Brief history

Cellular automata (often termed CA) are an idealization of a physical system in which space and time are discrete, and the physical quantities take only a finite set of values.

Although cellular automata have been reinvented several times (often under different names), the concept of a cellular automaton dates back from the late 1940s. During the following fifty years of existence, cellular automata have been developed and used in many different fields. A vast body of literature is related to these topics. Many conference proceedings), special journal issues and articles are available.

In this section, our purpose is not to present a detailed history of the developments of the cellular automata approach but, rather, to emphasize some of the important steps.

Self-reproducing systems

The reasons that have led to the elaboration of cellular automata are very ambitious and still very present. The pioneer is certainly John von Neumann who, at the end of the 1940s, was involved in the design of the first digital computers. Although von Neumann's name is definitely associated with the architecture of today's sequential computers, his concept of cellular automata constitutes also the first applicable model of massively parallel computation.

Von Neumann was thinking of imitating the behavior of a human brain in order to build a machine able to solve very complex problems. However, his motivation was more ambitious than just a performance increase of the computers of that time.

As long as possible we have postponed a general study of elasticity and its partial differential equations. Here they are!

The elastic equilibrium of a solid is generally treated in the continuum approximation. The strain satisfies certain equations in the bulk, and other equations at the surface. This set of equations has an infinite number of solutions, and the correct one is that which minimizes a given thermodynamic potential or free energy. This minimization is not needed for a semi-infinite solid because the good solution in this case is the one which vanishes at infinity.

The power of continuous elasticity theory is limited. In particular it is not appropriate to investigate the surface relaxation, i.e. the change in the atomic distance near the surface. Nevertheless, the continuum approximation allows for spectacular predictions, for instance the Asaro-Tiller-Grinfeld instability, which is one of the major obstacles to layer-by-layer heteroepitaxial growth.

Memento of elasticity in a bulk solid

In this section, the theory of linear elasticity in a homogeneous solid away from the surface will be recalled.

In order to write the condition for mechanical equilibrium, one has to consider the forces acting on a volume δV of the solid (Fig. 16.1). There may be an external force δfext, and there is a force produced by the part of the solid outside δV.

Why cellular automata are useful in physics

The purpose of this section is to show different reasons why cellular automata may be useful in physics. In a first paragraph, we shall consider cellular automata as simple dynamical systems. We shall see that although defined by very simple rules, cellular automata can exhibit, at a larger scale, complex dynamical behaviors. This will lead us to consider different levels of reality to describe the properties of physical systems. Cellular automata provide a fictitious microscopic world reproducing the correct physics at a coarse-grained scale. Finally, in a third section, a sampler of rules modeling simple physical systems is given.

Cellular automata as simple dynamical systems

In physics, the time evolution of physical quantities is often governed by nonlinear partial differential equations. Due to the nonlinearities, solution of these dynamical systems can be very complex. In particular, the solution of these equation can be strongly sensitive to the initial conditions, leading to what is called a chaotic behavior. Similar complications can occur in discrete dynamical systems. Models based on cellular automata provide an alternative approach to study the behavior of dynamical systems. By virtue of their simplicity, they are potentially amenable to easier analysis than continuous dynamical systems. The numerical studies are free of rounding approximations and thus lead to exact results.

Crudely speaking, two classes of problem can be posed. First, given a cellular automaton rule, predicts its properties.

While the first chapter of this book dwelt on the confirmation of the fundamental ideas of the philosophers of the fifth century BC, this last chapter will be devoted to the twentieth century and to the transformation of our materialistic everyday life brought about by electronics.

The early developments employed electronic tubes which owed nothing to surface physics and crystal growth. Then came the transistor and the realm of silicon, of which huge, dislocation-free crystals can be grown from the melt. In this end of the twentieth century, electromagnetic waves play an increasingly important part. Their production and detection require more and more complex semiconducting materials, often grown by molecular beam epitaxy.

Introduction

The modernity of surface science lies for a large part in the fact that new experimental techniques now make surfaces accessible to investigation. But the present interest in surface physics is also much motivated by the technological importance of crystals grown with a well-controlled composition and a high degree of perfection, and this depends very much on the state of the crystal surface during growth.

The purpose of this chapter is to outline the technological importance of materials with a controlled purity and morphology.

We shall mainly discuss semiconductors. It is not very easy to find simple articles or books explaining semiconductor electronics to the ignorant, but the book by Parker (1994) accomplishes this function.