Introduction

Diffusive phenomena play an important role in many areas of physics, chemistry and biology and still constitute an active field of research. There are many applications involving diffusion for which a particle based model, such as a lattice gas dynamics, could provide a useful approach and efficient numerical simulations.

For instance, processes such as aggregation, formation of a diffusion front, trapping of particles performing a random walk in some specific region of space, or the adsorption of diffusing particles on a substrate are important problems that are difficult to solve with the standard diffusion equation. A microscopic model, based on a cellular automata dynamics, is therefore of clear interest.

Diffusion is also a fundamental ingredient in reaction-diffusion phenomena that will be discussed in detail in the next chapter. Reaction processes such as A + B → C, as well as growth mechanisms are naturally implemented in the framework of a point particle description, often with the help of threshold rules. Consequently, microscopic fluctuations are often relevant at a macroscopic level of observation because they make symmetry breaking possible and are responsible for triggering complex patterns.

Cellular automata particles can be equipped with diffusive and reactive properties, in order to mimic real experiments and model several complex reaction-diffusion-growth processes in the same spirit as a cellular automata fluid simulates fluid flow: these systems are expected to retain the relevant aspects of the microscopic world they are modeling.

Superfluidity in liquid 3He, discovered in 1972, is due to pairing, as is the case for superconductivity in ordinary metals. However, the pair is in the 3P state so that a new aspect of superfluidity appears with internal degrees of freedom. It offers quite an interesting real example of ‘spontaneous symmetry breaking’. This aspect will be emphasised in the following discussion.

Fermi liquid 3He

A 3He atom is a fermion with nuclear spin ½. Although 3He gas liquefies at low temperatures due to the weak van der Waals force, it does not solidify even at T = 0 unless the pressure exceeds 33.5 atm (3.4 × 106 Pa) as shown in the phase diagram (Fig. 6.1 (a)). While Bose liquid 4He becomes superfluid at about 2K, 3He is Fermi-degenerate at less than TF ∼ 0.1 K and shows the properties of a normal Fermi gas. For example, the specific heat and the susceptibility are proportional to T at T < TF. It is only at an ultra-low temperature in the mK-region that the system undergoes a transition to the superfluid state.

Figure 6.1(b) is the phase diagram showing the superfluid phases with variables T,P and external magnetic field H in the mK-region. It should be noted that there are two thermodynamically distinct phases called the A phase and the B phase even at H = 0, and that for H ≠ 0 the A1 phase appears.

Superconductivity and superfluidity have already been studied for almost one century. In the case of superconductivity one may say it reached its peak with the dramatic appearance of BCS (Bardeen–Cooper–Schrieffer) theory in 1956. Since it was so successful, the research developed at an explosive rate, and by the beginning of the 1970s we thought the fundamental theories were well established. It is therefore logical that many of the well-known texts on superconductivity were written during this period. Superfluidity of liquid 3He was also studied most actively during the early 1970s. Many people therefore had the impression that research in the field of superconductivity and superfluidity had already reached its peak. However, much interesting progress has been made since then. First, the superfluidity of liquid 3He due to non-s-wave pairing was discovered in 1973, and became the subject of intense research, both experimental and theoretical. The generalised BCS theory again turned out to be quite successful, and over about ten years we obtained a basic understanding of this phenomenon. In the 1980s, stimulated by the superfluidity of liquid 4He, people started to look for non-s-wave superconductivity in the heavy fermion systems. Then, rather unexpectedly, we witnessed the biggest event of recent years, namely the discovery of copper oxide high temperature superconductors by J.G. Bednorz and K.A. Müller in 1986 ([G-1]).

With this rich history before us, it is clearly an extremely difficult task to decide which topics to cover.

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.

Condensate in confining potential

This subject is directly related to Chapter 1 of this book and will be briefly summarised here although it has already been treated in an appendix of [E-11].

Spin polarised hydrogen H↓ is listed in the right hand column of Table 1.1 as a system predicted to show Bose-type superfluidity. Bose–Einstein condensation was observed, somewhat unexpectedly, in spin polarised alkaline gases Li↓, Na↓, and Rb↓, with quite an ingenious method using laser light, prior to observation in H↓ gas (see [H-1] – [H-3]). It is unquestionable that the macroscopic wave function appears although superfluidity has yet to be observed. See [H-4] for works on H↓, and related systems preceding this breakthrough.

In a typical experiment the magneto-optical trap, which we can approximate by an anisotropic 3-dimensional harmonic oscillator potential, is used to provide the confining potential for atoms. Its spatial scale is given by R ∼ (ħ/mω)½ ∼ 10−3m, where m is the mass of the atom while ω is the frequency of the harmonic oscillator. There are N ∼ 106 atoms trapped in a volume ∼ R3, for which the Bose–Einstein condensation temperature TBE given by Eq. (1.3) is of the order of 10−7 K for Rb atoms. It should be added that evaporation cooling, which has been used to cool H↓ gas, is required in addition to laser cooling to reach this temperature.

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.