During molecular beam epitaxy, atoms arrive on the surface, and diffuse over a length ℓs before being incorporated into the crystal. Can we estimate ℓs The solution to this problem is contained in review articles by Stoyanov & Kashchiev (1981) and by Venables et al. (1984).

The simplest case is realized when two diffusing adatoms form an immobile and undissociable pair when they meet, and the resulting cluster grows with a compact shape. Stoyanov & Kashchiev have shown that in this case ℓs ∼ (D/F)1/6, where D is the diffusion constant of adatoms and F the deposition flux. More generally, they have shown that ℓs ∼ (D/F)γ, where γ is a function of the critical nucleus size. The critical nucleus is defined as the largest dissociable atom cluster. The experimental verification is not that easy: the critical nucleus size, and thus γ, depend in an unspecified way on the temperature and the deposition flux. A large host of computer simulation data (Monte Carlo) have anyway confirmed the relation between ℓs, D and F.

Orders of magnitude – The lengths should be less than the distances between defects on the surface, i.e. a few microns at the very best. The temperature must be low–which means below 0 °C for metals–otherwise ℓs becomes too long.

Experimental techniques– Microscopy (STM, LEEM, REM), atom-beam scattering, high-energy electron diffraction.

The one-dimensional diffusion automaton

In the previous chapter, we discussed several cellular automata rules which definitely had something to do with the description of physical processes. The question is of course how close these models are to the reality they are supposed to simulate?

In the real world, space and time are not discrete and, in classical physics, the state variables are continuous. Thus, it is crucial to show how a cellular automaton rule is connected to the laws of physics or to the usual quantities describing the phenomena which is modeled. This is particularly important if the cellular automaton is intended to be used as a numerical scheme to solve practical problems.

Lattice gas automata have a large number of potential applications in hydrodynamics and reaction-diffusion processes. The purpose of this chapter is to present the techniques that are used to establish the connection between the macroscopic physics and the microscopic discrete dynamics of lattice gas automata. The problem one has to address is the statistical description of a system of many interacting particles. The methods we shall discuss here are very close, in spirit, to those applied in kinetic theory: the N-body dynamics is described in terms of macroscopic quantities such as the particle density or the velocity field. The derivation of a Boltzmann equation is a main step in this process.

In order to illustrate and understand the main steps leading to a macroscopic description of a cellular automaton model, we first consider a linear rule for which a simple and rigorous calculation can be carried out.

This chapter presents a few more applications of the cellular automata and lattice Boltzmann techniques. We introduce some new ideas and models that have not been discussed in detail previously and which can give useful hints on how to address different problems.

We shall first discuss a lattice BGK model for wave propagation in a heterogeneous media and show how it can be applied to simulate a fracture process or make predictions for radio wave propagation inside a city. Second, we shall present how van der Waals and gravity forces can be included in an FHP cellular automata fluid in order to simulate the spreading of a liquid droplet on a wetting substrate. Then, we shall define a multiparticle fluid with a collision operator inspired by the lattice BGK method in order to avoid numerical instabilities and re-introduce fluctuations in a natural way. Finally, we shall explain how particles in suspension can be transported and eroded by a fluid flow and deposited on the ground. Snowdrift and sand dunes formation is a typical domain where this last model is applicable.

Wave propagation

One-dimensional waves

In this book, we have already encountered one-dimensional wave propagation. The chains of particles, or strings, discussed in section 2.2.9 move because of their internal shrinking and stretching. From a more physical point of view, this internal motion is mediated by longitudinal backward and forward deformation waves.

The cellular automata approach and the related modeling techniques are powerful methods to describe, understand and simulate the behavior of complex systems. The aim of this book is to provide a pedagogical and self-contained introduction to this field and also to introduce recent developments. Our main goal is to present the fundamental theoretical concepts necessary for a researcher to address advanced applications in physics and other scientific areas.

In particular, this book discusses the use of cellular automata in the framework of equilibrium and nonequilibrium statistical physics and in application-oriented problems. The basic ideas and concepts are illustrated on simple examples so as to highlight the method. A selected bibliography is provided in order to guide the reader through this expanding field.

Several relevant domains of application have been mentioned only through references to the bibliography, or are treated superficially. This is not because we feel these topics are less important but, rather, because a somewhat subjective selection was necessary according to the scope of the book. Nevertheless, we think that the topics we have covered are significant enough to give a fair idea of how the cellular automata technique may be applied to other systems.

This book is written for researchers and students working in statistical physics, solid state physics, chemical physics and computer science, and anyone interested in modeling complex systems. A glossary is included to give a definition of several technical terms that are frequently used throughout the text. At the end of the first six chapters, a selection of problems is given.

Scattering and transport

When scattering occurs in systems of large spatial extension like solids, a relationship appears between scattering and transport. This is particularly evident in the process of transport of thermal neutrons in nuclear reactors. On the one hand, neutrons may form beams for the study of matter and, on the other hand, as soon as the target presents spatially distributed scattering centres, the scattering process becomes a diffusion process, i.e., a process of transport as studied in kinetic theory. Clearly, both processes are similar and the difference only appears in the number and distribution of scatterers. Therefore, a fundamental connection exists between scattering theory and nonequilibrium statistical mechanics. The scattering approach to diffusion is also natural since diffusion is studied in finite pieces of material in the laboratory. Diffusion is a property of bulk matter which is extrapolated from experiments on finite samples to a hypothetical infinite sample.

Classically, the scattering on a spatially distributed target may be expected to be chaotic because the collisions on spherical scatterers have a defocusing character. Chaoticity will play an important role in such a connection. In the following, we shall elaborate in this direction with the tools developed in the previous chapters to obtain the so-called escape-rate formulas for the transport coefficients, which precisely express such a relationship (Gaspard and Nicolis 1990, Gaspard and Baras 1995).

We should mention here that Lax and Phillips (1967) proposed in the sixties a scattering theory of transport phenomena based on the properties of classical dynamics.