When grains are deposited on a sandpile, avalanches result. These have much in common with many other varieties of avalanche; for example, snow or rocks, or even the stress releases that result in earthquakes. The unifying phenomenon in all these cases is that of a threshold instability: an overburden builds up, typically that due to surface roughening, to the point where this threshold is crossed, and grains are released in an avalanche. Avalanches can be classed in two main categories; those that do not have intrinsic time or length scales, and those that do. Avalanches relevant to granular media belong to the second category, and we shall discuss their characteristics in depth. We will, however summarise some of the known characteristics of the first category, referring readers who are interested in more details to ref. on the subject.

Avalanches type I – SOC

Bak, Tang and Wiesenfeld, in their now famous theory of self-organised criticality (SOC), suggested that extended systems were marginally stable, such that the slightest overburdening would cause avalanching; a sandpile at its so-called ‘critical’ angle of repose was held to be paradigmatic of this. Although this turned out to be, in retrospect, the wrong paradigm, the explorations that surrounded it in fact greatly enriched the physics of granular avalanches. We touch briefly on the important features of SOC here.

Sand has many avatars – it can behave as a solid, liquid or gas, depending on external circumstances. This multiple identity is one of several reasons why the computer simulation of dry granular materials is difficult. Sand in the solid-like state responds to external stimuli on a very different timescale to sand in its liquidlike avatar – in contrast to most efficient computer simulation methods, which are typically tuned to one particular timescale such as a collision or relaxation time. Other features of sand which are difficult to simulate efficiently include complex, dissipative interparticle and particle–wall interactions, typically irregular grain shapes and strong hysteretic effects. Furthermore, the athermal nature of sand means that grains do not randomly sample all possible states ergodically – as a result, appropriate statistical averages can only be obtained by repeated (computationally demanding) simulations of a granular system.

For normal dry powders, interstitial fluid plays only a minor role – apart from exceptional cases when, say, there are small liquid pools at particle contacts which could seriously alter the pairwise nature of grain interactions. This is a clear distinction between granular systems and dense suspensions – for the former, interparticle interactions are restricted to short-ranged contact forces. In practice, the methods developed for granular simulations are quite similar to classical methods used to simulate simple liquids.

Introduction

Some general remarks

The two previous chapters have dealt, in different guises, with the post-avalanche smoothing of a sandpile which is expected to happen in nature. It is clear what happens physically: an avalanche provides a means of shaving off roughness from the surface of a sandpile by transferring grains from bumps to available voids, and thus leaves in its wake a smoother surface. However, surprisingly little research has been done on this phenomenon so far, despite its ubiquity in nature, ranging from snow to rock avalanches.

In particular, what has not attracted enough attention in the literature is the qualitative difference between the situations which obtain when sandpiles exhibit intermittent and continuous avalanches. In this chapter we examine both the latter situations, via coupled continuum equations of sandpile surfaces. These were originally envisaged as the local version of coupled equations that had been written down using global variables in; subsequently, many versions were introduced in the literature to model different situations. The use of these equations has also since been diversified into many areas, including ripple formation and the propagation of sand dunes, about which we will have something to say at the end of this chapter.

In order to discuss this, we introduce first the notion that granular dynamics is well described by the competition between the dynamics of grains moving independently of each other and that of their collective motion within clusters.

General definitions

The laws of thermodynamics are based on observations of macroscopic bodies, and encapsulate their thermal properties. On the other hand, matter is composed of atoms and molecules whose motions are governed by more fundamental laws (classical or quantum mechanics). It should be possible, in principle, to derive the behavior of a macroscopic body from the knowledge of its components. This is the problem addressed by kinetic theory in the following chapter. Actually, describing the full dynamics of the enormous number of particles involved is quite a daunting task. As we shall demonstrate, for discussing equilibrium properties of a macroscopic system, full knowledge of the behavior of its constituent particles is not necessary. All that is required is the likelihood that the particles are in a particular microscopic state. Statistical mechanics is thus an inherently probabilistic description of the system, and familiarity with manipulations of probabilities is an important prerequisite. The purpose of this chapter is to review some important results in the theory of probability, and to introduce the notations that will be used in the following chapters.

General definitions

Kinetic theorystudies the macroscopic properties of large numbers of particles, starting from their (classical) equations of motion.

Thermodynamics describes the equilibrium behavior of macroscopic objects in terms of concepts such as work, heat, and entropy. The phenomenological laws of thermodynamics tell us how these quantities are constrained as a system approaches its equilibrium. At the microscopic level, we know that these systems are composed of particles (atoms, molecules), whose interactions and dynamics are reasonably well understood in terms of more fundamental theories. If these microscopic descriptions are complete, we should be able to account for the macroscopic behavior, that is, derive the laws governing the macroscopic state functions in equilibrium. Kinetic theory attempts to achieve this objective. In particular, we shall try to answer the following questions:

1. How can we define “equilibrium” for a system of moving particles?

2. Do all systems naturally evolve towards an equilibrium state?

3. What is the time evolution of a system that is not quite in equilibrium?

The simplest system to study, the veritable workhorse of thermodynamics, is the dilute (nearly ideal) gas. A typical volume of gas contains of the order of 1023 particles, and in kinetic theory we try to deduce the macroscopic properties of the gas from the time evolution of the set of atomic coordinates.

General definitions

Statistical mechanicsis a probabilistic approach to equilibrium macroscopic properties of large numbers of degrees of freedom.

As discussed in chapter 1, equilibrium properties of macroscopic bodies are phenomenologically described by the laws of thermodynamics. The macrostate M depends on a relatively small number of thermodynamic coordinates. To provide a more fundamental derivation of these properties, we can examine the dynamics of the many degrees of freedom comprising a macroscopic body. Description of each microstate µ requires an enormous amount of information, and the corresponding time evolution, governed by the Hamiltonian equations discussed in chapter 3, is usually quite complicated. Rather than following the evolution of an individual (pure) microstate, statistical mechanics examines an ensemble of microstates corresponding to a given (mixed) macrostate. It aims to provide the probabilities PM(µ) for the equilibrium ensemble. Liouville's theorem justifies the assumption that all accessible microstates are equally likely in an equilibrium ensemble. As explained in chapter 2, such assignment of probabilities is subjective. In this chapter we shall provide unbiased estimates of PM(µ) for a number of different equilibrium ensembles. A central conclusion is that in the thermodynamic limit, with large numbers of degrees of freedom, all these ensembles are in fact equivalent. In contrast to kinetic theory, equilibrium statistical mechanics leaves out the question of how various systems evolve to a state of equilibrium.