K.-H. Heinig, G. Ódor, B. Liedke and K.-H. Heinig, PRE79, 051114 (2010 PRE79, 051114 (2010))

CUDA code for CUDA code for 2d2d KPZ KPZ

Each Each 3232-bit word stores-bit word storesthe slopes of the slopes of 4 x 44 x 4 sites sites

SpeedupSpeedup 230230 x x (Fermi)(Fermi) with respect with respect aa CPU CPU core of 2.8 GHz up to:core of 2.8 GHz up to:131072 x 131972131072 x 131972 size size

J. Kelling and G. Ódor Phys. Rev. E 84 (2011) 061150

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

Two-time aging observables

Time-dependent order-parameter field: (t; r) t : observation time, s : start timeScaling 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 >> sDifferent 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 pi = 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 = -0.483(2) C /z = 1.21(1) + oscillations due to kinematic vawessimulation 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 105 50001

Full agreement

Auto-response results

Fast oscillating decay, Low signal/noise ratio,Very slow convergence

GPU and CPU results agree and providea = 0.3, R/z= 1.25(1)

Fluctuation – Dissipation is broken weakly

Conclusions & outlookConclusions & outlook Fast parallel simulations due to mapping onto stochastic cellular automata Fast parallel simulations due to mapping onto stochastic cellular automata

(lattice gases)(lattice gases)

Extremely large scale Extremely large scale ((2215 15 x 2x 21515) ) simulations on GPUs and CPUs simulations on GPUs and CPUs

GPU speedup ~230 with respect to a single CPU coreGPU speedup ~230 with respect to a single CPU core

Ageing exponents of Ageing exponents of 2+1 d2+1 d KPZ are determined numerically KPZ are determined numerically

This also describes the behavior of driven lattice gas of dimersThis also describes the behavior of driven lattice gas of dimers

Lack of fluctuation-dissipation is shown explicitlyLack of fluctuation-dissipation is shown explicitly

Generalization to higher dimensions is straightforwardGeneralization to higher dimensions is straightforward

Local Scale Invariance hypothesis can be tested Local Scale Invariance hypothesis can be tested

AcknowledgementsAcknowledgements: : DAAD-MDAAD-MÖÖB, OTKA, OSIRIS FP7, NVIDIAB, OTKA, OSIRIS FP7, NVIDIA

Publications:Publications: H. Schulz, G. Ódor, G. Ódor, M. F. Nagy, Computer Physics Communications 182 (2011) 1467. 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), 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) 4186 G. Ódor, B. Liedke, K.-H. Heinig J. Kelling, Appl. Surf. Sci. 258 (2012) 4186 R. Juhász, G. Ódor, R. Juhász, G. Ódor, J. Stat. Mech. (2012) P08004J. Stat. Mech. (2012) P08004 J. Kelling, G. Ódor, M. F. Nagy, H. Schulz and K. -H. Heinig, EPJST 210 (2012) 175-187J. 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) G.Ódor, J. Kelling, S. Gemming, Phys. Rev. E 89, 032146 (2014)