Introduction

Granular Materials are a new frontier in Statistical Mechanics.

Being inspired to a long history of problems in engineer and industrial application, with roots in the 19th century, a large and heterogeneous family of experiments has demonstrated the richness of granular phenomenology. Moreover, the fundamental properties of granular media (mainly inelasticity of collisions and macroscopicity of grains) have motivated the born of an entire new branch of theoretical physics with its zoo of minimal models, the so-called granular gases, displaying an intriguing behavior in spite of their simplicity. The speculative spirit of this branch of theoretical granular physics must not be underestimated: often granular gases are very abstract models that can be observed only in the silicon cage of a computer simulation, but their importance for a substantial criticism of the basic assumptions (and limits) of Kinetic Theory, Hydrodynamics and general non-equilibrium Statistical Mechanics, is widely recognized. This thesis focuses on the particular subject of Kinetic Theory of Granular Gases and is devoted to two main tasks:

• a tentative review of recognized fundamental problems (experiments and theories);

• a study, by means of numerical and analytical tools, of some basic models.

In this introduction we give a brief presentation of granular materials and anticipate the content of the five chapters of this thesis.

What are granular materials?

A granular material is a substance made of grains, i.e. macroscopic particles with a spatial extension (average diameter) that ranges from the microns to the centimeters. In line of principle the size of grains is not limited as far as their behavior can be described by classical mechanics. For example, the physics of planetary rings (made of objects with a diameter far larger than centimeters) is sometimes approached using models of granular media. More often the term ``granular'' applies to industrial powders: in chemical or pharmaceutical industries the problem of mixing or separating different kinds of powders is well known; the problem of the transport of pills, seeds, concretes, etc. is also widely studied by engineers; the prevention of avalanches or the study of formation and motion of desert dunes are the subject of important studies all around the world, often involving granular theories; silos containing granular food sometimes undergo to dramatic breakages, or more often their content become irreversibly stuck in the inside, because of huge internal force chains; the problem of diffusion of fluids through densely packed granular materials (earths) is vital for the industry of natural combustibles; the study of ripples formations in the sand under shallow sea waters can solve important emergencies on many coasts of the world. Rough estimates of the losses suffered in the U.S. economy due to ``granular problems'' amount to ... billions of dollars a year.

The physicists usually have reduced the complexity of real situations, performing experiments to probe the fundamental behavior of granular media. The models proposed by theoretical physicists are even more idealized, in order to catch the essential ingredients of single phenomena. Therefore, in an experiment the grains are often all smooth spheres with the same size, same restitution coefficient, perfectly dry, in the void, and so on. In a numerical simulation the grains can become rods moving on a segment or disks without rotational freedom. However some ingredients are common in all the approaches to granular systems and, in some sense, can be considered the very definition (from the point of view of Physics) of the granular state of matter.

What are the basic properties of granular materials?

In the study of granular physics the properties that are usually shared by different models are the following:

• the grains are macroscopic: they are described by rules of classical mechanics and, moreover, the volume occupied by a grain is excluded by the volume available for the motion of all the other grains; the last property seems trivial but has many non-trivial consequences: mainly geometrical frustration, strong spatial correlations, relevance of collisional transport versus streaming transport, enhancement of re-collisions in the kinetic equations (breakdown of molecular chaos), in a few words: a failure of mean field approximations; of course these consequences are observables when the excluded volume is a relevant percentage of the total volume, otherwise they can be neglected;

• as a consequence of macroscopicity, the grains interact (with each other as well as with the boundaries) by means of dissipative interactions: this means that static and dynamic friction is always at work and that collisions are inelastic; the energy lost is transferred to internal degrees of freedom, i.e. heat, and subsequently dispersed to the environment;

• the environmental thermal temperature plays no role in the dynamics of the grains, i.e. they are almost always considered at $T=0$; this is due again to the macroscopic nature of grains and in particular to their masses, which are usually of the order of an Avogadro number of molecular masses: the kinetic or potential energy of a grain is therefore many orders of magnitude larger than the thermal energy conserved in the internal degrees of freedom; in the kinetic theory of granular gases the role of ``microscopic degrees'' is played by the grains themselves, so that a granular temperature can be introduced in terms of the random motion of grains.

What are the open problems in the physics of granular materials?

It is useful to stress here the existence of a main division between two different ``states'' in which the granular materials can be, depending upon the external conditions (available volume, intensity of the driving, degree of inelasticity of the collisions, presence of fluids, and so on):

• Stable or metastable granular systems: this family of problems comprehends the study of the distribution and the analysis of correlations in the internal forces in a pile or silo of grains, the characterization of the propagation of mechanical perturbations (sound) inside densely packed arrays of grains, the very slow compaction dynamics observed when a box full of grains is vibrated (the grains can rest in a metastable state, in the absence of vibration, which is far from the minimum packing fraction attainable), the study of time and size distributions of avalanches in a pile which has reached its critical slope.

• Flowing granular systems: this set of problems is instead composed of all the situations where an uninterrupted flow is present. Typically granular flows are divided in slow dense flows and rapid dilute flows. When the stationary velocity of the flow increases (due to an increase of external driving forces) the shear work induced by internal friction generates granular temperature and granular pressure, which in exchange produces a decrease of volume fraction occupied by grains [52]. This ensures that, almost always, a rapid flow is also dilute and that theoretical methods belonging to kinetic theory, as well as a hydrodynamic description, can be tried and are sometimes successful. Every kind of typical fluid experiment has been performed on granular systems: from Couette cells to inclined channels to rotating drums, finding non-linear constitutive relations. High amplitude vibrations can generate interesting convection phenomena in a box containing grains, always associated to size and density segregation (apparently violating entropic principles). Patterns (two dimensional standing waves) can form on the free surface of a vibrated granular layer. The study of simulated models have arisen new questions on the constitutive behavior in rapid flows, and recently new experiments have focused on this subject, measuring the velocity probability distribution functions and finding that in a wide set of situations this distribution is not Gaussian. The study of internal stress fluctuations and of velocity structure factors has given further elements to the investigation of kinetics of granular flows. A strong debate is still alive, on the limits of application of hydrodynamic formalism (and on the possibility of its derivation through kinetic theory).

What are granular gases?

In a word, they are ``ideal'' inelastic gases. The paradigm of inelastic gases is the gas of smooth hard spheres with restitution coefficient $r < 1$. They are often considered in two and even in one dimension. It could be said that a simple change in a parameter ($r$, from $r=1$ to $r < 1$) has awakened again the interest in hard spheres gases, which were deeply investigated in their elastic form in the 60's and 70's. The interest on granular gases is born from the necessity of probing granular kinetics theories elaborated in the 80's (for a review see [52]): this program was begun by means of simple simulations (the first simulation of granular materials are described in [52]). As long as new phenomena were observed (clustering [96], inelastic collapse [157], breakdown of equipartition [77], and so on) the interest increased and a better and better formalization of the theory was achieved.

The attention has been initially directed toward cooling granular gases, that is inelastic gases without external energy sources. A cooling granular gas starts from an initial state which is usually homogeneous in density and has some kind of distribution of velocities (for example Gaussian). After a number of collisions of the order of the size (number of particles) of the system, the effects of inelasticity begin to appear: the total kinetic energy (that, if calculated in the center of mass frame, is equivalent to the global granular temperature) decreases as an inverse power of the time $E(t) \sim t^{-2}$ and the distribution of velocities, even if scaled with its variance, is no more the initial distribution. After some time, if the system is large enough, velocity correlations start to appear: in fact two particles exit from an inelastic collision with a reduction of their relative velocity in the direction perpendicular to their distance, i.e. they travel more parallel than before the collision. This velocity correlations appear, for example, as vortices. When this happens the decay of the energy with time changes, in particular the energy decays more slowly, because collisions among such correlated particles dissipate less energy. Furtherly, some time later, correlations in the density emerge, the particles clusterize and the energy decay changes again. A numerical study of these late stages of the cooling gas evolution usually requires good resources and optimization of the algorithms (these instabilities appear at large scale and at late times, furtherly the aging of the system prevents from accumulating statistics over time), but also special regularizations for the collision rules: in a cluster of particles, in fact, the time between collisions may reduce to zero (numerical zero), that is the collision rate may diverge. This situation is known as inelastic collapse and usually prevents the simulation from going on. A restitution coefficient that depends on the relative velocity of the colliding particles is commonly used to circumvent the problem.

After the seminal paper of Du, Li, and Kadanoff [77], part of the effort has been devoted to understand the problem of driven granular gases. Different models have been proposed, in order to reproduce realistic situations. The driving can be in-homogeneous, for example a vibrating wall, leading to an in-homogeneous stationary state, or otherwise homogeneous, for example a vibrating plate under the layer of grains. A good success has been obtained by a class of idealized driven models, usually referred to as homogeneously heated granular gases: the velocities of the particles are periodically ``kicked'' by a random amount. A debate exists on the necessity of introducing also a systematic friction proportional to the velocities of the particle. If the friction is considered, the model can be modeled in terms of a Langevin equation for the velocity of each particle with the addition of inelastic collisions. The friction term regularizes the evolution of the system, which otherwise would present undesirable phenomena (very large fluctuations of the total momentum and, depending on the dimensionality, also an increase of total kinetic energy). The numerical and analytical study of homogeneously heated granular gases have demonstrated that the stationary granular state is usually very different from that of an ideal gas, with non-Gaussian distribution of velocities and stationary spatial correlations. These features have been recently confirmed by experiments which are reasonable realizations of the model (for example spheres traveling on the surface of a rapidly vibrating plate, considering only horizontal velocities).

What is the content of this thesis?

This thesis consists of a review of existing experiments and theories (Chapter 1, Chapter 2 and first section of Chapter 5), and of a presentation of results obtained by the author with his coworkers.

We anticipate that we have devoted a part of the review work (Chapter 2) to a very concise summary of kinetic theory, with the ambition of offering a survey of the logical path from the Newton laws of mechanics to the Navier-Stokes-like hydrodynamic theories, giving in parallel the ``granular'' (inelastic) version of the theory. This give us the possibility to stress the assumptions underlying the theories of ordinary fluids, in order to better evaluate the limits of existing granular kinetic and hydrodynamic models.

This thesis is organized as follows:

• The First Chapter concerns recent experiments on granular matter. It focuses on the difference between granular statics (or quasi-statics) and granular dynamics. In particular many experimental results on the rapid flow of dilute granular systems are reviewed.

• The Second Chapter, as already stated, contains a review of kinetic theory of standard fluids and a comparison with the kinetic theory of granular fluids. It begins with a discussion of different binary interactions (central potentials, hard spheres, Maxwell molecules, inelasticity) and statistical tools (mean free path or time and differential cross section). The standard derivation of kinetic Boltzmann equation from Liouville equation follows, with a discussion of the H-theorem, of ring kinetics theories (for dense systems), Enskog corrections and of granular kinetic ``Boltzmann'' or ``Boltzmann-Enskog'' equations. The hydrodynamic limit is taken in the third section of the chapter, reviewing the Chapman-Enskog expansion and the Grad method of moments, obtaining general equations for the time evolution of conserved quantities (density, moment and energy) and showing the approximations needed to come to the well-known Navier-Stokes equations. A presentation of granular hydrodynamic theories (from the 80's first theories to the most recent ones) comes after, followed, in the last section, by a discussion of the main criticisms to these theories and their basic assumptions.

• The Third Chapter is devoted to the study of the Randomly Driven Granular Gas (or Homogeneously Heated Granular Gas) with friction, a model that we have introduced [183] and investigated. We have collected results from simulations and analysis of toy models [184]. The model has two parameters (the restitution coefficient and the ratio between collision time and heating time). In some particular limits the model reproduces the properties of an ideal gases, but in general it shows a more complex behavior: the velocity distributions are non-Gaussian and the particle clusterize in fractal stationary structures.

• The Fourth Chapter discusses non-homogeneous driving in the presence of gravity. Two model have been proposed [15] in order to reproduce typical experimental situations: the grains rolling on an inclined plane with a vibrating bottom side, and the grains rolling down an inclined channel with rough bottom. The profiles of coarse-grained fields (density, velocity, temperature) agree with the experiments, as well as the distribution of velocities and the density correlations. A solvable hydrodynamic theory is presented and the problems related to the boundary conditions are discussed.

• The Fifth Chapter is devoted to the study of cooling granular gases and in particular focuses on the velocity distributions and on the emergence of velocity correlations. This chapter takes into account three different models: a very simplified mean field kinetic model (an inelastic version of the Maxwell molecules) called pseudo-Maxwell model, the $1D$ gas of inelastic hard spheres, and finally a $1D$ and $2D$ lattice gas model wit inelastic interactions among sites. The study of the pseudo-Maxwell model has led to the discovery of an exact stationary solution of the kinetic equation in $1D$, which in turn appears to be very different from the velocity distribution observed in the simulations of inelastic hard spheres. The study of the lattice gas model instead is capable of reproduce some of the main features of the inelastic hard spheres, in the homogeneous as well as in the velocity-correlated regimes.

• The Appendix A concerns the Direct Simulation Monte Carlo algorithm (due to Bird [30]) which is used in many simulations in Chapter 3 and Chapter 4. The Appendix B contains the numerical coefficients needed in the granular hydrodynamics as formulated by Brey et al. [39].