1. Kinetic and Hydrodynamic Theory
of Granular Gases
A thesis towards the degree of Doctor of Philosophy submitted by
Oded Bar-Lev
Submitted to the senate of Tel Aviv University
Under the supervision of
Prof. Philip Rosenau
Prof. Isaac Goldhirsch
November 2005
2. This work was done under the supervision of:
Prof. Philip Rosenau
Department of Applied Mathematics, School of Mathematics
The Raymond and Beverly Sackler Faculty of Exact Sciences
and
Prof. Isaac Goldhirsch
Department of Fluid Mechanics and Heat Transfer
The Iby and Aldar Fleischman Faculty of Engineering.
4. Thanks
Many people accompanied me through this work, during good and bad times. First and
formost I wish to thank Prof. Isaac Goldhirsch who is an inspiration to all of his students,
past and present. His physical insights and vast knowledge in any conceivable subject are
a model to me. His guidance, the wonderful sense of humor everything is done with and
above all his caring for his students at all levels are virtues I would like to adopt. I wish
to thank Prof. Philip Rosenau for accepting me for who I am, rendering his support when
needed.
I wish to thank Dr. Henri Noskowicz. It is rare to find an excellent scientist with whom
one can have such communication and partnership as we did. This is a perfect example
of a sum that is greater than its parts.
I wish to thank all the administrative staff at the School of Mathematics, especially Sigal,
and the Faculty of Engineering, especially Varda.
I wish to thank my parents for supporting me whenever I needed support, and for bringing
me up with thirst for knowledge. I wish to thank Anat, who was there at the beginning
even during rough times. Thanks to Esti for being there - loving and caring. And finally
I wish to convey my love to Adili - who is always on my mind
9. List of Figures
3.1 Development of the distribution function as a function of the number of
accumulated collisions per particle, c. Shown are the negatives of the second
derivative of the logarithm of the rescaled distributions versus the rescaled
speed ξu for three values of the coefficient of restitution, e. Notice the
crossover from the Maxwellian value, 2, to the value 0, corresponding to
the exponential decay. Also notice the fast convergence of the bulk versus
the slow convergence of the tail. The initial distribution is Maxwellian. . . 39
3.2 Development of the distribution function as a function of the number of
accumulated collisions per particle, c . Shown are − ∂2
∂˜u2 log ˜f (˜u) versus
the rescaled speed ˜u = ξu for e = 0.2. Notice the crossover from the
Maxwellian value, 2, to the value 0, corresponding to the exponential decay.
Also notice the fast convergence of the bulk versus the slow convergence of
the tail. The initial distribution is Maxwellian. . . . . . . . . . . . . . . . 43
3.3 Development of the distribution function as a function of the number of
accumulated collisions per particle, c . Shown are − ∂2
∂˜u2 log ˜f (˜u) versus
the rescaled speed ˜u = ξu for e = 0.5. Notice the crossover from the
Maxwellian value, 2, to the value 0, corresponding to the exponential decay.
Also notice the fast convergence of the bulk versus the slow convergence of
the tail. The initial distribution is Maxwellian. . . . . . . . . . . . . . . . 44
9
10. 10 LIST OF FIGURES
3.4 Development of the distribution function as a function of the number of
accumulated collisions per particle, c . Shown are − ∂2
∂˜u2 log ˜f (˜u) versus
the rescaled speed ˜u = ξu for e = 0.7. Notice the crossover from the
Maxwellian value, 2, to the value 0, corresponding to the exponential decay.
Also notice the fast convergence of the bulk versus the slow convergence of
the tail. The initial distribution is Maxwellian. . . . . . . . . . . . . . . . 45
3.5 The average number of collisions per particle needed to cover 95% of the
difference between the initial condition (Maxwellian) and the stationary
distribution function versus the rescaled speed ˜u = ξu for e = 0.2. Notice
the fast convergence of the bulk versus the slow convergence of the tail. . 46
3.6 The average number of collisions per particle needed to cover 95% of the
difference between the initial condition (Maxwellian) and the stationary
distribution function versus the rescaled speed ˜u = ξu for e = 0.5. Notice
the fast convergence of the bulk versus the slow convergence of the tail. . 47
3.7 The average number of collisions per particle needed to cover 95% of the
difference between the initial condition (Maxwellian) and the stationary
distribution function versus the rescaled speed ˜u = ξu for e = 0.7. Notice
the fast convergence of the bulk versus the slow convergence of the tail. . 48
3.8 The ratio of the loss term to the gain term extracted from a converged
numerical solution of the Boltzmann equation for three values of e, versus
the rescaled speed. For uξ < 0.1 the ratio is smaller than unity, indicating a
dominance of the gain, when 0.1 < uξ < 1/
√
1 − e2 the ratio is proportional
to uξ the loss dominates the gain for uξ > 1), and for uξ > 1/
√
ǫ the ratio
is proportional to (ξu)γ(ǫ)
, with γ ≈ 3, in conformity with the theoretical
calculation (C.3) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 49
3.9 Comparison of the first order perturbation theory result for the HCS ver-
sus the numerical solution of the Boltzmann equation for e = 0.8. The
(rescaled) distributions are shown versus the rescaled speed, ξu. . . . . . . 53
11. LIST OF FIGURES 11
3.10 Comparison of the numerical solution of the Boltzmann equation for e = 0.8
with the numerical solution of the reduced equation, Eq. (3.46). Shown
are the logarithms of the distribution functions. Notice the close corre-
spondence, even for small speeds and not-so-small ǫ. . . . . . . . . . . . . 54
3.11 Plot of the argument of the exponent in Eq. (3.50) versus x. The value of
ǫ is 0.1. The left plot corresponds to z = 5 and the right plot corresponds
to z = 50. These results are extracted from a numerical solution of the
reduced equation, Eq. (3.46). . . . . . . . . . . . . . . . . . . . . . . . . . 61
4.1 The rescaled spin distribution, F(ζ), for the homogeneous cooling state
(full line) and a Gaussian distribution (dashed) with the same value of
Trot/T, for the following values of the small parameters: ǫ = ǫ3 = 0.1.
Recall that the rescaled spin ζ equals s/
√
ǫ3T. . . . . . . . . . . . . . . . . 77
4.2 The spin distribution, ρ(ζ), for a general homogeneous steady state (full
line) and a Gaussian distribution (dashed) with the same value of Trot/T,
for the following values of the small parameters: ǫ = ǫ3 = 0.1. Recall that
the rescaled spin ζ equals s/
√
ǫ3T. . . . . . . . . . . . . . . . . . . . . . . 82
4.3 Sz
Sz(B)
(full line) in a wall-bounded shear flow described in the text, with
ǫ = ǫ3 = 0.1, 1
ν0
d
∆
= 0.2, ϕ = 0.15 and Φ = 0.01, and results of MD
simulations (crosses) of 131072 spheres with same parameters (and rough
walls). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 83
4.4 Theoretical profile (full line) and MD results (circles) for T
TB
(dashed line)
in a wall-bounded shear flow described in the text, with ǫ = ǫ3 = 0.1,
1
ν0
d
∆
= 0.2, ϕ = 0.15 and Φ = 0.01. The MD temperature profile is slightly
below the theoretical prediction (and exhibits a “plug”); this is a finite
density effect. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 84
4.5 Theoretical profile (full line) and MD results (crosses) for Vx
Vp
in the wall
bounded shear flow described in the text for ǫ = ǫ3 = 0.1, 1
ν0
d
∆
= 0.2,
ϕ = 0.1 and Φ = 0.01 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 85
12. 12 LIST OF FIGURES
G.1 Plot of ϕV (u2
) e−u2
as a function of u, demonstrating the convergence of
the expansion in Sonine polynomials. Each line corresponds to a specific
number of Sonine polynomials retained. The highest curve corresponds to
keeping only S0
5
2
(u2
). In the next highest curve S1
5
2
(u2
) is included, an so
on up to S9
5
2
(u2
). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 129
G.2 Plot of ϕT (u2
) e−u2
as a function of u, demonstrating the convergence of
the expansion in Sonine polynomials. Each line corresponds to a specific
number of Sonine polynomials retained. The lowest curve corresponds to
keeping only S0
3
2
(u2
). In the next lowest curve S1
3
2
(u2
) is included, an so on
up to S9
3
2
(u2
). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 130
G.3 Plot of ϕn (u2
) e−u2
as a function of u, demonstrating the convergence of
the expansion in Sonine polynomials. Each line corresponds to a specific
number of Sonine polynomials retained. The highest curve corresponds to
keeping only S0
3
2
(u2
). In the next highest curve S1
3
2
(u2
) is included, an so
on up to S9
3
2
(u2
). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 131
H.1 Plot of φǫ (u2
) e−u2
as a function of u, demonstrating the convergence of
the expansion in Sonine polynomials. Each line corresponds to the number
of Sonine polynomials retained. The highest curve corresponds to keeping
only S0
1
2
(u2
). In the next highest curve S1
1
2
(u2
) is included, an so on up to
S9
1
2
(u2
). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 135
13. Abstract
Several novel results pertaining to the kinetic theory and hydrodynamics of granular
gases are presented in this thesis. The properties of the homogeneous cooling state (HCS)
of granular gases are studied first. This important state serves as a basis of many other
studies on granular gases. The distribution function for the HCS was known to possess
an exponential tail. Here a theoretical analysis as well as a direct numerical solution of
the HCS are presented. They are in very good agreement with each other and help reveal
the rather complex structure of the HCS distribution function. It is shown that the HCS
distribution function comprises four regions in the speed. (i) A very high velocity tail
which ‘remembers’ the initial distribution (and decays in time); and a universal domain
which consists of three sub domains: (ii) A high velocity tail which decays exponentially
with the speed; (iii) A near-Maxwellian low velocity universal ‘head’ and (iv) an inter-
mediate region. A two stage reduction of the pertinent Boltzmann equation is presented
leading to a transcendental equation. Although the analysis is carried out for small values
of the “degree of inelasticity”, ǫ ≡ 1 − e2
, where 0 < e < 1 is the coefficient of restitution,
and the reduction is, of asymptotic nature, the results are remarkably good (less then 1%
difference) compared to the exact numerical solution of the pertinent Boltzmann equation.
The second part of this thesis presents the first systematic study of the consequences of
friction for granular gases: to this end a pertinent Chapman-Enskog (CE) expansion has
been developed and carried out. In the near smooth limit it is shown that the required
hydrodynamic fields include (in addition to the standard density, velocity and transla-
tional granular temperature) an infinite set of number densities, n (s, r, t), corresponding
to the continuum of values of the angular velocities. The derived hydrodynamic equa-
tions and constitutive relations have been tested against MD simulations with remarkable
agreement.
In order to derive the hydrodynamic equations and the constitutive relations using
the CE expansion one needs to invert the linearized Boltzmann operator. A novel com-
13
14. puter aided method, based on the exploitation of the pertinent generating functions was
developed in order to overcome the difficulties of this inversion. This method comprises
the third part of this thesis.
14
15. Chapter 1
Introduction
1.1 Granular Materials
The fact that collections of macroscopic grains, such as sand, salt, cereals or coal, can
flow is no surprise to anyone. Whether these flows can be described by hydrodynamic
equations is not a-priori clear [1, 2, 3].
Collections of macroscopic grains, also known as granular matter, are commonplace
in Nature and industry. The handling, conveying and storage of grains (and the frequent
malfunction of grain handling facilities, or even potentially catastrophic events such as
the collapse of a silo) are of great industrial importance. Snow avalanches, rock and land
slides, and sand dune dynamics, comprise examples of naturally occurring granular flows.
The planetary rings are mostly composed of ice particles.
In addition to the practical and environmental importance of granular materials, there
are excellent scientific reasons to study them. On one hand, fluidized granular materials
exhibit almost every known hydrodynamic flow (and instability), such as Rayleigh-B´enard
convection, Taylor-Couette flows, Faraday crispations, shear flows and more. On the other
hand they possess a rich rheology which does not parallel “regular” hydrodynamics. For
instance, vertically vibrated shallow layers of grains exhibit “oscillons” [4] which are stable
geyser-like excitations. Granular fluids often posses non-trivial microstructures, which
15
16. 16 CHAPTER 1. INTRODUCTION
affect other properties of theirs [5]. The normal stress in these fluids is often anisotropic,
much like in other non-Newtonian materials (but for a somewhat different reason, see
more below). Their unique properties led some to consider them to constitute “a new
state of matter”. To the theoretician granular materials pose significant challenges, many
of which pertain to the fact that they lack scale separation (see below and [1]); in a way
their hydrodynamic description can be considered to be an extreme case of the application
of hydrodynamics [3]. Surprisingly, the field of granular flows became part of the discipline
of fluid mechanics only about 25 years ago.
1.2 Granular Gases
Interstellar dust and planetary rings are collections of interacting discrete solid particles,
cf. eg. [6, 7]. Under the influence of the earth’s gravity, collections of solid particles can
be fluidized by the application of sufficiently strong forcing, e.g. by vibration, shear or
other means [8]. In all of these cases the grains interact by nearly instantaneous collisions
(compared to the mean free time) in a way that is reminiscent of the classical picture of
a molecular gas. This is the reason the fluidized phase of grains is known as a ‘granular
gas’ [9]. The ‘older’ term “rapid granular flows” is short for “rapidly sheared granular
flows” and it refers to the same class of systems. Except for the astrophysical realizations
of granular gases or experiments performed in vacuum, e.g. [4], the grains are embedded
in an ambient fluid. Following Bagnold (see e.g., review [10]), when the stress due to the
grains is far larger than that exerted by the fluid one may ignore the effects of the ambient
fluid and consider direct grain interactions alone [10]. Otherwise the grain-fluid system
should be considered a suspension.
In spite of the similarity of a granular gas to that of a (classical) molecular gas, there
are significant differences between the two kinds of “gases”: (i). The size of a grain in a
granular gas is much larger (say, larger than about a micron) to render Brownian motion
practically inconsequential (when the grains are embedded in an ambient fluid), and (ii).
17. 1.2. GRANULAR GASES 17
Most or all of the energy that is considered to be “lost” into the internal degrees of
freedom of a grain during a collision, is not retrievable in a practically finite time due to
the large number of these degrees of freedom. The term “collision” implies that the grain
interactions are short ranged, and that the contact time during a collision (the collision
time) is short with respect to e.g., the mean free time. In addition, it is often assumed
that all collisions are binary (see however [11]). This is actually true for the idealized case
of hard sphere collisions, but only an approximation in reality, that becomes better the
more dilute the gas.
The concept of a granular gas is superficially similar to that of the classical model of
a molecular gas. There are however important differences between these two “kinds” of
gases. The first difference is of practical nature: due to the fact that the grains are of
macroscopic dimensions, their typical number in an actual granular system is far smaller
than the Avogadro number. Therefore fluctuations in granular gases are expected to be
more prominent than in molecular gases (c.f. [1]). However, if size were the only difference
a (theoretically) sufficiently large sized granular gas would exhibit properties similar to
those of a molecular gas, albeit on larger scales. The inelasticity of the grain interactions
dictates much deeper differences, part of which are the following:
(i). All granular gases which have finite kinetic energy must be in non-equilibrium states,
since energy must be pumped into the granular gas to compensate for the losses
incurred in the inelastic collisions.
(ii). Statistical fluctuations, instabilities or external forces may create density inhomo-
geneities in a granular gas. Since in relatively dense domains the rate of collisions
(proportional to the square of the number density) is higher than in dilute domains,
the kinetic energy in the dense domains decays at a higher rate than in the dilute
ones. The ensuing pressure difference leads grains from the dilute into the dense
domains, thereby further increasing the density of the latter, and giving rise to
dense clusters [12, 9, 13, 14, 15, 16, 17, 18, 19]. It follows that no granular gas can
18. 18 CHAPTER 1. INTRODUCTION
be found in a truly homogeneous state (such as molecular equilibrium, in the ab-
sence of gravity). In unforced granular systems the clusters may merge in a process
of coarsening [20]. In sheared granular gases [14] an instability creates a density
nonuniformity that begets clusters. The latter collide with each other and disperse,
and are then recreated by the same mechanism. Sufficiently small granular sys-
tems will not exhibit clustering [9], but even then there are linear instabilities which
render the system inhomogeneous [12, 9, 13, 18].
(iii). Most states of granular matter in general, and granular gases, in particular, are
metastable and history dependent. This property can be related in part to clus-
tering. For instance, upon increasing the shear rate in a uniform granular Couette
system, one injects energy at the boundaries, thus raising the granular temperature
(defined as the average of the square of the fluctuating velocity), hence the pressure,
there. As a result, material moves away from the boundaries toward the “center” of
the system, where clusters and/or a central plug are formed. More states of granular
Couette systems can be created by varying the initial conditions [14]. However this
is not just a boundary conditions related property. As is demonstrated in chapter
3 [21], even in a non-bounded system the distribution function can have parts that
“remember” the initial conditions for infinite time.
(iv). A ball hitting a floor with vertical velocity v is known to bounce off with a velocity
e v, where e is the coefficient of restitution. When the ball is dropped from rest at
height h0, its n−th maximal height is e2n
h0. The time that elapses between positions
hn and hn+1 of the ball, is given by: τn = τ0 en
. Since the sum of τn is finite (as
0 < e < 1), it follows that an infinite number of collisions can occur in a finite
time. A similar process, known as ‘inelastic collapse’, may take place in granular
gases [22, 23, 24], leading (via a theoretically infinite number of collisions) to the
emergence of strings of particles whose relative velocities vanish. For a review,
see [2]. Clearly, ‘collapse’ is a non-hydrodynamic phenomenon. In reality, e is
19. 1.2. GRANULAR GASES 19
not a constant but rather velocity dependent. When the relative velocity of the
colliding particles is sufficiently low e(v) → 1, therefore the collapse process is never
completed.
(v). Scale separation, between the microscopic (grain) scales and the macroscopic scales,
in granular gases is weak or nonexistent [1]. This is demonstrated here using the
example of a simple shear flow of a frictionless monodisperse collection of spheres,
with a fixed coefficient of normal restitution, e. The velocity field is given by V =
γyˆx, where γ is the shear rate, x is the streamwise coordinate, and y a spanwise
coordinate. In the absence of gravity, γ−1
provides the only ‘input’ variable that has
dimensions of time. The granular temperature has dimensions of squared velocity
(see below for definitions), it follows from dimensional analysis (also, from kinetic
calculations, see below) that T ∝ γ2
ℓ2
, where ℓ is the mean free path (the only
relevant microscopic length scale). Define the degree of inelasticity, ǫ, by ǫ ≡ 1 −e2
.
Clearly, T should be larger, for a given value of γ, the smaller ǫ. Furthermore, when
ǫ = 0, the shear work raises T indefinitely (in molecular gases the excess energy
escapes in the form of a heat flux through the boundaries; there is no full equivalent
to this in granular systems). Therefore T → ∞ as ǫ → 0. While this argument does
not provide the precise form of this divergence, an initial guess (corroborated by
mean field, as well as detailed kinetic calculations [25, 26, 27]) would be that T ∝ 1
ǫ
.
All in all, one obtains: T = C γ2ℓ2
ǫ
. The value of C is about 1 in two dimensions
and 3 in three dimensions. It follows that γℓ√
T
=
√
ǫ√
C
, i.e. the change of speed over
a mean free path (in the spanwise direction) is comparable to the thermal speed
(unless ǫ is very small), hence the shear rate can be considered ‘large’, and the
system is typically supersonic. Indeed, shocks are a frequent occurrence in granular
gases, see [5] for references. Consider next the mean free time, τ, i.e. the ratio of
the mean free path and the thermal speed: τ ≡ ℓ√
T
. Clearly, in the simple shear
flow τ and γ−1
are the microscopic and macroscopic time scales characterizing the
system, respectively. Since τ
γ−1 = τγ = ℓγ√
T
=
√
ǫ√
C
, there is no good temporal scale
20. 20 CHAPTER 1. INTRODUCTION
separation except in near-elastic cases. Consequently, one cannot a-priori employ
the assumption of “fast local equilibration” and/or use local equilibrium as a zeroth
order distribution function (both for solving the Boltzmann equation and for the
study of generalized hydrodynamics of these systems) unless the system is nearly
elastic.
(vi). It follows from (v) that in the Chapman-Enskog (CE) expansion method (in “pow-
ers of the gradients”) one cannot neglect higher order gradient contributions (e.g.,
Burnett [27]) to granular hydrodynamics, except when ǫ << 1. The Burnett equa-
tions are ill posed (but one can use them for steady states, else a resummation may
be needed [28, 29]). At finite densities the Burnett coefficients diverge [30], pos-
sibly implying that the correct theory is non-analytic in the gradients [31], hence
nonlocal.
(vii). A molecular fluid that is not subject to strong thermal or velocity gradients, pos-
sesses a range, or plateau, of scales (larger than the mean free path and far smaller
than the scales characterizing macroscopic gradients) which can be used to define
“scale independent” fields densities (e.g., mass density, stress). Such plateaus do
not exist in systems which lack scale separation. Consequently some of the fields
(e.g., the stress tensor) characterizing granular gases may be scale (or resolution)
dependent [32].
(viii). In the example of the simple shear flow, consider the ratio of the xx component
of the stress, τxx, and the pressure, p ≡ τxx+τyy+τzz
3
. As this is a (γ dependent)
dimensionless entity, which must be even in γ by symmetry, it follows that to second
order in γ: τxx
p
= 1
3
1 + cxx
γ2ℓ2
T
, in three dimensions, where cxx is dimensionless.
Notice that the O (γ2
) correction is a Burnett contribution (second order in the
gradients). A similar result holds for τyy. The two constants, cxx and cyy, are
not required to be equal; indeed detailed kinetic calculations show that they are
both O (1) and different from each other. This results in significant normal stress
21. 1.3. SIMULATIONS AND EXPERIMENTS 21
differences. Notice that in molecular gases the typical value of γ2ℓ2
T
is negligible
small (for air at STP conditions and γ = 0.1 sec−1
, the value is about 10−18
). This
shows yet another facet of granular gases: they may ‘amplify’ some negligible effects
(in molecular gases) to the level of O(1) phenomena.
(ix). As granular gases at finite granular temperature are in non-equilibrium states, one
should not expect them to possess Maxwellian velocity distributions or exhibit
equipartition. On the contrary, one should probably be surprised when the dis-
tribution is (close to) Maxwellian or equipartition is approximately obeyed.
As mentioned, one of the practical implications of the macroscopic grain sizes is the
enhancement of fluctuations in granular gases. There is also an experimental advantage,
viz. the fact that (at least part of) the dynamics of a granular gas is visible to the naked
eye; therefore one can e.g., study the inner structure of a shock by ‘just’ using a camera
[33] (in air the typical shock thickness is of the order of a tenth of a micron).
1.3 Simulations and Experiments
The introduction of the event driven simulation method [34] in the field of granular gases
[9, 35] has revolutionized the computational capabilities in this field. Whereas in early
simulations cited in [10] systems consisting of only O(100) particles could be studied, it
is now possible to simulate O(106
) particles even on frontend PCs and it is now com-
monplace. This capability is important as granular gases lack the (Reynolds) scaling
properties of Newtonian fluids; important features, such as clusters and other microstruc-
tures, arise only when the system is sufficiently large [12, 9, 13, 32] and their properties
change further, possibly saturating at sufficiently large system sizes [36].
In molecular dynamic (MD) simulations (also known as Discrete Element Methods)
it is common to divide the computational domain into boxes and define the velocity
corresponding to a given box as its center of mass velocity. The fluctuating velocity of a
particle whose center of mass resides in a given box is defined as the difference between its
22. 22 CHAPTER 1. INTRODUCTION
velocity and the velocity corresponding to this box. This definition allows one to compute
the granular temperature, stress and other characterizations of the granular system. It is
different from the definition presented in section 4.1 whereby the fluctuating velocity is
measured with respect to the (ensemble) averaged velocity at the position of the particle.
When there is strong scale separation, as is the case in most molecular fluid flows, the
two definitions yield practically the same result. This is not the case in granular systems
[1, 32]. Therefore it is important that results of MD simulations are presented alongside
the sizes of the boxes employed to obtain these results; even better, results should be
presented as functions of the resolution.
Granular materials in general and granular gases in particular are notorious for the
difficulties they pose to the experimenter. This is the reason [10] that many theories in
this field are compared to results of numerical simulations. Clearly, computations are not
substitutes for physical experiments, no matter how clever they are.
A variety of experimental methods ranging from the ‘bucket method’ for collecting
grains falling off a chute [37] in order to find the flow rate, to sophisticated optical and
nuclear tracking techniques have been employed in this field. Among the recent additions
to the experimental tool-kit in the field of granular gases are NMR methods [38], the
Positron Emission Particle Tracking method [39], Particle Image Velocimetry [40] and
ultrafast video recording systems (e.g. in [41] the use of a rate of 10000 frames per
6 seconds has been reported). In many cases experimentalists are now able to follow
the motions of each (or most) of the grains thus enabling, among other things, direct
experimental tests of theoretical predictions.
1.4 The Homogeneous Cooling State (HCS)
Due to the inelasticity of the collisions granular gases lack an unforced steady state at
finite granular temperature. There have been two proposals for ‘equilibrium-like’ granular
systems:
23. 1.4. THE HOMOGENEOUS COOLING STATE (HCS) 23
(i). A statistically homogeneous (in density) and isotropic (all grains’ velocity directions
have equal probability) state in which random noise replenishes the energy lost in
collisions, referred to as the “heated state”, cf. [42] and refs therein.
(ii). An unforced, homogeneous and isotropic granular gas [43, 12, 9], now known as the
Homogeneous Cooling State (HCS).
The heated state has proven useful in the study of a number of properties of inelastic
gases and the prediction [44] that the tail of its velocity distribution should be a stretched
exponential corresponds to experimental findings [45, 46, 39]. Following the above argu-
ments the HCS is unstable and may thus be considered mostly a theoretical construct; it
has proven to be a useful concept and it is a solution of the pertinent Boltzmann equation;
the exponential tail [47] its distribution function is predicted to possess has been observed
experimentally [45].
The HCS was first considered (not by this name, though) by Haff [43] who discovered
that its granular temperature decays as 1
t2 , where t denotes time. This result, though
obtained by Haff from a hydrodynamic description can be understood on the basis of
a simple physical/dimensional argument. Assume that the precise details of the initial
state in which this system is prepared are ‘forgotten’ with time 1
. As the dimension of the
granular temperature is
length2
time2 one expects T ∝ 1
t2 . An improved dimensional argument
is that by dimensional analysis ˙T ∝ −T
3
2 if the equation of motion for the homogeneous
state is assumed to be local in time. The latter result can also be obtained by a mean field
argument (the loss of energy per particle in a collision is proportional to T and the rate of
collisions experienced by a particle is proportional to
√
T, i.e. the typical grains speed).
1
Molecular gases are known to practically equilibrate in a matter of 3-4 collisions per particle. When
the coefficient of restitution is very close to unity one expects a similar rate of ‘equilibration’. Therefore,
a near elastic granular system can be thought to be well described as having an ‘equilibrium’ distribution
(except for the tail of the distribution) whose temperature ‘slowly’ decays by inelasticity. Were it not for
the effects of long lived hydrodynamic modes and clustering this argument would have been correct for
long times.
24. 24 CHAPTER 1. INTRODUCTION
Assuming that the only parameter that determines the (long time) distribution of the
HCS is its granular temperature, one obtains that the single particle distribution function
of this state (for times which are sufficiently long, i.e. a few mean free times following the
initial state) can be written in the following scaling form: f(v, r, t) = nT− 3
2 f∗
( v√
T
), where
f∗
is a ‘scaling function’ [48]. One often refers to the state defined by this distribution as
the HCS. One should remember that the un-scaled distribution of an unforced granular
gas tends to a delta function in v (up to a constant and a power of v) as the system decays
to a state of zero granular temperature.
Rigorous results concerning the HCS can be found in refs. [49, 50].
1.5 Frictional Granular Gases
All macroscopic grains are frictional. Frictional interactions, including frictional (tan-
gential) restitution in granular collisions, are important and consequential (as examples
consider the recent study of the effects of friction on granular patterns [51]; note also the
hysteretic effects induced by friction, [52]). Furthermore, it is known that friction induces
non-equipartition, cf. e.g., [53, 54, 55, 56, 57]. It is therefore curious that a rather small
proportion of the granular literature, cf. e.g., [53, 54, 55, 56, 57, 58, 59, 37, 60, 61, 62, 63,
64, 65, 66, 67, 48, 68, 69, 70, 71, 72] is concerned with the modeling of frictional granular
gases. Even fewer articles are devoted to direct kinetic theoretical based studies of the full
(i.e., not only e.g., the homogeneous cooling state) frictional granular hydrodynamics, cf.
e.g., [58, 59, 61, 72] and references therein. Perhaps the aforementioned fact that friction-
less models have been rather successful in explaining many of the observed phenomena in
granular gases [5], or the difficulties one encounters in the theoretical study of frictional
granular gases are responsible for this state of affairs, and perhaps there are other reasons.
In any case, we believe it is important to study the full kinetics and hydrodynamics of
frictional granular gases [73, 74].
The study of gases whose constituents experience frictional interactions started in the
25. 1.6. STRUCTURE OF THE THESIS 25
realm of molecular gases. Indeed, the classic book by Chapman and Cowling [75] presents
references on the kinetics of gases composed of “rough molecules” that date back to the
year 1894. A Chapman-Enskog approach to the derivation of the hydrodynamics of such
gases is presented in the same book. Later works are described in a review article [76];
see also studies of celestial granular systems [77, 78].
In previous kinetic theoretical based studies of frictional granular hydrodynamics [59,
61, 72] it is assumed that the basic distribution function is Maxwellian in both the velocity
and angular velocity (and usually different rotational and translational temperatures are
allowed for), and corrections due to gradients are identified (on the basis of symmetry).
The assumed distribution function is substituted in the Enskog equations, resulting in
a closure for the constitutive relations. The above Maxwellian distribution corresponds
(when both the rotational and translation temperature are taken to be equal) to the limit
of rough molecules, in which there is equipartition between the rotational and translational
kinetic energies. Here granular gases near the smooth limit are studied, and a systematic
approach to the problem, i.e., a perturbative scheme is developed.
1.6 Structure of the Thesis
The next chapter is devoted to a general formulation of the kinetic theory of granular
gases. Frictional interactions (or tangential restitution) are included in the description.
Chapter 3 describes an analysis of the Boltzmann Equation for the HCS of frictionless
granular gases. The distribution function corresponding to this case is obtained and char-
acterized. Chapter 4 is concerned with the development of the hydrodynamic equations
and constitutive relations for nearly smooth frictional granular gases. A novel computer
aided method for the inversion of the linear Boltzmann operator is described in Chapter
5. Chapter 6 briefly summarizes and concludes the thesis. Some technical details are
relegated to Appendices. Appendices A – C provide details concerning the HCS prob-
lem studied in Chapter 3. Finally, Appendices D – K provide details concerning the
26. 26 CHAPTER 1. INTRODUCTION
calculations that pertain to the nearly smooth case of Chapter 4.
27. Chapter 2
Kinetic Theory of Granular Gases
This chapter presents a brief exposition of the kinetic theory of granular gases, geared to
the needs of this thesis. General background on the kinetics of Hamiltonian systems can
be found in [75, 79, 80].
2.1 Kinematics
The basic model for granular gases employed in this thesis is one of colliding hard spheres.
Consider a monodisperse system of homogeneous spherical grains of mass m = 1, diameter
d, and moment of inertia I (for homogeneous spheres, I = 2
5
d
2
2
), each. Let ˜I be a
dimensionless moment of inertia, defined by ˜I ≡ 4I
d2 . The velocity of particle ‘i’ is denoted
by vi, and its angular momentum by ωi. It is convenient to define a ‘spin variable’, si,
whose dimension is that of velocity, by si ≡ d
2
ωi.
Consider a binary collision between sphere ‘1’ and sphere ‘2’. Let k be a unit vector
pointing from the center of sphere ‘2’ to the center of sphere ‘1’ (notice that an opposite
convention is used in some papers). The relative velocity of sphere ‘1’ with respect to
sphere ‘2’, at the point of contact (when they are in contact), is given by:
g12 = v12 + k × s12 (2.1)
27
28. 28 CHAPTER 2. KINETIC THEORY OF GRANULAR GASES
where
v12 ≡ v1 − v2
s12 ≡ s1 + s2.
In the following and throughout the thesis, precollisional entities are primed. During a
collision the normal component of the relative velocity changes according to:
k · g12 = −ek · g′
12 (2.2)
where e is the coefficient of normal restitution. The effect of the tangential impulse at a
collision is modeled in the same way as in [55, 72], with slight notational differences. Let
J = v1 − v′
1 be the change in the momentum of particle ”1” in a collision. Linear and
angular momentum conservation imply that: v1 = v′
1 + J, v2 = v′
2 − J, s1 = s′
1 − 1
˜I
k × J,
and s2 = s′
2−1
˜I
k × J. Next, consider the decomposition of J into its normal and tangential
components: J = Ak + B(k×(k×g′
12))
|k×g′
12|
. (Pay attention that J lies in the plane defined by k
and g12). Let γ be the angle between the direction, −k, and the relative velocity of the
grains at the point of contact (when they are in contact):
cos γ ≡ −
k · g′
12
g′
12
= −
k · v′
12
g′
12
(2.3)
where g′
12 ≡ ||g′
12||; it follows that 0 ≤ γ ≤ π
2
. Define γ0 such that if γ > γ0 there is
sliding (Coulomb friction) during the collision and B = µf A, where µf is the friction
coefficient, while if γ ≤ γ0 there is sticking (or the grain is ‘rough’) and k× (k × g12) =
−β0k× (k × g′
12), with 1 ≥ β0 ≥ −1. In both cases A = −1+e
2
k · v′
12. In the case of
sliding one obtains:
k× (k × g12) = 1 −
1 + ˜I
˜I
(1 + e) µf cot γ k× (k × g′
12) (2.4)
It follows that in both cases:
k× (k × g12) = −β (γ) k× (k × g′
12) (2.5)
29. 2.1. KINEMATICS 29
where (requiring β (γ) to be a continuous function of γ):
β (γ) = min β0, −1 +
1 + ˜I
˜I
(1 + e) µf cot γ (2.6)
The transition angle, γ0, between the two ranges of γ is determined from Eq. (2.6) to
be: cot γ0 =
˜I
1+˜I
1+β0
1+e
1
µf
. Using the conservation laws for the linear and angular mo-
menta and Eqs. (2.4,2.6) one obtains the transformation between the precollisional and
postcollisional velocities and spins of a colliding pair of grains:
v1 = v′
1 −
1 + e
2
(k · g′
12) k +
˜I
1 + ˜I
1 + β(γ)
2
k× (k × g′
12)
v2 = v′
2 +
1 + e
2
(k · g′
12) k −
˜I
1 + ˜I
1 + β(γ)
2
k× (k × g′
12)
s1 = s′
1 +
1
1 + ˜I
1 + β(γ)
2
k × g′
12
s2 = s′
2 +
1
1 + ˜I
1 + β(γ)
2
k × g′
12 (2.7)
The Jacobian of this transformation is given by:
J (γ) ≡
∂ (v1, v2, s1, s2)
∂ (v′
1, v′
2, s′
1, s′
2)
=
eβ2
0 γ < γ0
e |β (γ)| γ > γ0
(2.8)
For future reference the loss of translational kinetic energy in a collision as a function of
both precollisional (a) and postcollisional variables (b) is presented:
2∆E = v′2
1 + v′2
2 − v2
1 + v2
2
=
1 − e2
2
(k · g′
12)
2
+ (1 + β (γ))
˜I
1 + ˜I
1 + β (γ)
2
˜I
1 + ˜I
− 1 (k · g′
12)
2
−
1
2
(1 + β (γ))2
˜I
1 + ˜I
2
g′2
12 + (1 + β (γ))
˜I
1 + ˜I
(g′
12·v′
12) (2.9a)
=
1 − e2
2e2
(k · g12)2
+
1 + β (γ)
β (γ)
˜I
1 + ˜I
1 −
1 + β (γ)
2β (γ)
˜I
1 + ˜I
(k · g12)2
+
1
2
1 + β (γ)
β (γ)
2 ˜I
1 + ˜I
2
g2
12 −
1 + β (γ)
β (γ)
˜I
1 + ˜I
g12·v12 (2.9b)
30. 30 CHAPTER 2. KINETIC THEORY OF GRANULAR GASES
2.2 The Boltzmann Equation
This subsection is devoted to the derivation of the Boltzmann equation corresponding to
the above model. There are several ways of obtaining this equation, the most basic of
which is a truncation of the corresponding BBGKY hierarchy (needless to say, all methods
yield the same result). Here a phenomenological derivation (which parallels standard
textbook derivations [79] for regular gases) is presented, since it is physically transparent.
Notice that the derivation is valid for time dependent and (possibly) inhomogeneous and
anisotropic situations.
Let f(r, v, s, t) denote the single particle distribution function at point r at time t.
Let n(r, t) be the (particle) number density at point r at time t. The quantity f(r,v,s,t)
n(r,t)
is
the (normalized) probability distribution of the linear and angular velocities at point r at
time t, i.e. f(r, v, s, t) satisfies the normalization condition:
f(r, v, s, t)dvds = n(r, t) (2.10)
A standard procedure [75, 79, 48, 81, 27] yields the following equation for the single
particle distribution function:
∂
∂t
f(r, v, s, t) + v · ∇f(r, v, s, t) + F·
∂f(r, v, s, t)
∂v
=
∂f(r, v, s, t)
∂t c
(2.11)
where F is an external (velocity independent) force and ∂
∂t c
represents the effect of the
collisions. In the derivation of Eq. (2.11) the assumption of molecular chaos has been
invoked. This assumption does not necessarily hold for strongly inelastic granular gases
(see e.g. [82]). The l.h.s. of Eq. (2.11) is independent of the nature of the collisions and
its physical meaning is obvious. Following standard practice [75, 79], it is convenient to
separate the r.h.s. of Eq. (2.11) into a gain term ∂f(r,v,s,t)
∂t
g
≥ 0, which represents all
collisions that increase the number density of particles having velocity v and spin s, and
a loss term ∂f(r,v,s,t)
∂t
ℓ
≥ 0, which represents all collisions that decrease this number
density. Hence:
∂f(r, v, s, t)
∂t c
=
∂f(r, v, s, t)
∂t g
−
∂f(r, v, s, t)
∂t ℓ
(2.12)
31. 2.2. THE BOLTZMANN EQUATION 31
For notation simplicity define the six-vector
τ ≡ (v, s)
The number of particles per unit volume having velocities and spin in the differential vol-
ume dτ1 near τ1 (denoted below by (τ1, dτ1)) is f(r, τ1, t)dτ1 and the total flux of particles
(impinging on the particle “1”) having velocity and spin (τ2, dτ2) is: σT |g12| f(r, τ2, t)dv2
where: σT = πd2
is the total cross-section and |g12| is the norm of g12. It follows that
the rate of collisions per unit volume between particles with velocities and spins (τ1, dτ1)
and (τ2, dτ2), respectively is: σT |g12| f(r, τ1, t)f(r, τ2, t)dτ1dτ2. Since every collision with
a particle whose velocity is v1 and spin is s1 changes these properties of the particle (with
probability one) it follows that the loss term is given by:
∂f(r, τ1, t)
∂t ℓ
= σT f(r, τ1, t) |g12| f(r, τ2, t)dτ2 (2.13)
Notice that the form of the loss term is not affected by the inelasticity or roughness of
collisions. Next, consider ∂f(r,τ,t)
∂t
g
. Consider collisions between particles “1” and “2”
with respective incoming velocities and spins τ′
1 and τ′
2 such that the outgoing velocity
and spin of “1” is τ1. The flux of “2” particles impinging on “1” is |g′
12| f(r, τ′
2, t)dτ′
2.
In order for the collision to take place the particles must approach each other prior to the
collision. This condition is equivalent to (k · g12) > 0. The total number of collisions per
unit volume among particles with (τ′
1, dτ′
1) and particles (τ′
2, dτ′
2) is:
σ (k · g12) f(r, τ′
1, t)f(r, τ′
2, t)dkdτ′
1dτ′
2
where σ = d2
and σ (k · g12) dk is the differential cross section. It follows that the rate of
collisions resulting in particles having velocity and spin τ is given by:
∂f(r, τ, t)
∂t g
=
(k·g′
12)>0
σ (k · g′
12) f(r, τ′
1, t)f(r, τ′
2, t)δ (τ1 − τ) dkdτ′
1dτ′
2 (2.14)
The velocity and spin τ1 in Eq. (2.14) is related to the velocities and spins τ′
1 and τ′
2 as in
Eqs.(2.7). Eq. (2.14) is referred to as the first form of the gain term. Next transform the
32. 32 CHAPTER 2. KINETIC THEORY OF GRANULAR GASES
integration over τ′
1, τ′
2 and k to an integration over unprimed (postcollisional) variables
so that the integration over v1 (i.e. the delta-function) can be trivially performed. To
this end, use Eqs. (2.2) and (2.8) to obtain
∂f(r, τ, t)
∂t g
= −
(k·g12)<0
σ
eJ (γ)
(k · g12) f(r, τ′
1, t)f(r, τ′
2, t)δ (τ1 − τ) dkdτ1dτ2 .
Upon changing the sign of k and noting that (since k is a unit vector) d (−k) = dk, one
obtains:
∂f(r, τ1, t)
∂t g
= σ
(k·g12)>0
1
eJ (γ)
(k · g12) f(r, τ′
1, t)f(r, τ′
2, t)dkdτ2 (2.15)
This is the “second form” of the gain term. Combining now Eq. (2.11), Eq. (2.12), Eq.
(2.13) and Eq. (2.15) one obtains:
∂
∂t
f(r, τ1, t) + v1·∇f(r, τ1, t) + F·
∂f(r, τ1, t)
∂v1
= B(f, f; e, β) ≡
σ
(k·g12)>0
(k · g12)
1
eJ (γ)
f(r, τ′
1, t)f(r, τ′
2, t) − f(r, τ1, t)f(r, τ2, t) dkdτ2(2.16)
The r.h.s. of Eq. (2.16) is the nonlinear Boltzmann collision operator. It defines a
functional of f (the Boltzmann collision operator) which is denoted by B(f, f; e, β) or
B(f, f), in short. Eq. (2.16) is the Boltzmann equation describing the dynamics of a
system of frictional spheres whose collisions are inelastic (in the above defined sense).
Notice that in the singular case, β0 = 0 or β (γ) = 0, when the Jacobian vanishes, the
relation between the precollisional and postcollisional variables is not invertible (same
as when e = 0, even in the absence of friction), and the Boltzmann equation has to be
revised.
33. Chapter 3
The Homogeneous Cooling State
This chapter is devoted to a study of the distribution function for the HCS of smooth
(β0 = −1) inelastic particles. Both analytical and numerical methods are employed. Since
this is an isotropic and homogeneous state, the Boltzmann equation can be reduced to a
one dimensional equation. Denote the speed variable by u, the granular temperature by
T and define the ’degree of inelasticity’ by
ǫ ≡ 1 − e2
(3.1)
It turns out that the distribution function assumes different forms in each of the following
four ranges of u. For u∗
< u the distribution function retains memory of the initial condi-
tions. For u < u∗
the distribution function converges to a universal scaling function. The
latter range can be subdivided into three subranges. For u < O(
√
T/
√
ǫ) the distribution
function is near Maxwellian. The subrange where O(
√
T/
√
ǫ) < u < O(
√
T/ǫ) is a tran-
sition range and for O(
√
T/ǫ) < u < u∗
the distribution function is basically a decaying
exponential. These results follow from a direct solution of the pertinent 1-dimensional
Boltzmann equation, heuristic considerations and each of three levels of analytic reduction
of the Boltzmann equation:
(i). An exact reduction to an equation in the speeds.
(ii). An asymptotic reduction to a relatively simple one dimensional integrodifferential
33
34. 34 CHAPTER 3. THE HOMOGENEOUS COOLING STATE
equation.
and
(iii). An asymptotic reduction of the latter to a simple transcendental equation which
leads to an approximate analytic solution for the distribution function of the HCS.
The distribution functions obtained from the reduced description are practically indis-
tinguishable from the corresponding numerical solution of the full Boltzmann equation.
3.1 The Boltzmann Equation for the HCS
In the smooth limit, β0 = −1, and one can ignore the spin variables (since there is
no interaction between the angular and linear velocities). From Eq. (2.6) one obtains
β (γ) = −1, and from Eq. (2.1) one finds that g12 = v12. The other kinematic equations
change accordingly. Eq. (2.7) reduces to:
v1 = v′
1 −
1 + e
2
(k · v′
12) k
v2 = v′
2 +
1 + e
2
(k · v′
12) k (3.2)
The Jacobian (2.8) is now a constant J = e and the energy loss in a collision (2.9) is
reduced to
2∆E =
1 − e2
2
(k · v′
12)
2
=
1 − e2
2e2
(k · v12)2
(3.3)
Since this is an unforced, homogeneous and isotropic state, the external force filed, F, must
be taken to vanish. The distribution function f, at time t, depends only on the magnitude
of the speed v, and the number density is a constant. In this case the Boltzmann equation
can be reduced to an equation involving speeds alone.
Consider the loss term first. Let µ = v1·v2
v1v2
be the cosine of the angle between v2 and
v1. Using v2
12 = v2
1 + v2
2 − 2µv1v2, one obtains from Eq. (2.13):
∂f(v1, t)
∂t ℓ
= 2πσT f(v1)
∞
0
1
−1
2π
0
v2
2 + v2
2 − 2µv1v2f(v2)v2
2dv2dµ
=
2πσT
3v1
f(v1)
∞
0
(v1 + v2)3
− |v1 − v2|3
v2f(v2)dv2 (3.4)
35. 3.2. HEURISTIC ANALYSIS 35
Hence:
∂f(u, t)
∂t ℓ
= 4πσT f(u)
∞
0
uv2
+
v4
3u
f(v)dv +
∞
u
v3
+
1
3
u2
v − uv2
−
v4
3u
f(v)dv
(3.5)
Let < vk
; t > denote the k-th moment of the velocity (at time t):
< vk
; t >=
4π
n
∞
0
v′k+2
f(v′
, t)dv′
(3.6)
The above will be denoted by < vk
> for simplicity when no confusion is expected. It
follows from Eq. (3.5) that:
∂f(u, t)
∂t ℓ
= 4πσT f(u)
n
4π
u +
< v2
>
3u
+
∞
u
v3
+
1
3
u2
v − uv2
−
v4
3u
f(v)dv
(3.7)
The angular integrations for the ∂f
∂t g
term are rather tedious, and they are relegated to
Appendix A. The result is most conveniently written as:
∂f(u, t)
∂t g
=
9
i=1
Gi(u) (3.8)
where Gi are given in Eqs.(A.34) to (A.42).
3.2 Heuristic Analysis
In this section, a heuristic study of the properties of the HCS is presented and validated by
numerical results (presented in the last part of this section). Two time dependent velocity
ranges are identified: one that remembers the initial distribution function f(v, t = 0), and
another that does not. The latter assumes a universal scaling form and subdivides into a
near-Maxwellian, a transient and an exponential range.
3.2.1 Large Speeds
At this stage one makes the plausible assumption that f(u) decays (for large enough
values of u) faster then algebraically in u (in the elastic case, the decay is Gaussian). It
36. 36 CHAPTER 3. THE HOMOGENEOUS COOLING STATE
follows that the second integral in equation Eq. (3.7) is negligible with respect to the first
integral. Thus one obtains:
∂f(u, t)
∂t loss
≈ nσT f(u) u +
< v2
>
3u
(3.9)
It is argued, at this point, that for u ≫
√
< v2 > , the loss term dominates over the gain
term. In the elastic case the gain and loss terms are equal (in equilibrium, which is the
corresponding homogeneous and isotropic solution). In both cases, elastic and inelastic,
the rare particles that belong to the above range of velocities collide chiefly with “typical”
particles, whose velocities are O(
√
< v2 >). However, in an inelastic collision the probabil-
ity for a fast particle to produce an even faster particle is diminished, thereby reducing the
gain term. For a more quantitative explanation, consider a collision (v′
1, v′
2) → (v1, v2)
of a fast particle with velocity v′
1 with a typical particle with velocity v′
2. There are two
possibilities for this collision to produce higher speeds: either v1 > v′
1 or v2 > v′
1. Using
the kinematic equations Eqs. (2.2) and (2.7), as well as v′
12 · k < 0 this is equivalent to
v1 > v′
1 ⇒ v′
2 · k > v′
1 · k > −
1 + e
3 − e
(v′
2 · k)
v2 > v′
1 ⇒ (v′
1 · k − v′
2 · k) v′
1 · k +
3 − e
1 + e
v′
2 · k >
2
1 + e
2
v′2
1 − v′2
2
Since, by assumption |v′
2 · k| ≤ v′
2 = O
√
< v2 > the first inequality can be satisfied
only if v′
1 ·k = O(
√
< v2 >) which is a (very rare) grazing collision, with impact parameter
very close to 2a. This is true also in the elastic case, and the effect of the inelasticity is
minor. In the inelastic case the second inequality sets a limit on the magnitude of v′
1 for a
given v′
2, while in the elastic case the inequality is satisfied by v′
1 · ˆt
2
< v′
2 · ˆt
2
, where
ˆt is the unit tangent vector, which is clearly satisfied by a finite fraction of the collisions.
In order to prove this fact, rewrite the second inequality as
(3 + e)(1 − e)
4
(v′
1 · k)
2
−
(1 − e2
)
2
(v′
2 · k) (v′
1 · k) +
(3 − e)(1 + e)
4
(v′
2 · k)
2
< v′2
2 − v′
1 · ˆt
2
The maximum value of v′
1 for which there is a solution to the above inequality is v′
1,max =
3−e
2(1−e)
v′
2 = (3−e)(1+e)
2
1√
ǫ
v′
2 with v′
1 · ˆt = 0 and v′
2 · k = 1
2(3−e)(1+e)
√
ǫv′
2, thus for
ǫv′2
1 > O(< v2
>) (3.10)
37. 3.2. HEURISTIC ANALYSIS 37
the gain term, in the inelastic case, starts to diminish with respect to the loss term. Hence,
there must be a value of the speed u∗
≥ O <v2>
ǫ
, such that for u > u∗
the gain term
is subdominent to the loss term. Therefore the Boltzmann equation simplifies to:
∂f(u, t)
∂t
∼ −nσT u f(u, t) (3.11)
(upon keeping the first term in the r.h.s. of Eq. (3.9)) whose solution is
f(u, t) ∼ f(u, 0)e−nσT ut
(3.12)
Therefore, for u > u∗
the HCS distribution ’remembers’ the initial condition. Notice that
as the initial distribution f(u, 0) is u dependent, the part of the distribution, which retains
memory of the initial condition, is not necessarily exponential. (although, for very long
times, the distribution function decays exponentially).
3.2.2 The speed ranges and universality
As is well known [83, 84, 80], the distribution function for an unforced system of elas-
tically colliding spheres (or any other elastic system) converges very rapidly (in a time
corresponding to a few collisions per particle) from practically any initial distribution to
a near-Maxwellian distribution (except in the high speed tail [83, 84], where convergence
is much slower). The corresponding convergence rate can be estimated, for example, from
the value of the leading nonzero eigenvalue of the linearized Boltzmann operator, which
is O(1) in units of the inverse mean free time. Next, consider a quasielastic system. If the
degree of inelasticity, ǫ, is sufficiently small, the outcome of a small number of collisions
per particle is very close to the outcome of a small number of elastic collisions. Therefore
a near elastic system is expected to develop a near Maxwellian distribution for speeds of
the order of v2 (or smaller) with a granular temperature (i.e., v2
) slowly decaying
with time. As the energy loss in a single collision is proportional to (3.3) ǫ (v12·k)2
, one
expects the near-Maxwellian distribution to extend to speeds of the order of 1/
√
ǫ. This
implies that the statistical weight of the non-Maxwellian (tail) part of the distribution
38. 38 CHAPTER 3. THE HOMOGENEOUS COOLING STATE
function is small. Similarly, one expects the memory of the details of the initial distribu-
tion to fade following a few collisions per particle, even when ǫ is not small, although it
is not a-priori clear that in this case the bulk of the distribution function becomes near-
Maxwellian. As a matter of fact it has been checked (by solving the pertinent Boltzmann
equation numerically) that this is the case down to quite low values of e (e.g., for e = 0.2).
The development of the distribution function in time is demonstrated in Fig. (3.1). As
the form of the distribution function emerging after a few collisions per particle is inde-
pendent of the form of the initial distribution (except in the tail), it does not change in
time any more, i.e., it becomes universal (but still dependent on the value of e). The
universal distribution, being normalized, is characterized by e and its (shrinking) width
alone. Define the inverse width, ξ, as follows:
ξ2
(t) ≡
3
2 v2; t
(3.13)
In the universal range, dξ/dt has dimensions of inverse length; the only available parameter
having this dimension is nσT (the inverse mean free path). It follows that dξ/dt = CnσT ,
where C is a constant. To find C a simplified mean field approach is used, by assuming
that the distribution function is Maxwellian for all speeds:
fM (u, t) = nξ3
π−3/2
e−ξ2u2
(3.14)
Multiplying the Boltzmann equation (2.16) by v2
1 and integrating over v1 using the above
distribution function yields:
˙ξ = ǫ
√
2
3
nσT π− 1
2 (3.15)
i.e., C ∝ ǫ, in conformity with systematic studies (cf. e.g., Sec. 3.3). This is easy to
understand since ξ does not change in time when ǫ = 0. Defining λ by C ≡ ǫλπ− 1
2 (thus,
for the mean field λ =
√
2/3), one obtains:
ξ(t) = ǫλnσT π− 1
2 (t − t0) + ξ(t0) (3.16)
where t0 is a time beyond which the distribution is largely (except at the far tail) universal.
39. 3.2. HEURISTIC ANALYSIS 39
0 1 2 3 4 5 6 7 8 9
0
0.5
1
1.5
2
2.5
ξ u
e = 0.2 ; c = ∞
e = 0.5 ; c = ∞
e = 0.7 ; c = ∞
e = 0.2 ; c = 3
e = 0.5 ; c = 3
e = 0.7 ; c = 3
Figure 3.1: Development of the distribution function as a function of the number of
accumulated collisions per particle, c. Shown are the negatives of the second derivative
of the logarithm of the rescaled distributions versus the rescaled speed ξu for three values
of the coefficient of restitution, e. Notice the crossover from the Maxwellian value, 2, to
the value 0, corresponding to the exponential decay. Also notice the fast convergence of
the bulk versus the slow convergence of the tail. The initial distribution is Maxwellian.
With this definition, one can express the bulk of the distribution function as a scaling
function: f(u, t) = nξ3
(t) ˜f(uξ(t)), where the prefactor is determined by dimensional
considerations, and where ˜f is universal (its dependence on e has been suppressed).
Let u be a speed in the range corresponding to the universal (and scaling) form of f,
40. 40 CHAPTER 3. THE HOMOGENEOUS COOLING STATE
at time t. Clearly, after a sufficient time elapses, u will be in the tail of the distribution,
since the entire distribution shrinks. Let t(u) be a time at which u is sufficiently larger
than v2
; t(u) , so that its subsequent dynamics is dominated by the loss term. Following
the result presented at the end of Sec. 3.2.1: f(u, t) = exp [−nσT u(t − t(u))] f(u, t(u)) for
t > t(u), hence, using Eq. (3.16): f(u, t) = exp −
√
πuξ(t)−ξ(t(u))
ǫλ
f(u, t(u)). Substituting
now the scaling form of f, multiplying both sides of the equation by u3
and rearranging
terms, one obtains:
u3
ξ3
(t) ˜f (uξ(t)) exp
√
πuξ (t)
ǫλ
= exp
uξ (t(u))
√
π
ǫλ
u3
ξ3
(t(u)) ˜f (uξ (t(u))) (3.17)
Eq. (3.17) implies that there must be a constant, A, such that uξ (t(u)) = A. Clearly,
A ≫ 1 since u is in the tail of the distribution at t = t(u). It follows that, for t > t(u)
the distribution, f(u), is given by:
f(u, t) = n A3 ˜f(A) exp
A
√
π
λǫ
1
u3
exp −
ξ(t)u
√
π
λǫ
(3.18)
Fig. (3.1) depicts the convergence of a Maxwellian initial condition to a universal dis-
tribution for several values of e (a similar result is obtained for non-Maxwellian initial
conditions). The crossover between the Maxwellian-like range and the exponential range
is visible. As the values of ǫ used here are O(1) (for technical reasons) one cannot read
the scaling of the crossover ‘points’, in terms of ǫ, off Fig. (3.1).
To reiterate, the exponential tail is, strictly speaking, an intermediate, universal tail
(developing out of the scaling part of the distribution at an earlier time), since the true
tail retains the memory of the initial condition.
Assume for the moment that the near-Maxwellian range of speeds crosses over smoothly
to the exponential range. Equating exp (−ξ2
u2
) with the exponential form, one obtains
that the crossover must occur at uξ ∝ 1/ǫ. However, as explained above (see also Sec.
3.3), the Maxwellian range is limited to uξ < 1/
√
ǫ. Therefore there must be a crossover
region, ‘starting’ at uξ ∝ 1/
√
ǫ, and ending where the exponential decay ‘starts’. The
crude matching of the Maxwellian and exponential ranges may be taken to suggest that
the exponentially decaying range starts at uξ ∝ 1/ǫ, a conclusion corroborated below.
41. 3.2. HEURISTIC ANALYSIS 41
3.2.3 Numerical Results
Denote the (universal) scaled distribution function by ˜f(˜u), where
˜u ≡ ξ(t)u (3.19a)
f(u, t) ≡ nξ3
(t) ˜f(˜u) (3.19b)
In the numerical computations of f(u, t), it is convenient to replace the basic variables
(u, t) by (˜u, c) where c is a monotonously increasing function of t. A natural choice is the
average accumulated number of collisions per particle in an elastic system in equilibrium.
Upon following the same steps leading to Eq. (3.15) one has ˙c = 8
π
nσT
ξ
and by integration
c =
6
ǫ
ln(
ξ
ξ(0)
) (3.20)
In the large c limit ˜f(˜u, c) is expected to be the universal function ˜f(˜u). Using Eq. (3.20),
it follows from Eq. (3.19b) that:
∂f
∂t
= n ˙ξξ2
(3 ˜f + ˜u
∂ ˜f
∂˜u
+
6
ǫ
∂ ˜f
∂c
) (3.21)
Next, the non-dimensional (and rescaled) gain and loss parts ˜rg and ˜rℓ respectively are
defined by:
∂f(v, t)
∂t ℓ,g
≡ n2
σT ξ2
˜rℓ,g(˜v, c) (3.22)
The Boltzmann equation can be written as:
˙ξ 3 ˜f + ˜u
∂ ˜f
∂˜u
+
6
ǫ
∂ ˜f
∂c
= nσT (˜rg − ˜rℓ) (3.23)
There remains to find an expression for ˙ξ. Clearly:
∂ < v2
; t >
∂t
=
4π
n
∞
0
∂f
∂t
v4
dv =
4πnσT
ξ3
∞
0
˜r (˜v, c)g − ˜r (˜v, c)ℓ ˜v4
d˜v
Hence:
˙ξ = −
4πnσT
3
∞
0
(˜r(˜v, c)g − ˜r(˜v, c)ℓ) ˜v4
d˜v (3.24)
42. 42 CHAPTER 3. THE HOMOGENEOUS COOLING STATE
Substituting Eq. (3.24) in Eq. (3.23) and rearranging terms one obtains an equation of
motion for ˜f in terms of c:
∂ ˜f
∂c
= −
ǫ
2
˜f −
ǫ
6
˜u
∂ ˜f
∂˜u
−
ǫ
8π
˜rg − ˜rℓ
∞
0
(˜rg − ˜rℓ) ˜v4d˜v
(3.25)
Equation 3.25 has been solved numerically for several values of e ranging from 0 to
0.9 and for a variety of initial distributions, using a fourth-order Runge-Kutta scheme.
Due to the smallness of the distribution function for large ˜u it was practically impossible
to go beyond ˜u ≈ 10 − 11. In this velocity range, the distribution function for e > 0.9
did not exhibit any transition region towards exponential behavior, remaining Maxwellian
throughout.
In Figures 3.2–3.4 y = − ∂2
∂˜u2 log ˜f (˜u) is plotted vs. ˜u for e = 0.2, 0.5 and 0.7 and for
different values of the average number of collisions per particle c. For a Maxwellian y = 2
and for an exponential y = 0. The initial distribution function is a Maxwellian. The tran-
sition region is clearly observed, and the existence of an exponential tail is demonstrated.
Following the discussion in the beginning of this section, the transition region should be
situated between ˜u = O 1√
ǫ
and ˜u = O 1
ǫ
. For the current values of e these numbers
are too close to warrant a definitive conclusion. To clearly distinguish the limits of the
transition region one should go to unattainably small values of ǫ (i.e e very close to 1).
In section 3.4 this problem is overcome using a model equation.
Non-Maxwellian initial conditions were considered as well (e.g. ˜f (0, c) ∝ ˜ue−˜u2
). In
all cases it takes less the one collision per particle for the head of the distribution to
become Maxwellian. A derived quantity of interest is ξ, found to be linear in time its
slope being consistent with Eq. (3.16) and perturbation theory presented in section 3.3.
Figures 3.5–3.7 show the convergence rate conveniently defined as the average number
of collisions per particle needed to cover 95% of the difference between the initial condition
and the stationary distribution ˜f (c95) ≡ 95% ˜f (∞, u) − ˜f (0, u) .
In figure 3.8 the ratio r = Loss
Gain
vs. ˜u for the numerical ˜f for three values of e is
plotted. For ˜u < 0.1 r < 1 (the gain dominates), for 0.1 < ˜u < 1√
ǫ
r ∝ ˜u and or 1√
ǫ
≪ ˜u
r ∝ ˜uγ
here γ ≈ 3
43. 3.3. THE NEAR-MAXWELLIAN RANGE OF SPEEDS 43
0 1 2 3 4 5 6 7 8 9
0
0.5
1
1.5
2
2.5
3
c = 0
c = 1.25
c = 2.5
c = 3.75
c = 5
c = 6.25
c = 15
ξ u
Figure 3.2: Development of the distribution function as a function of the number of
accumulated collisions per particle, c . Shown are − ∂2
∂˜u2 log ˜f (˜u) versus the rescaled
speed ˜u = ξu for e = 0.2. Notice the crossover from the Maxwellian value, 2, to the value
0, corresponding to the exponential decay. Also notice the fast convergence of the bulk
versus the slow convergence of the tail. The initial distribution is Maxwellian.
3.3 The near-Maxwellian range of speeds
This section is devoted to a study of the near-Maxwellian range of speeds. To this end it
is convenient to perform a perturbative expansion of the pertinent Boltzmann equation in
powers of the degree of inelasticity, ǫ = 1 − e2
. Only the universal distribution is sought.
44. 44 CHAPTER 3. THE HOMOGENEOUS COOLING STATE
0 1 2 3 4 5 6 7 8 9
0
0.5
1
1.5
2
2.5
3
c = 0
c = 1.6
c = 3.2
c = 4.8
c = 6.4
c = 8
c = 20
ξ u
Figure 3.3: Development of the distribution function as a function of the number of
accumulated collisions per particle, c . Shown are − ∂2
∂˜u2 log ˜f (˜u) versus the rescaled
speed ˜u = ξu for e = 0.5. Notice the crossover from the Maxwellian value, 2, to the value
0, corresponding to the exponential decay. Also notice the fast convergence of the bulk
versus the slow convergence of the tail. The initial distribution is Maxwellian.
Using the (universal) scaled distribution function (3.19) ˜f(˜u), where ξ(t) is given by
Eq. (3.16). Notice that the parameter λ, is left to be determined by the perturbation
theory. With this scaling, the Maxwellian solution transforms to
˜fM ≡ π−3/2
exp −˜u2
(3.26)
45. 3.3. THE NEAR-MAXWELLIAN RANGE OF SPEEDS 45
0 1 2 3 4 5 6 7 8 9
0
0.5
1
1.5
2
2.5
3
c = 0
c = 1.2
c = 3.5
c = 6
c = 8.2
c = 10.6
c = 29
ξ u
Figure 3.4: Development of the distribution function as a function of the number of
accumulated collisions per particle, c . Shown are − ∂2
∂˜u2 log ˜f (˜u) versus the rescaled
speed ˜u = ξu for e = 0.7. Notice the crossover from the Maxwellian value, 2, to the value
0, corresponding to the exponential decay. Also notice the fast convergence of the bulk
versus the slow convergence of the tail. The initial distribution is Maxwellian.
and the (rescaled) Boltzmann equation reads:
3 ˜f(˜v1) + ˜v1
d ˜f(˜v1)
d˜v1
=
1
ǫλ
√
π (k·˜v12)>0
d˜v2dk (k · ˜v12)
1
e2
˜f (˜v′
1) ˜f (˜v′
2) − ˜f (˜v1) ˜f (˜v2)
≡
1
ǫλ
√
π
˜B( ˜f, ˜f; e) (3.27)
46. 46 CHAPTER 3. THE HOMOGENEOUS COOLING STATE
0 1 2 3 4 5 6 7 8 9 10
0
2
4
6
8
10
12
14
ξ u
c95
Figure 3.5: The average number of collisions per particle needed to cover 95% of the differ-
ence between the initial condition (Maxwellian) and the stationary distribution function
versus the rescaled speed ˜u = ξu for e = 0.2. Notice the fast convergence of the bulk
versus the slow convergence of the tail.
thereby defining the (bilinear) operator ˜B. Define the function φ by
˜f(˜v) = ˜fM (˜v)(1 + φ(˜v))
The correction φ is calculated perturbatively, in powers of ǫ; it can be shown that this
is the only choice of a small parameter which is a function of ǫ, up to a multiplicative
constant, that yields a consistent perturbation theory. Notice that when ǫ = 0, φ = 0.
The zeroth order (Maxwellian) distribution is conveniently chosen to possess the correct
47. 3.3. THE NEAR-MAXWELLIAN RANGE OF SPEEDS 47
0 1 2 3 4 5 6 7 8 9 10
0
2
4
6
8
10
12
14
ξ u
c95
Figure 3.6: The average number of collisions per particle needed to cover 95% of the differ-
ence between the initial condition (Maxwellian) and the stationary distribution function
versus the rescaled speed ˜u = ξu for e = 0.5. Notice the fast convergence of the bulk
versus the slow convergence of the tail.
kinetic energy (average of ˜u2
). Therefore φ must satisfy:
∞
0
v4
e−v2
φ(v)dv = 0 (3.28a)
∞
0
v2
e−v2
φ(v)dv = 0 (3.28b)
(so that it does not affect the average kinetic energy, and preserve the normalization of
48. 48 CHAPTER 3. THE HOMOGENEOUS COOLING STATE
0 1 2 3 4 5 6 7 8 9 10
0
2
4
6
8
10
12
14
ξ u
c95
Figure 3.7: The average number of collisions per particle needed to cover 95% of the differ-
ence between the initial condition (Maxwellian) and the stationary distribution function
versus the rescaled speed ˜u = ξu for e = 0.7. Notice the fast convergence of the bulk
versus the slow convergence of the tail.
˜f). Define now the following expansions (formally for ǫ ≪ 1):
φ(u, e) =
∞
k=1
ǫk
φk(u) (3.29a)
λ(e) =
∞
k=0
ǫk
λk (3.29b)
˜B(f, g, e) =
∞
k=0
ǫk
Bk(f, g) (3.29c)
49. 3.3. THE NEAR-MAXWELLIAN RANGE OF SPEEDS 49
0.1 0.2 0.4 0.6 0.8 1 2 3 4 5 6
10
0
10
1
10
2
10
3
ξ u
Loss
−−−−−−−−
Gain
e = 0.2
e = 0.5
e = 0.7
Figure 3.8: The ratio of the loss term to the gain term extracted from a converged
numerical solution of the Boltzmann equation for three values of e, versus the rescaled
speed. For uξ < 0.1 the ratio is smaller than unity, indicating a dominance of the gain,
when 0.1 < uξ < 1/
√
1 − e2 the ratio is proportional to uξ the loss dominates the gain for
uξ > 1), and for uξ > 1/
√
ǫ the ratio is proportional to (ξu)γ(ǫ)
, with γ ≈ 3, in conformity
with the theoretical calculation (C.3)
The last definition for the nonlinear Boltzmann operator does not include an expansion
of f or g in powers of ǫ; the only sources of ǫ dependence in the expansion of ˜B are
the explicit 1/e2
term in the definition of this operator and the ǫ dependence of {v′
1, v′
2}
on {v1, v2}. Therefore the full expansion of ˜B in powers of ǫ, when ˜fM (1 + k ǫk
φk) is
substituted for ˜f, requires a further expansion. This, slightly cumbersome definition, is
calculationally convenient, as seen below. The tilde superscripts are removed below for
sake of notational simplicity. The zeroth order in ǫ, now including the expansion of f in
50. 50 CHAPTER 3. THE HOMOGENEOUS COOLING STATE
powers of ǫ, of the Boltzmann equation, reads:
1
λ0
√
π
[B0(fM φ1, fM ) + B0(fM , fM φ1)] = 3fM + u
dfM
du
−
1
λ0
√
π
B1(fM , fM ). (3.30)
It is convenient to multiply both sides of the above equation by uλ0/4fM , thus obtaining:
L ∗ φ1(u) ≡
u
4fM
√
π
[B0(fM φ1, fM ) + B0(fM , fM φ1)]
=
uλ0
4fM
3fM + u
dfM
du
−
1
λ0
√
π
B1(fM , fM ) (3.31)
The left hand side of Eq. (3.31) defines the action of the linearized Boltzmann operator,
L, and the right hand side is a source term, s1, whose functional dependence on u is
easy to calculate, cf. Eq. (C.5), as fM is given and B1 is easily obtained from B(f, f; e).
Only the prefactor, λ0, is not a-priori known (see below). The integrations defining the
operator L can be straightforwardly carried out, yielding:
L ∗ φ =
√
π
u
0
dvvφ(v)erf(v) +
√
πeu2
erf(u)
∞
u
dvvφ(v)e−v2
−
u
4
e−u2
+
√
π
4
1
2
+ u2
erf(u) φ(u)
−
u
0
dv v2
u2
+
v4
3
e−v2
φ(v) −
∞
u
dv uv3
+
vu3
3
e−v2
φ(v) (3.32)
Thus, the correction φ1 satisfies: L ∗ φ1(u) = s1(u). It is easy to show that the n-th order
in perturbation theory is of the form: L ∗ φn = sn.
The operator L is self-adjoint in a Hilbert space in which the scalar product of two
real functions, a(v) and b(v), is:
(a, b) ≡
∞
0
vfM (v)a(v)b(v)dv
Also, L ∗ vk
= 0 for k = 0, 2 [80]: the functions 1 and v2
are the only normalizable zero
eigenfunctions of L; they clearly correspond to the conservation of particle number and
energy (in the elastic limit). Hence:
vk
, L ∗ φn = vk
, sn =
∞
0
dvvk+1
fM (v)sn(v) = 0 k = 0, 2 (3.33)
51. 3.3. THE NEAR-MAXWELLIAN RANGE OF SPEEDS 51
These are the two solubility conditions at order n. In addition, the two normalization
conditions (3.28) that φn must satisfy, read:
(vk
, φn) =
∞
0
dvvk+1
fM (v)φn(v) = 0 k = 1, 3. (3.34)
The general solution of L ∗ φn = sn is
φn(u) = an + bnu2
+ Φn(u)
where, an + bnu2
is the solution of the homogeneous equation, L ∗ φn = 0. Without loss
of generality one may impose Φn(0) = Φ′′
n(0) = 0 on the inhomogeneous solution.
All in all φn must satisfy two conditions, Eq. (3.34), and these determine the param-
eters an and bn. The source term, sn, which depends on one free parameter, λn−1 (e.g.,
s1 depends on λ0), must also satisfy two conditions, Eq. (3.33). This does not lead to
an overdetermination of λn−1, because, as shown in C.2, the condition corresponding to
k = 0 in Eq. (3.33) is identically satisfied by sn. In the corresponding elastic case the
condition k = 2 is also identically satisfied by the source term, in conformity with the
fact that there is no additional free parameter in this case. The condition k = 2, applied
to s1, reads:
v2
, s1 = π−3/2
∞
0
dvv3
e−v2 λ0
4
3v − 2v3
− I2(v) = 0 (3.35)
where I2 is given by Eq. (C.4). This integral can be carried out analytically yielding:
λ0 =
√
2
3
, in conformity with the mean field result, Eq. ( 3.15).
The correction φ1 can be straightforwardly found from the equation L ∗ φ1 = s1 in
terms of a polynomial expansion:
φ1(u) = a1 + b1u2
+ A4u4
+ A6u6
+ . . . (3.36)
where a1 = −0.320, b1 = 0.574, A4 = −0.246, A6 = 6.6075 × 10−2
, A8 = −1.337 × 10−2
and A10 = 1.001 × 10−3
. One can check that this perturbative solution agrees well with
the numerical solution of the Boltzmann equation up to u = O (1/
√
ǫ) and slightly beyond
(for values of ǫ which are O(1)). Notice the rather rapid decrease of these coefficients.
52. 52 CHAPTER 3. THE HOMOGENEOUS COOLING STATE
A straightforward asymptotic analysis of the equation L ∗ φ1 = s1 (using Eq. (3.32)
for u ≫ 1), reveals that:
φ1(u) ∼ −u2
ln(u) −
2
√
2
√
π
u + 3.9538u2
i.e., the ‘asymptotic’ solution of the Boltzmann equation for large u is, to linear order in
ǫ:
f(u) ∼ π−3/2
e−u2
1 − ǫu2
ln u
this limit makes sense only for ǫ ≪ 1 and u < O (1/
√
ǫ). This, and the above results
imply that the perturbation expansion is valid for u < 1/
√
ǫ, which is also the range of
the near-Maxwellian distribution. The reduced equation presented in the next section
shares this property.
3.4 Reduction of the Boltzmann equation for the HCS
The goal of the present section is to present a derivation of a reduced equation for the
HCS, on the basis of the Boltzmann equation. The reduction is achieved by an asymptotic
analysis of the Boltzmann equation at high speeds, u ≫ 1 (in the rescaled variables).
The error allowed in the non-linear Boltzmann operator is algebraically subdominant to
the loss term at large speeds. It turns out that the error of the resulting equation at
low speeds is formally O(ǫ). The solutions of the resulting equation, much like those of
other equations derived by asymptotic methods, correctly reproduce the HCS distribution
function, far beyond the nominal limits of its derivation. As a matter of fact, the solutions
of the reduced equation are practically indistinguishable from those of the full Boltzmann
equation for all speeds, even at values of the coefficient of restitution which are as low as
0.4. A formal perturbative analysis of the reduced equation in powers of ǫ yields a leading
order correction which is precisely that obtained in the previous section for 1 ≪ u ≪ 1/
√
ǫ.
Possible explanations for these features are proposed. Needless to say, the three universal
speed ranges are fully reproduced by the reduced equation (analytically and numerically).
53. 3.4. REDUCTION OF THE BOLTZMANN EQUATION FOR THE HCS 53
This section ends with a second reduction of the Boltzmann equation, which is of the form
of a transcendental equation. The latter yields solutions which are in excellent agreement
with the numerical solutions (and, of course, with the first reduced equation). Thus this
section provides a nearly analytic and uniform solution for the HCS. Finally, it is stressed
that the methods presented below are non-perturbative.
0 1 2 3 4 5
0
0.02
0.04
0.06
0.08
0.1
0.12
0.14
0.16
0.18
ξ u
f
Numerical
Perturbation
Figure 3.9: Comparison of the first order perturbation theory result for the HCS versus
the numerical solution of the Boltzmann equation for e = 0.8. The (rescaled) distributions
are shown versus the rescaled speed, ξu.
54. 54 CHAPTER 3. THE HOMOGENEOUS COOLING STATE
0 1 2 3 4 5
0
5
10
15
20
25
30
ξ u
−log[f(ξu)]
Simplified Equation
Full Equation
Figure 3.10: Comparison of the numerical solution of the Boltzmann equation for e = 0.8
with the numerical solution of the reduced equation, Eq. (3.46). Shown are the logarithms
of the distribution functions. Notice the close correspondence, even for small speeds and
not-so-small ǫ.
3.4.1 Derivation of the reduced equation
The goal of the present subsection is to derive a simplification of the Boltzmann equation,
which is formally valid at large values of the speed.
As in the above, it is convenient to employ a non-dimensional representation with
˜u = ξu, f(u, t) = nξ3 ˜f(˜u) and all lengths rescaled by the particle radius. The integrals,
{Gi}, Eqs (A.34 – A.42), are rescaled as follows: ˜Gi ≡ Gi/ (4πσT n2
ξ2
). It follows that
55. 3.4. REDUCTION OF THE BOLTZMANN EQUATION FOR THE HCS 55
the gain term is given by:
∂f
∂t g
=
9
i=1
Gi = 4πσT n2
ξ2
·
9
i=1
˜Gi,
and the loss term is:
∂f
∂t l
≡ 4πσT n2
ξ2 ∂f
∂t l
= 4πσT n2
ξ2
· ˜f (u)
˜u
0
˜u˜v2
+
˜v4
3˜u
˜f (˜v) d˜v +
∞
˜u
˜v3
+
˜u2
˜v
3
˜f (˜v) d˜v
so that
n ˙ξξ2
3 ˜f + ˜u
d ˜f
d˜u
= 4πσT n2
ξ2
9
i=1
Gi −
∂f
∂t l
Recalling that ˙ξ = ǫλnσT π−1/2
(following Eq. (3.35): λ =
√
2/3 + O(ǫ)), one obtains:
ǫ
λπ− 3
2
4
3 ˜f + ˜u
d ˜f
d˜u
=
9
i=1
Gi −
∂f
∂t l
(3.37)
In the sequel all entities are assumed dimensionless and the tilde signs are dropped for
notational simplicity.
As shown in B, only G2 is non-negligible with respect to the loss term in the large-u
limit. It is convenient to define (the dimensionless) G∗
2 as follows:
G∗
2(ǫ, u) ≡
8
(1 + e)
3
2 (3 − e)u
u
0
dv1v1f(v1) ×
∞
√
(u2−v2
1)(3−e)/(1+e)
dv2 (3 − e)(v2
1 − u2) + (1 + e)v2
2 v2 f(v2) (3.38)
where the dependence of G∗
2 on ǫ has been spelled out. The difference between G∗
2 and
G2 is in the ranges of integration. Denote this difference by: G2 − G∗
2 = −δ1G2 − δ2G2,
where:
δ1G2(ǫ, u) ≡
8
(1 + e)
3
2 (3 − e)u
u(1−e)/2
0
dv1v1f(v1) ×
[2u−(1−e)v1]/(1+e)
√
(u2−v2
1)(3−e)/(1+e)
dv2 (3 − e)(v2
1 − u2) + (1 + e)v2
2 v2 f(v2) (3.39)
56. 56 CHAPTER 3. THE HOMOGENEOUS COOLING STATE
and
δ2G2(ǫ, u) ≡
8
(1 + e)
3
2 (3 − e)u
u
0
dv1v1f(v1) ×
∞
[2u−(1−e)v1]/(1+e)
dv2 (3 − e)(v2
1 − u2) + (1 + e)v2
2 v2 f(v2) (3.40)
It is shown in B.2 that δG1 and δG2 are algebraically subdominant to the loss term for
u ≫ 1. It follows that the gain term can be substituted for by G∗
2, the error being
subdominant to the loss term. Consider the latter next. The large-u form of the loss term
is:
uf(u)
∞
0
dv v2
f(v) =
1
4π
uf(u)
where the normalization of the dimensionless form of the distribution function has been
used. Therefore, for large u, the Boltzmann equation for the HCS reduces to:
ǫλ
4π
3
2
3f + u
df
du
= G∗
2 −
1
4π
uf(u) (3.41)
Interestingly, Eq. (3.41) becomes an identity for ǫ = 0 and f(u) = π−3/2
e−u2
, as can
be checked by direct substitution. Thus, Eq. (3.41) can be regarded to be a uniform
approximation of the Boltzmann equation, whose error is algebraically small (with respect
to the loss term) for u ≫ 1, and O(ǫ) for u ≤ O(1). Since the bulk of the distribution is
near-Maxwellian for practically all physical values of e, one expects Eq. (3.41) to comprise
a rather faithful approximation of the pertinent Boltzmann equation. It is further reduced
below.
Define the following variable transformations for Eq. (3.38), x ≡ v2
1, y ≡ v2
2 and
z ≡ u2
. Define a shift: t ≡ y − 3−e
1+e
(z − x) and let µ ≡ 3−e
1+e
. Also define: ˆh(v2
) ≡ f (v).
With these transformations, one obtains from Eq. (3.38):
G∗
2 (ǫ, u) =
2
(3 − e) (1 + e)
√
z
z
0
ˆh(z − x)dx
∞
0
ˆh(t + µx)
√
tdt (3.42)
Next, Eq. (3.42) is further simplified. Let ˆh (z) ≡ e−ψ(z)
. The next step is showing that
ˆh(t + µx) in Eq. (3.42) can be replaced by exp [−ψ(µx) − tψ′
(µx)], where ψ′
denotes
57. 3.4. REDUCTION OF THE BOLTZMANN EQUATION FOR THE HCS 57
the derivative of ψ with respect to its argument. To this end note that for z = O(1)
and t = O(1), ˆh(t + µx) is near-exponential in its argument (i.e., near-Maxwellian in the
original variables) and the above expansion is justified. The contribution of t > O(1) is at
least exponentially small with respect to that of t = O(1) and will therefore be neglected.
Consider next the case when z > O(1). Upon splitting the integration over x, from 0 to
Λ = O(1) and from Λ to z one obtains two contributions. In the first range one is justified
in expanding ψ, and the contribution of the second range is at least exponentially smaller
than that of the first range, for every value of t. Using this expansion in Eq. (3.42), one
obtains:
G∗
2 (ǫ, u) ≃
√
π
(3 − e) (1 + e)
√
z
z
0
ˆh(z − x)ˆh (µx)
1
(ψ′ (µx))
3
2
dx (3.43)
Further straightforward considerations (taking into account that the distribution crosses
over from a near-Maxwellian to an exponential for u > O(1/ǫ) and that ˆh(µz) < ˆh(z),
since µ > 1), reveal (in agreement with numerical tests) that the integration over x in
Eq. (3.43) is dominated by the ‘small’ x range, where the distribution is near-Maxwellian,
hence ˆh is near exponential and ψ′
(µx) = 1 in this range. One can therefore replace ψ′
in
Eq. (3.43) by unity, obtaining:
G∗
2 (ǫ, u) ≃
√
π
(3 − e) (1 + e)
√
z
z
0
ˆh (z − x) ˆh (µx) dx (3.44)
Next, define: ˆh (z) ≡ (3−e)(1+e)
4
π(−3/2)
h (z). Substituting this definition into Eq. (3.44)
and using Eq. (3.41), one obtains:
ǫ
2λ
√
π
z
3
2 h′
(z) + z +
3ǫλ
√
π
√
z h (z) =
z
0
h (z − x) h (µx) dx (3.45)
As a final step, neglect ǫ
√
z with respect to z since this contribution is only important
when z < ǫ2
, i.e., for values of z at which the asymptotic evaluation is not valid. Also the
equation is valid only to within O(ǫ). Therefore the final form of the reduced equation is:
ǫ
2λ
√
π
z
3
2 h′
(z) + zh (z) =
z
0
h (z − x) h (µx) dx (3.46)
58. 58 CHAPTER 3. THE HOMOGENEOUS COOLING STATE
One can easily check that the reduced equation admits a Maxwellian solution (i.e., h(z) =
e−z
) when ǫ = 0, and that the leading term (in z = u2
) of its perturbative expansion to
first power in ǫ reads:
h(z) = e−z
(1 −
1
2
ǫzlnz)
in agreement with the large speed limit (1 ≪ u, 1/
√
ǫ, for ǫ ≪ 1) of the perturbative
solution of the full equation. Thus, although the error of the equation is O(ǫ) at low
speeds, it is correct to order ǫu2
. As the numerical results show, cf. Fig. 3.10, the
equation faithfully reproduces the distribution function for the HCS. Further analysis of
the equation is given in the next subsection.
3.4.2 Qualitative Analysis of the Reduced Boltzmann Equation
Here it is shown that the solution of the simplified equation has the same structure (three
regions in velocity space) as the solution of the full Boltzmann equation. It is convenient
to write Eq. (3.46) in terms of φ (z) ≡ − ln h (z). Clearly,
φ
′
(z) =
√
π
2λ
1
ǫ
z− 3
2
z
0
1 − eφ(z)−φ(µx)−φ(z−x)
dx (3.47)
Let us assume, for the time being, that the gain is much smaller than the loss every-
where. That is
z ≫
z
0
e−φ(µx)−φ(z−x)+φ(z)
dx. (3.48)
In this case the normalized solution would be
h (z) =
π
µ
3
2
4
(3 − e) (1 + e)
√
π
8λ3
1
ǫ3
e−
√
π
λǫ
√
z
(3.49)
Upon substitution into the r.h.s. of Eq. (3.48) the condition is (the O(1) constant in
brackets in Eq. (3.49) is omitted)
z ≫
1
ǫ3
z
0
e
√
π
λǫ [
√
z−
√
µx−
√
z−x]dx
59. 3.4. REDUCTION OF THE BOLTZMANN EQUATION FOR THE HCS 59
The function L(x) ≡
√
z −
√
µx −
√
z − x is non-positive, L(0) = 0, L(z) =
√
z −
√
µz
and it has a minimum at x0 = µ
µ+1
z. The integrand is exponentially small except near
x = 0 and x = z. Thus the condition is:
z ≫
1
ǫ3
z
0
e
√
π
λǫ [
√
z−
√
µx−
√
z−x]dx ≃
1
ǫ3
∞
0
e−
√
π
λǫ
√
µt
+ e
√
z−
√
µz
e−
√
π
λǫ
t
tdt ∝
1
ǫ
Not surprisingly, the solution breaks down for sufficiently small values of z (since by
construction the distribution is Maxwellian for O(1) values). Next, assume that:
h (z) = 4
(3−e)(1+e)
µ− 3
2 e−z
; z < O(1
ǫ
)
h (z) = Ae−
√
π
λǫ
√
z
; z ≥ O(1
ǫ
)
Proceeding as above then, upon substitution into the r.h.s. of Eq. (3.48), the inequality
reads
z ≫
4
(3 − e)(1 + e)
µ− 3
2
1
ǫ
0
e
√
π
λǫ [
√
z−
√
z−x]−µx
dx + A
z
1
ǫ
e
√
π
λǫ [
√
z−
√
µx−
√
z−x]dx
The second integral is exponentially small and may be ignored. Defining L1(x) =
√
π
λǫ
√
z − x+
µx, then L1(0) =
√
π
λǫ
√
z, L′
1(x) = −
√
π
2λǫ
1√
z−x
+ µ and L′
1(x0) = 0 for x0 = z − π
µ2λ2ǫ2 . Two
cases are possible: z < O(ǫ−2
) or z ≥ O(ǫ−2
). If z < O(ǫ−2
), L1(x) is monotonously
decreasing and the integrand is maximal at x = 1
ǫ
. Hence
1
ǫ
0
e
√
π
λǫ [
√
z−
√
z−x]−µx
dx ∝ e
√
π
λǫ
h√
z−
√
z− 1
ǫ
i
− µ
ǫ
≃ e
1
ǫ
“ √
π
λǫ
√
z
−µ
”
≫ z
(exponentially large) while if z ≥ O(ǫ−2
), the maximum of L1(x) is at x = 0 and
1
ǫ
0
e
√
π
λǫ [
√
z−
√
z−x]−µx
dx ∝
∞
0
e−µx
dx = O(1) ≪ z
To summarize: the assumption of a uniformly valid exponential distribution led to a
contradiction. The way to resolve that contradiction was to assume that the gain was
negligible only for z ≥ O(ǫ−1
). Furthermore, assuming a Maxwellian for z < O(ǫ−1
), led
to a second contradiction which further restricted the range in which the gain term is
negligible compared to the loss to z ≥ O(ǫ−2
).
60. 60 CHAPTER 3. THE HOMOGENEOUS COOLING STATE
The present analysis yields a result in full agreement with the analysis of the full Boltz-
mann equation in section 3.2. One distinguishes three regions: 1) (Pseudo) Maxwellian
z < O(ǫ−1
) ⇔ u < O(ǫ− 1
2 ) 2) Intermediate range:O(ǫ−1
) < z < O(ǫ−2
) ⇔ O(ǫ− 1
2 ) < u <
O(ǫ−1
) 3) Exponential region z > O(ǫ−2
) ⇔ u > O(ǫ−1
).
3.4.3 Second reduction of the Boltzmann equation
Although Eq. (3.46) represents an enormous simplification of the full Boltzmann equation
it still requires a certain numerical effort to solve it. A further reduction is therefore useful.
As in the previous subsection, let φ(z) ≡ − ln h. With this definition, Eq. (3.46)
transforms to (c.f. Eq. (3.47) ):
z − ǫ
2λ
√
π
z
3
2 φ
′
(z) =
z
0
eφ(z)−φ(µx)−φ(z−x)
dx (3.50)
The arguments presented in subsection 3.4.1, as well as numerical studies (cf. Fig. 3.11),
show that the integral in Eq. (3.50), like the integral in Eq. (3.43), is dominated by
the small values of x when z > O(1
ǫ
), hence one can affect the following (approximate)
replacement for z > O(1
ǫ
):
φ(z) − φ(z − x) − φ(µx) ≈ φ′
(z)x − φ(µx) ≈ φ′
(z)x − φ(0) − µx, (3.51)
where the second approximate equality follows from the fact that the distribution is near
Maxwellian for small values of x (i.e., φ(z) ≈ φ(0) + z for small z). Consider now the
range z < O(1
ǫ
). In this range the (approximate) linearity of φ in its argument, renders
Eq. (3.51) valid as well. Finally, it is straightforward to deduce from Eq. (3.46) that
h(0) is close to unity (to within O(ǫ), hence one can substitute φ(0) = 0. With these
approximations, the integral on the right hand side of Eq. (3.50) (i.e., the integral of an
exponential) can be immediately performed and Eq. (3.50) reduces to:
z − ǫ
2λ
√
π
z
3
2 φ
′
(z) =
1 − e−q(z)
q (z)
z (3.52)
where
q (z) ≡ z (µ − φ′
(z)) (3.53)
61. 3.4. REDUCTION OF THE BOLTZMANN EQUATION FOR THE HCS 61
0 1 2 3 4 5
−0.8
−0.6
−0.4
−0.2
0
x
φ(z)−φ(µx)−φ(z−x)
0 10 20 30 40 50
−8
−6
−4
−2
0
x
φ(z)−φ(µx)−φ(z−x)
Figure 3.11: Plot of the argument of the exponent in Eq. (3.50) versus x. The value of ǫ
is 0.1. The left plot corresponds to z = 5 and the right plot corresponds to z = 50. These
results are extracted from a numerical solution of the reduced equation, Eq. (3.46).
Substituting φ′
(z) = µ − q(z)/z, which follows from the definition of q, into Eq. (3.52),
one obtains the following transcendental equation for q(z):
2ǫλ
√
π
µ
√
z −
q(z)
√
z
− 1 +
1 − e−q(z)
q(z)
= 0 (3.54)
Once q(z) is known, the value of φ(z) can be obtained by direct integration, using the
definition of q(z). A straightforward analysis of Eq. (3.54) shows that for z < O(1/ǫ),
q(z) is approximately given by:
q(z) ≈
4ǫλ√
π
µz
√
z + 4ǫλ√
π
It follows that in this range φ(z) = µz + O(
√
z), i.e. the distribution is near-Maxwellian.
When z > O(1/ǫ2
), the solution of Eq. (3.54) is given by:
q(z) ≈ µz −
√
πz
2ǫλ
hence φ(z) ≈
√
πz/ (ǫλ), which corresponds (as expected) to an exponential distribution.
A full numerical evaluation of q(z) from Eq. (3.54) and the calculation of φ(z) on the
62. 62 CHAPTER 3. THE HOMOGENEOUS COOLING STATE
basis of this result yield a distribution function (i.e., exp [−φ(z)]), which is essentially
indistinguishable from the numerical solution of the Boltzmann equation. Therefore there
is no point in presenting a graph showing this correspondence. It is somewhat surprising
that the second reduction still yields such good agreement with the numerical solution.
3.5 Concluding remarks
A study of the properties of the HCS has been presented. Among the surprising features
is the fact that the reduced descriptions so faithfully agree with the numerical solution
(even in terms of perturbative corrections), in spite of the fact that the asymptotic anal-
yses leading to the reductions are based on the large speed properties of the Boltzmann
equation. It is possible that this good correspondence is due to the fact that the bulk
of the distribution function is not significantly removed from a Maxwellian (except at
values of e which are near zero) and that the asymptotics sets in (as numerical results
show) at values of u which are basically of the thermal speed (for e not close to 1); when
e is close to unity the approximate equation is a faithful approximation of the Boltz-
mann equation anyway. Another interesting feature is the nontrivial structure of the
transition region, between the near-Maxwellian and the exponential ranges. This region
is prominent for near-elastic values of the coefficient of restitution but it does exist (for
O(1/
√
ǫ) < ˜u < O(1/ǫ) at every value of e. It is perhaps important to mention that the
dominance of the loss over the gain term in the Boltzmann equation is ‘just’ algebraic
(proportional to u3
for the tail of the distribution function and linear in u in the transition
region, cf. C.3 and Fig. (3.8)). Note that the Boltzmann equation without corrections for
precollisional correlations [82, 85] at low values of the coefficient of restitution is merely
an academic exercise. However, I believe that even in this case, it is interesting to uncover
the structure of its solutions.
63. Chapter 4
Hydrodynamics of nearly smooth
granular gases
In this chapter hydrodynamic equations of motion for a monodisperse collection of nearly
smooth homogeneous spheres are derived from the corresponding Boltzmann equation, us-
ing a Chapman-Enskog expansion around the elastic smooth spheres limit. Since, in the
smooth limit the rotational degrees of freedom are uncoupled from the translational ones,
it turns out that the required hydrodynamic fields include (in addition to the standard
density, velocity and translational granular temperature fields) the (infinite) set of num-
ber densities, n(s, r, t), corresponding the continuum of values of the angular velocities.
The Chapman-Enskog expansion was carried out to high (up to 10th) order in a Sonine
polynomial expansion by using the method introduced in chapter 5. One of the conse-
quence of these equations is that the asymptotic spin distribution in the homogeneous
cooling state for nearly smooth, nearly elastic spheres, is highly non-Maxwellian. The
simple sheared flow possesses a highly non-Maxwellian distribution as well. In the case of
wall-bounded shear, it is shown that the angular velocity injected at the boundaries has
a finite penetration length.
63
