The dynamic approach to fractional Brownian motion (FBM) establishes a link between non-Poisson renewal process with abrupt jumps resetting to zero the system's memory and correlated dynamic processes, whose individual trajectories keep a non-vanishing memory of their past time evolution. It is well known that the recrossing times of the origin by an ordinary 1D diffusion trajectory generates a distribution of time distances between two consecutive origin recrossing times with an inverse power law with index m=1.5. However, with theoretical and numerical arguments, it is proved that this is the special case of a more general condition, insofar as the recrossing times produced by the dynamic FBM generates process with m=2-H. Later, the model of ballistic deposition is studied, which is as a simple way to establish cooperation among the columns of a growing surface, to show that cooperation generates memory properties and, at same time, non-Poisson renewal events. Finally, the connection between trajectory and density memory is discussed, showing that the trajectory memory does not necessarily yields density memory, and density memory might be compatible with the existence of abrupt jumps resetting to zero the system's memory.

Non-equilibrium surface growth for competitive growth models in (1+1) dimensions, particularly mixing random deposition (RD) with correlated growth process which occur with probability $p$ are studied. The composite mixtures are found to be in the universality class of the correlated growth process, and a nonuniversal exponent $\delta$ is identified in the scaling in $p$. The only effects of the RD admixture are dilations of the time and height scales which result in a slowdown of the dynamics of building up the correlations. The bulk morphology is taken into account and is reflected in the surface roughening, as well as the scaling behavior. It is found that the continuum equations and scaling laws for RD added, in particular, to Kardar-Parisi-Zhang (KPZ) processes are partly determined from the underlying bulk structures. Nonequilibrium surface growth analysis are also applied to a study of the static and dynamic load balancing for a conservative update algorithm for Parallel Discrete Event Simulations (PDES). This load balancing is governed by the KPZ equation. For uneven load distributions in conservative PDES simulations, the simulated (virtual) time horizon (VTH) per Processing Element (PE) and the imulated time horizon per volume element $N_{v}$ are used to study the PEs progress in terms of utilization. The width of these time horizons relates to the desynchronization of the system of processors, and is related to the memory requirements of the PEs. The utilization increases when the dynamic, rather than static, load balancing is performed.

Lattice Monte Carlo methods are used to investigate far from and out-of-equilibrium systems, including surface growth, spin systems and solid mixtures. Applications range from the determination of universal growth or aging behaviors to palpable systems, where coarsening of nanocomposites or self-organization of functional nanostructures are of interest. Such studies require observations of large systems over long times scales, to allow structures to grow over orders of magnitude, which necessitates massively parallel simulations. This work addresses the problem of parallel processing introducing correlations in Monte Carlo updates and proposes a virtually correlation-free domain decomposition scheme to solve it. The effect of correlations on scaling and dynamical properties of surface growth systems and related lattice gases is investigated further by comparing results obtained by correlation-free and intrinsically correlated but highly efficient simulations using a stochastic cellular automaton (SCA). Efficient massively parallel implementations on graphics processing units (GPUs) were developed, which enable large-scale simulations leading to unprecedented precision in the final results. The primary subject of study is the Kardar–Parisi–Zhang (KPZ) surface growth in (2 + 1) dimensions, which is simulated using a dimer lattice gas and the restricted solid-on-solid model (RSOS) model. Using extensive simulations, conjectures regard- ing growth, autocorrelation and autoresponse properties are tested and new precise numerical predictions for several universal parameters are made.:1. Introduction 1.1. Motivations and Goals 1.2. Overview 2. Methods and Models 2.1. Estimation of Scaling Exponents and Error Margins 2.2. From Continuum- to Atomistic Models 2.3. Models for Phase Ordering and Nanostructure Evolution 2.3.1. The Kinetic Metropolis Lattice Monte-Carlo Method 2.3.2. The Potts Model 2.4. The Kardar–Parisi–Zhang and Edwards–Wilkinson Universality Classes 2.4.0.1. Physical Aging 2.4.1. The Octahedron Model 2.4.2. The Restricted Solid on Solid Model 3. Parallel Implementation: Towards Large-Scale Simulations 3.1. Parallel Architectures and Programming Models 3.1.1. CPU 3.1.2. GPU 3.1.3. Heterogeneous Parallelism and MPI 3.1.4. Bit-Coding of Lattice Sites 3.2. Domain Decomposition for Stochastic Lattice Models 3.2.1. DD for Asynchronous Updates 3.2.1.1. Dead border (DB) 3.2.1.2. Double tiling (DT) 3.2.1.3. DT DD with random origin (DTr) 3.2.1.4. Implementation 3.2.2. Second DD Layer on GPUs 3.2.2.1. Single-Hit DT 3.2.2.2. Single-Hit dead border (DB) 3.2.2.3. DD Parameters for the Octahedron Model 3.2.3. Performance 3.3. Lattice Level DD: Stochastic Cellular Automaton 3.3.1. Local Approach for the Octahedron Model 3.3.2. Non-Local Approach for the Octahedron Model 3.3.2.1. Bit-Vectorized GPU Implementation 3.3.3. Performance of SCA Implementations 3.4. The Multi-Surface Coding Approach 3.4.0.1. Vectorization 3.4.0.2. Scalar Updates 3.4.0.3. Domain Decomposition 3.4.1. Implementation: SkyMC 3.4.1.1. 2d Restricted Solid on Solid Model 3.4.1.2. 2d and 3d Potts Model 3.4.1.3. Sequential CPU Reference 3.4.2. SkyMC Benchmarks 3.5. Measurements 3.5.0.1. Measurement Intervals 3.5.0.2. Measuring using Heterogeneous Resources 4. Monte-Carlo Investigation of the Kardar–Parisi–Zhang Universality Class 4.1. Evolution of Surface Roughness 4.1.1. Comparison of Parallel Implementations of the Octahedron Model 4.1.1.1. The Growth Regime 4.1.1.2. Distribution of Interface Heights in the Growth Regime 4.1.1.3. KPZ Ansatz for the Growth Regime 4.1.1.4. The Steady State 4.1.2. Investigations using RSOS 4.1.2.1. The Growth Regime 4.1.2.2. The Steady State 4.1.2.3. Consistency of Fine-Size Scaling with Respect to DD 4.1.3. Results for Growth Phase and Steady State 4.2. Autocorrelation Functions 4.2.1. Comparison of DD Methods for RS Dynamics 4.2.1.1. Device-Layer DD 4.2.1.2. Block-Layer DD 4.2.2. Autocorrelation Properties under RS Dynamics 4.2.3. Autocorrelation Properties under SCA Dynamics 4.2.3.1. Autocorrelation of Heights 4.2.3.2. Autocorrelation of Slopes 4.2.4. Autocorrelation in the SCA Steady State 4.2.5. Autocorrelation in the EW Case under SCA 4.2.5.1. Autocorrelation of Heights 4.2.5.2. Autocorrelations of Slopes 4.3. Autoresponse Functions 4.3.1. Autoresponse Properties 4.3.1.1. Autoresponse of Heights 4.3.1.2. Autoresponse of Slopes 4.3.1.3. Self-Averaging 4.4. Summary 5. Further Topics 5.1. Investigations of the Potts Model 5.1.1. Testing Results from the Parallel Implementations 5.1.2. Domain Growth in Disordered Potts Models 5.2. Local Scale Invariance in KPZ Surface Growth 6. Conclusions and Outlook Acknowledgements A. Coding Details A.1. Bit-Coding A.2. Packing and Unpacking Signed Integers A.3. Random Number Generation / Gitter-Monte-Carlo-Methoden werden zur Untersuchung von Systemen wie Oberflächenwachstum, Spinsystemen oder gemischten Feststoffen verwendet, welche fern eines Gleichgewichtes bleiben oder zu einem streben. Die Anwendungen reichen von der Bestimmung universellen Wachstums- und Alterungsverhaltens hin zu konkreten Systemen, in denen die Reifung von Nanokompositmaterialien oder die Selbstorganisation von funktionalen Nanostrukturen von Interesse sind. In solchen Studien müssen große Systemen über lange Zeiträume betrachtet werden, um Strukturwachstum über mehrere Größenordnungen zu erlauben. Dies erfordert massivparallele Simulationen. Diese Arbeit adressiert das Problem, dass parallele Verarbeitung Korrelationen in Monte-Carlo-Updates verursachen und entwickelt eine praktisch korrelationsfreie Domänenzerlegungsmethode, um es zu lösen. Der Einfluss von Korrelationen auf Skalierungs- und dynamische Eigenschaften von Oberflächenwachtums- sowie verwandten Gittergassystemen wird weitergehend durch den Vergleich von Ergebnissen aus korrelationsfreien und intrinsisch korrelierten Simulationen mit einem stochastischen zellulären Automaten untersucht. Effiziente massiv parallele Implementationen auf Grafikkarten wurden entwickelt, welche großskalige Simulationen und damit präzedenzlos genaue Ergebnisse ermöglichen. Das primäre Studienobjekt ist das (2 + 1)-dimensionale Kardar–Parisi–Zhang- Oberflächenwachstum, welches durch ein Dimer-Gittergas und das Kim-Kosterlitz-Modell simuliert wird. Durch massive Simulationen werden Thesen über Wachstums-, Autokorrelations- und Antworteigenschaften getestet und neue, präzise numerische Vorhersagen zu einigen universellen Parametern getroffen.:1. Introduction 1.1. Motivations and Goals 1.2. Overview 2. Methods and Models 2.1. Estimation of Scaling Exponents and Error Margins 2.2. From Continuum- to Atomistic Models 2.3. Models for Phase Ordering and Nanostructure Evolution 2.3.1. The Kinetic Metropolis Lattice Monte-Carlo Method 2.3.2. The Potts Model 2.4. The Kardar–Parisi–Zhang and Edwards–Wilkinson Universality Classes 2.4.0.1. Physical Aging 2.4.1. The Octahedron Model 2.4.2. The Restricted Solid on Solid Model 3. Parallel Implementation: Towards Large-Scale Simulations 3.1. Parallel Architectures and Programming Models 3.1.1. CPU 3.1.2. GPU 3.1.3. Heterogeneous Parallelism and MPI 3.1.4. Bit-Coding of Lattice Sites 3.2. Domain Decomposition for Stochastic Lattice Models 3.2.1. DD for Asynchronous Updates 3.2.1.1. Dead border (DB) 3.2.1.2. Double tiling (DT) 3.2.1.3. DT DD with random origin (DTr) 3.2.1.4. Implementation 3.2.2. Second DD Layer on GPUs 3.2.2.1. Single-Hit DT 3.2.2.2. Single-Hit dead border (DB) 3.2.2.3. DD Parameters for the Octahedron Model 3.2.3. Performance 3.3. Lattice Level DD: Stochastic Cellular Automaton 3.3.1. Local Approach for the Octahedron Model 3.3.2. Non-Local Approach for the Octahedron Model 3.3.2.1. Bit-Vectorized GPU Implementation 3.3.3. Performance of SCA Implementations 3.4. The Multi-Surface Coding Approach 3.4.0.1. Vectorization 3.4.0.2. Scalar Updates 3.4.0.3. Domain Decomposition 3.4.1. Implementation: SkyMC 3.4.1.1. 2d Restricted Solid on Solid Model 3.4.1.2. 2d and 3d Potts Model 3.4.1.3. Sequential CPU Reference 3.4.2. SkyMC Benchmarks 3.5. Measurements 3.5.0.1. Measurement Intervals 3.5.0.2. Measuring using Heterogeneous Resources 4. Monte-Carlo Investigation of the Kardar–Parisi–Zhang Universality Class 4.1. Evolution of Surface Roughness 4.1.1. Comparison of Parallel Implementations of the Octahedron Model 4.1.1.1. The Growth Regime 4.1.1.2. Distribution of Interface Heights in the Growth Regime 4.1.1.3. KPZ Ansatz for the Growth Regime 4.1.1.4. The Steady State 4.1.2. Investigations using RSOS 4.1.2.1. The Growth Regime 4.1.2.2. The Steady State 4.1.2.3. Consistency of Fine-Size Scaling with Respect to DD 4.1.3. Results for Growth Phase and Steady State 4.2. Autocorrelation Functions 4.2.1. Comparison of DD Methods for RS Dynamics 4.2.1.1. Device-Layer DD 4.2.1.2. Block-Layer DD 4.2.2. Autocorrelation Properties under RS Dynamics 4.2.3. Autocorrelation Properties under SCA Dynamics 4.2.3.1. Autocorrelation of Heights 4.2.3.2. Autocorrelation of Slopes 4.2.4. Autocorrelation in the SCA Steady State 4.2.5. Autocorrelation in the EW Case under SCA 4.2.5.1. Autocorrelation of Heights 4.2.5.2. Autocorrelations of Slopes 4.3. Autoresponse Functions 4.3.1. Autoresponse Properties 4.3.1.1. Autoresponse of Heights 4.3.1.2. Autoresponse of Slopes 4.3.1.3. Self-Averaging 4.4. Summary 5. Further Topics 5.1. Investigations of the Potts Model 5.1.1. Testing Results from the Parallel Implementations 5.1.2. Domain Growth in Disordered Potts Models 5.2. Local Scale Invariance in KPZ Surface Growth 6. Conclusions and Outlook Acknowledgements A. Coding Details A.1. Bit-Coding A.2. Packing and Unpacking Signed Integers A.3. Random Number Generation