Annual Reviews of Computational Physics VIII

The following sections are included:

• PREFACE

• CONTENTS

• INTRODUCTION

These notes describe a set of pedagogical lectures delivered in July 1999 at the Regional Summer School on Scaling and Disordered Systems in Zanjan, Iran. Lecture 1 is an introduction to the Langevin method for stochastic dynamics. The general ideas are first introduced in the context of the Brownian motion of a single particle, and then elaborated for the case of a fluctuating surface. Lecture 2 provides a brief review of dynamic scaling phenomena for growing surfaces. A number of simple discrete growth models are described, and compared with continuum Langevin equations that capture their underlying symmetries and conservation laws. Applications of the method to other nonequilibrium situations such as moving flux-lines or drifting polymers are explained in lecture 3. These examples involve the coupling of fields describing fluctuations parallel and perpendicular to the direction of motion. The coupling of different fields is also encountered in lecture 4 which is a brief account of our ongoing research on the growth of “ordered” films. The coupling of surface roughness to the order parameter leads to interesting new phenomena and textures:

Molecular-Beam-Epitaxy is one of the most effective techniques for growing a wide variety of high purity materials. These lecture notes present an introductory review of some of the central ideas in the kinetics of submonolayer and multilayer epitaxial growth. The concepts of a critical island size, dynamical scaling of the island-size distribution, and the barrier to interlayer diffusion (Ehrlich-Schwoebel barrier) are introduced. The results of kinetic Monte Carlo simulations of a realistic model of submonolayer epitaxial growth as well as an analytical expression for the scaled island-size distribution are presented and compared with rate-equation analyses and recent experiments. The results provide a quantitative explanation for the variation of the submonolayer island density, critical island size, island-size distribution and morphology as a function of temperature and deposition rate found in recent experiments. A realistic model for multilayer homoepitaxial growth on fcc and bcc lattices which takes into account the correct crystal structure is presented. The effects of instabilities which lead to mound formation and coarsening are discussed and a unified picture of the effects of attractive and repulsive interactions at ascending and descending steps on surface morphology and island nucleation is presented. An accurate prediction of the mound angle is obtained analytically and by kinetic Monte Carlo simulations, and is compared with experiments. The general dependence of the mound angle, and mound coarsening behavior on temperature, deposition rate, and strength of the step barrier is also presented and compared with recent experiments.

We describe wavelet transformations (WTs) and their basic properties, and discuss their application to data processing. These transformations provide a powerful method for data and image processing by (1) providing a flexible spatial- scale (and time-scale) window which narrows when focusing on small-scale features and widens on large-scale features, and (2) providing the capability for analyzing special characteristics of the data or image around specific points. We then discuss application of WTs to developing a unified and highly efficient method for characterization and simulation of porous media problems, from pore to field scale. The method uses WTs for data mining and characterization, scale-up of the fine-scale model of porous media, and interpretation of their transient flow properties. The data that can be treated by WTs include the direct data, such as various well logs and the permeability distributions, and the indirect data, such as seismic signals, and other characteristics of porous media. Wavelet transformations denoise the data, uncover their special features, and discover the structure of their distribution. They also provide an efficient method for processing seismic data, which are typically in huge amounts, and help interpret them in a meaningful manner. From these analyses one determines the spatial distribution of the fractures and construct a fine-grid model for the porous media. Wavelet transformations then coarsen the fine-grid description of the porous media, such that finer resolution is maintained in regions of high flow rates, whereas coarser property description is applied to the rest of the media. Wavelet transformations can also analyze time-dependent flow data for the porous media, such as long-term production data for oil reservoirs, as well as time-dependent seismic data (i.e. repeated measurements over regular time intervals). Therefore, one has, for the first time, a unified approach to characterization, modeling, and scale-up of models of porous media and extraction of information from their flow properties, from pore to field scale. The computational cost of the method can be orders of magnitude less than those of the most efficient methods currently available.

The following sections are included:

• Introduction

• Numerical Methods

• Experimental Evidence for Correlated Heterogeneity

• Simulation of rate-Controlled Mercury Injection Experiments

• Implications of Correlated Heterogeneity to Two-Phase Flow

• Discussion

• Acknowledgments

• References

The fixed scale transformation (FST) is a rather general method for computing a steady state, self-similar probability measure on an infinite set of configurations generated by evolution rules which typically incoporate both deterministic and stochastic aspects. A basic feature of our approach consists of projecting the state space onto a set of scale invariant product states. We can thus obtain our measure in terms of a finite number of distinct amplitudes which can be determined self-consistently from the evolution rules by means of a steady state condition. For non-Markovian processes lattice path sums enable us to cast the problem in Markovian form. We incorporate these features of the FST into a renormalization group calculation for the correlation length exponents of directed percolation.

We review the statistical mechanics and scaling properties of macromolecular (polymer) solutions. A pedagogic introduction to the modern theory of polymer solutions is given focusing on the link to experimental results. A short review of the analytic and numerical techniques used to study these systems is presented. Whilst the classical approach to polymer physics has been to focus on universal properties, recent work has tended to look at specific properties and how they change the relatively well understood universal properties. We describe in detail some new work on two facets of this nonuniversal behavior, namely the effect of electrostatics and rigidity on polymer conformations.

A generic model of a random polypeptide chain, with discrete torsional degrees of freedom and Hookean springs connecting pairs of hydrophobic residues, reproduces the energy probability distribution of real proteins over a very large range of energies. We show that this system with harmonic interactions, under dissipative dynamics driven by random noise, leads to a distribution of energy states obeying a one-dimensional Ornstein–Uhlenbeck process and giving rise to the so-called Wigner distribution. We find stretched exponential relaxation under Metropolis dynamics at low temperatures with the exponent β ≃ 1/4, in agreement with the best experimental results on proteins. The time dependent correlation functions for fluctuations about the native state, computed in the Gaussian approximation for real proteins, have also been found to have the same functional form. Our results indicate that the energy landscape of polypeptide chains exhibits universal features over a very large range of energies and is relatively independent of the specific dynamics.

The static Casimir effect describes an attractive force between two conducting plates, due to quantum fluctuations of the electromagnetic (EM) field in the intervening space. Thermal fluctuations of correlated fluids (such as critical mixtures, super-fluids, or liquid crystals) are also modified by the boundaries, resulting in finite-size corrections at criticality, and additional forces that effect wetting and layering phenomena. Modified fluctuations of the EM field can also account for the “van der Waals” interaction between conducting spheres, and have analogs in the fluctuation-induced interactions between inclusions on a membrane. We employ a path integral formalism to study these phenomena for boundaries of arbitrary shape. This allows us to examine the many unexpected phenomena of the dynamic Casimir effect due to moving boundaries. With the inclusion of quantum fluctuations, the EM vacuum behaves essentially as a complex fluid, and modifies the motion of objects through it. In particular, from the mechanical response function of the EM vacuum, we extract a plethora of interesting results, the most notable being: (i) The effective mass of a plate depends on its shape, and becomes anisotropic. (ii) There is dissipation and damping of the motion, again dependent upon shape, and direction of motion. (iii) There is a continuous spectrum of resonant cavity modes that can be excited by the motion.

In these lectures, scaling is taken as a symmetry, the symmetry of inflation. Quasicrystals are real materials with inflation symmetry. Their generic physical properties (electrical resistivity, nonstick, nonwet surface, etc.) are caused directly by inflation symmetry. Inflation symmetry is not only esthetically pleasing, but is also a group of specific operations enabling us to calculate the physical properties of the material, a real-space renormalization group.

Tricks are explained to simulate percolation lattices with one program for variable dimensionality, and to look at large lattices in two (or three) dimensions.

In the array of Josephson-junctions at golden mean frustration there exists time-temperature superposition principle similar to that of supercooled liquids for temperatures close to 0.3J/k_B. Below T* such that 0.2 < k_BT* /J < 0.3 this scaling is lost. We propose that T* marks the temperature where well-defined metastable states are developed shaping a rugged landscape and causing the loss of scaling.

Computer simulations with random numbers, that is, Monte Carlo methods, have been considerably applied in recent years to model the fluctuations of stock market or currency exchange rates. Here we concentrate on the percolation model of Cont and Bouchaud, to simulate, not to predict, the market behavior.

A time-delayed stochastic model has been introduced for the predator -prey problem in which the prey are immobile edible plants and the predators are diffusing herbivors. In the stationary state, clusters of predators and preys are formed, although they do not see each other. It is shown that the interaction rate should be lowered by a correction factor because of this clustering.

The phenomenological ageing theory of Azbel as well as many Monte Carlo simulations based on Penna's mutation accumulation model are reviewed. We emphasize the work of Suzana and Paulo Murilo de Oliveira on the growth of life expectancy in the 20th century.