Contents Part I Kinetic Theory Asymptotic solutions of the nonlinear Boltzmann equation for dissipative systems Matthieu H. Ernst, Ricardo Brito .................................. 3 The Homogeneous Cooling State revisited Isaac Goldhirsch, S. Henri Noskowicz, O. Bar-Lev ................... 37 The Inelastic Maxwell Model Eli Ben-Naim, Paul L. Krapivsky .................................. 63 Cooling granular gases: the role of correlations in the velocity field Andrea Baldassarri, Umberto Marini Bettolo Marconi, Andrea Puglisi .. 93 Self-similar asymptotics for the Boltzmann equation with inelastic interactions Alexander V. Bobylev, Carlo Cercignani ............................ 117 Kinetic Integrals in the Kinetic Theory of Dissipative Gases Thorsten P¨ oschel, Nikolai V. Brilliantov ............................ 129 Kinetics of fragmenting freely evolving Granular Gases Ignacio Pagonabarraga and Emmanuel Trizac ....................... 161 Part II Granular Hydrodynamics Shock waves in granular gases Alexander Goldshtein, Alexander Alexeev, Michael Shapiro ............ 185
Isaac Goldhirsch1, S. Henri Noskowicz1, and O. Bar-Lev2
1 Department of Fluid Mechanics and Heat Transfer, Faculty of Engineering,Tel-Aviv University, Ramat-Aviv, Tel-Aviv 69978, Israel
2 School of Mathematics, Faculty of Exact Sciences, Tel-Aviv University,Ramat-Aviv, Tel-Aviv 69978, Israel
Summary. The distribution function for the homogeneous cooling state (HCS) ofa monodisperse collection of inelastically colliding spheres has more structure thanhitherto thought. Define the following notation: e is the (fixed) coefficient of normalrestitution, ε ≡ 1− e2 the ‘degree of inelasticity’, T the granular temperature, andu the speed variable. There is an initial-state-dependent speed, u∗, such that foru∗ < u the HCS distribution function retains memory of the initial distribution.For u < u∗ the initial distribution converges to a universal scaling function. In the
latter range of speeds, there is a near-Maxwellian subrange for u < O(√
T/√ε),
a ‘transition’ subrange for O(√
T/√ε)< u < O
(√T/ε
), and a near-exponential
subrange for u > O(√
T/ε). These speed ranges are more distinct for near-elastic
collisions, but still observable for any value of 0 ≤ e < 1. These results followfrom numerical studies, heuristic considerations and each of three levels of analyticreduction of the Boltzmann equation: (i). an exact reduction to an equation inthe speeds; (ii). an asymptotic reduction to a relatively simple one dimensionalintegrodifferential equation, and (iii). an asymptotic reduction of the latter to asimple transcendental equation; the latter leads to an approximate analytic solutionfor the distribution function of the HCS. The distribution functions obtained fromthe reduced descriptions are practically indistinguishable from the correspondingnumerical solutions of the full Boltzmann equation. Possible implications of theresults beyond the realm of the HCS are briefly mentioned.
1 Introduction
Granular gases lack a steady unforced homogeneous (of uniform macroscopicdensity) and isotropic (the directions of the particle velocities are isotropicallydistributed) state, i.e., an equilibrium-like state, since the kinetic energy of afree granular gas decreases with time as a result of the inelasticity of the graincollisions. A free, homogeneous and isotropic granular gas, whose energy doesdecay (hence its state is not stationary) is conceptually possible. This state,which is probably the ‘simplest’ state of a granular gas one can conceive, is
38 I. Goldhirsch, S. H. Noskowicz, and O. Bar-Lev
known as the homogeneous cooling state (HCS). The Boltzmann equationcorresponding to a granular gas indeed possesses an HCS solution. However,this solution is known to be unstable to clustering [1] and collapse [2], henceit is only a theoretical construct. Nevertheless, it turns out that the HCS is auseful concept in many respects, part of which are mentioned below.
First, a direct study of the HCS enables one to learn some properties ofthe inelastic Boltzmann equation. The HCS serves (more accurately, the lo-cal HCS does) as a zeroth order in some perturbation theories (based on theChapman-Enskog expansion) that result in constitutive relations for granulargases [3]. Furthermore, the HCS has been employed in derivations of Green-Kubo relations for granular gases [4]. In addition, the study of the HCS revealsthe breaking of equipartition [5, 6] in granular gases, in its simplest manifes-tation.
In spite of its basic significance in the field of granular gases, it seemsthat the HCS has not been thoroughly studied. Haff [7] was probably thefirst to show that the energy (or granular temperature) of an HCS of smoothinelastic spheres (with a fixed coefficient of normal restitution) decays as 1
t2 ,where t denotes time (corrections to Haff’s law were observed numericallyin ([1]-iv,v,vi) and theorized upon in ([8]-i,iii) and ([1]-vi,v,vi); as we onlyconsider the HCS, we shall not dwell on this problem below). The velocitydistribution of the HCS is not Maxwellian. Its departure from a Maxwelliandistribution (in the bulk, i.e., not the high speed tail) has been studied per-turbatively [8], and its tail has been shown [9] to decay exponentially (in thespeed). It has also been discovered that the HCS distribution function canbe appropriately scaled so that it is described (asymptotically in time) by atime-independent scaling function [8]. This property has been exploited in theabove mentioned derivations of constitutive relations and Green-Kubo rela-tions. All of the above results pertain to collections of smooth spheres. TheHCS for e.g., frictional spheres has not been studied in great detail: mean fieldapproaches [6] have revealed that it is (expectedly) more complex than thecorresponding HCS for smooth spheres (see also ([8]-i) for an early account ofthe breakdown of equipartition in these systems).
Clearly the HCS deserves careful and detailed study. Due to the abovementioned instabilities, it is impractical to study it by molecular dynamicsmethods except in very small and nearly elastic systems (although Monte-Carlo methods, such as the DSMC allow for such a study, e.g. ([1]-v), theysolve the underlying Boltzmann equation and do not provide a more basicvalidation as MD would have). The goal of the work presented below is toprovide a detailed analytic description of the HCS for smooth spheres in threedimensions. Among other results, it is shown that the pertinent Boltzmannequation can be reduced to a one-dimensional integrodifferential equation inthe speed, which quantitatively captures essentially all properties of the HCS.
Prior to embarking on the main goal of this article, it is perhaps impor-tant to stress that the coefficient of normal restitution for real grains is trulyvelocity dependent and that this property has important consequences, such
The Homogeneous Cooling State revisited 39
as the elimination of the collapse phenomenon [2] and the destabilization ofclusters [10]. This and other facts pertaining to realistic grains do not dimin-ish the importance of the HCS described here as the simplest (and perhapsof academic nature) case of a granular gas.
2 Setting the problem up
2.1 The collision model
The model considered in the present work is that of a monodisperse collec-tion of spheres of mass m = 1 and radius a. The system is assumed to beof infinite extent. The spheres interact by (binary) instantaneous collisionscharacterized by a fixed coefficient of normal restitution, denoted by e, with0 ≤ e ≤ 1. The basic collision process between two spheres, whose prec-ollisional (‘incoming’) velocities are ~vin,1 and ~vin,2, and whose correspondingpostcollisional (‘outgoing’) velocities are ~vout,1 and ~vout,2, respectively, is given
by: ~vout,21 ·~n = −e ~vin,21 ·~n, where ~vout,21 ≡ ~vout,2−~vout,1, ~vin,21 ≡ ~vin,2−~vin,1,and ~n is the unit vector pointing from the center of sphere “1” to the centerof sphere “2”. Combining the collision rule with momentum conservation oneobtains:
~vout,i = ~vin,i +1 + e
2
(~vin,ji · ~n
)~n , (1)
and
~vin,i = ~vout,i +1 + e
2e
(~vout,ji · ~n
)~n , (2)
where i, j ∈ 1, 2, and ~vin,12 · ~n ≥ 0 must hold.
2.2 The Boltzmann equation
Let f(~r,~v, t) denote the single particle (velocity) distribution function atpoint ~r at time t, where ~v denotes the (particle) velocity. By definition,∫f(~r,~v, t)d~v = n(~r, t), where n(~r, t) is the (particle) number density at point
~r at time t. For low densities the distribution function, f , for a granular gas,is believed to satisfy the inelastic Boltzmann equation [8, 11]:
∂f(~r,~v, t)
∂t+ ~v · ∇f(~r,~v, t) + ~F · ∂f(~r,~v, t)
∂~v=
(∂f(~r,~v, t)
∂t
)
c
, (3)
where ~F is an external (velocity independent) force and (∂f/∂t)c representsthe effect of the collisions. It is convenient [12] to separate the right handside of Eq. (3) into a ‘gain term’, (∂f(~r,~v, t)/∂t)g ≥ 0, which represents allcollisions that increase the number density of particles having velocity ~v, and
a ‘loss term’,(∂f(~r,~v,t)
∂t
)`≥ 0, which represents all collisions that decrease
40 I. Goldhirsch, S. H. Noskowicz, and O. Bar-Lev
this number density, such that(∂f(~r,~v,t)
∂t
)c=(∂f(~r,~v,t)
∂t
)g−(∂f(~r,~v,t)
∂t
)`. A
standard procedure [6, 11, 12] yields:(∂f(~r, ~v1, t)
∂t
)
`
= σT f(~r,~v1, t)
∫|~v12| f(~r,~v2, t) d~v2, (4)
where σT = 4πa2 is the total collision cross section. Notice that the form ofthe loss term is not affected by the inelasticity of collisions. The gain term isgiven by:
(∂f (~r, ~v1, t)
∂t
)
g
=1
e2
∫|~v12| f(~r,~v ′1, t)f(~r,~v ′2, t) d~v2 b db dφ , (5)
where b is the (outgoing) impact parameter (0 ≤ b ≤ 2a), φ the (outgoing)azimuthal angle of the collision and ~v ′1, ~v ′2 are the precollisional (‘incoming’)velocities corresponding to the postcollisional (‘outgoing’) velocities ~v1, ~v2,respectively. Notice that the dependence of Eq. (5) on the coefficient of resti-tution, e, is both explicit, through the factor 1/e2, and implicit, through thedependence of ~v ′1 and ~v ′2 on ~v1 and ~v2.
2.3 The Boltzmann equation for the homogeneous and isotropiccase
In the homogeneous and isotropic case, the distribution function, f , at timet, depends only on the speed, v (and t), and the number density is a con-
stant. Below, the external force field, ~F , is assumed to vanish. In this case theBoltzmann equation can be reduced to an equation involving speeds alone, asshown below.
Consider first the loss term. Let µ be the cosine of the angle between ~v2and ~v1 in Eq. (4). Using v221 = v21 + v22 − 2µv1v2, and integrating over µ andthe corresponding azimuthal angle (the latter yielding a factor of 2π), oneobtains from Eq. (4):(∂f(v1, t)
∂t
)
`
=2πσT3v1
f(v1, t)
∫ ∞
0
[(v1 + v2)
3 − |v1 − v2|3]v2f(v2, t)dv2 (6)
Further straightforward algebra yields from Eq. (6):
(∂f(u, t)
∂t
)
`
= 4πσT f(u)
[∫ ∞
0
(uv2 +
v4
3u
)f(v)dv
+
∫ ∞
u
(v3 +
1
3u2v − uv2 − v4
3u
)f(v)dv
], (7)
The angular integrations for the (∂f/∂t)g term are rather tedious. A detailedderivation will be presented elsewhere. The result is that the gain term canbe expressed as a sum of nine expressions, Gi(u); 1 ≤ i ≤ 9 (presented in
Appendix A-1): (∂f(u, t)/∂t)g =∑9i=1Gi(u, t).
The Homogeneous Cooling State revisited 41
3 Heuristic analysis
In this section, a heuristic study of the HCS distribution function is presentedand validated by numerical solutions of the pertinent Boltzmann equation. Itturns out that the values of the speed can be divided into a range in whichmemory of the the initial condition is retained, and one in which it is not.The latter range can be described by a scaling function and it consists of anear-Maxwellian, an intermediate and an exponential subrange.
3.1 Large speeds
Assuming that for asymptotically large values of u, the distribution f(u),decays faster than algebraically in u (a rather weak assumption, corroboratedby the results below), it follows from Eq. (7) that the second integral on ther.h.s. of Eq. (7) is negligible with respect to the first integral for sufficientlylarge speeds. Hence, asymptotically:
(∂f(u, t)
∂t
)
`
≈ nσT f(u, t)[u+
⟨v2; t
⟩
3u
], (8)
where⟨vk; t
⟩denotes the value of the k-th moment of the velocity (at time t):⟨
vk; t⟩= 4π
n
∫∞0v′
k+2
f (v ′, t) dv ′. The asymptotic properties of the gain termare considered next.
Notice that both for a granular and elastic gas, ‘high speed’ particles arerare and they collide chiefly with particles of typical speed. When the fasterparticle emerging from such a collision is slower than the faster ‘parent’ parti-cle, one can say that the collision contributes to a speed flux from high to lowspeeds, i.e., a ‘downward’ flux. Had there been no opposite flux in the elasticcase, a state of equilibrium could not have existed. Therefore there must be(at least in the elastic case) collisions which contribute to an upward fluxin the speed space to compensate for the downward flux. In contrast, in theinelastic case, the distribution function eventually shrinks to a delta functionin the speed. Therefore, the downward flux must dominate over the upwardflux. These observations are put on a more quantitative basis, by consideringthe kinematics of collisions, in the next paragraph.
Consider a collision (~v ′1, ~v′2)→ (~v1, ~v2) of a ‘fast’ particle ‘1’ with a typical
particle ‘2’: v ′1 À v ′2 = O(√〈v2〉). This collision can contribute to the upward
flux, i.e., ‘produce’ a particle whose speed is larger than v ′1, if either v1 > v ′1(henceforth, Case I) or v2 > v ′1 (henceforth, Case II). Using the kinematic
equations, Eqs. (1,2), as well as ~v ′12 · ~n > 0, this is equivalent to
Case I: ~v ′2 · ~n < ~v ′1 · ~n < −1 + e
3− e(~v ′2 · ~n
)
Case II:(~v ′1 · ~n− ~v ′2 · ~n
)(~v ′1 · ~n+
3− e1 + e
~v ′2 · ~n)>
(2
1 + e
)2 (v′21 − v′22
)
42 I. Goldhirsch, S. H. Noskowicz, and O. Bar-Lev
Since, by assumption,∣∣∣~v ′2 · ~n
∣∣∣ ≤ v ′2 = O(√〈v2〉
), inequality I can be satisfied
only if∣∣∣~v ′1 · ~n
∣∣∣ = O(√〈v2〉
), hence
∣∣∣~v ′12 · ~n∣∣∣ = O
(√〈v2〉
). This is possible
only for (the relatively rare) near grazing collisions (impact parameter close to2a). This result holds equally for both the elastic and inelastic cases. Considernext case II. When e = 1, the corresponding inequality is satisfied by v′21t < v′22t,where the subscript t denotes the tangential component of a vector: vt ≡√v2 − (~v · ~n)2. The latter inequality can be rewritten as v12t×(v1t+v2t) < 0;
this inequality is clearly satisfied by a finite fraction of the collisions. On theother hand, when e < 1, one obtains from condition II:
(3 + e)(1− e)4
(~v ′1 · ~n
)2− (1− e2)
2
(~v ′2 · ~n
)(~v ′1 · ~n
)
+(3− e)(1 + e)
4
(~v ′2 · ~n
)2< v′22 − (~v ′1t)
2(9)
A straightforward analysis of the inequality, Eq. (9), reveals that it can be sat-
isfied only when v ′1 is smaller than v ′1,max =√
3−e2(1−e)v
′2 =
√(3−e)(1+e)2(1−e2) v ′2 and
that this maximal value is attained when v ′1t = 0 and ~v ′2 · ~n =√
1−e22(3−e)(1+e)v
′2.
It follows that in the inelastic case there is virtually no upward speed flux forv ′1 > v1,max, i.e., when (1− e2) v′21 > O(
⟨v2⟩). As mentioned, the downward
flux is hardly affected by inelasticity. Therefore, the gain term must be sub-dominant to the loss term at values of the speed which exceed the order of
v ′1max. Hence, there must be a value of the speed, u∗ ≥ O(√
〈v2〉1−e2
), such
that for u > u∗ the gain term is subdominant to the loss term. In this range
of speeds the Boltzmann equation reduces to: ∂f(u,t)∂t ∼ −nσTu f(u, t) (uponkeeping the first term in the r.h.s. of Eq. (8)), or f(u, t) ∼ f(u, 0)e−nσTut.Therefore, for u > u∗ the HCS distributions ‘remembers’ the initial condi-tion. Notice that as the initial distribution, f(u, 0), is u dependent, the (largeu) part of the distribution, which retains memory of the initial condition, isnot necessarily exponential (although, as t→∞, the exponential dominates).
3.2 The speed ranges and universality
As is well known [13, 14], the distribution function for an unforced systemof elastically colliding spheres (or any other elastic system) converges veryrapidly (in a time corresponding to a few collisions per particle) from practi-cally any initial distribution to a near-Maxwellian distribution (except in thehigh speed tail [13], where convergence is much slower). The correspondingconvergence rate can be estimated, for example, from the value of the leadingnonzero eigenvalue of the linearized Boltzmann operator, which is O(1) inunits of the inverse mean free time. Next, consider a quasielastic system. Ifthe degree of inelasticity, ε ≡ 1 − e2, is sufficiently small, the outcome of a
The Homogeneous Cooling State revisited 43
small number of collisions per particle is very close to the outcome of a smallnumber of elastic collisions. Therefore a near elastic system is expected todevelop 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 ε(~v12 ·~n)2, one expectsthe near-Maxwellian distribution to extend to speeds of the order of 1/
√ε.
This implies that the statistical weight of the non-Maxwellian (tail) part ofthe distribution function is small. Similarly, one expects the memory of thedetails of the initial distribution to fade following a few collisions per particle,even when ε is not small, although it is not a-priori clear that in this case thebulk of the distribution function becomes near-Maxwellian. As a matter offact we have checked (by solving the pertinent Boltzmann equation numeri-cally) that this is the case down to quite low values of e (e.g., for e = 0.2). Thedevelopment of the distribution function in time is demonstrated in Fig. (1).As the form of the distribution function emerging after a few collisions perparticle is independent of the form of the initial distribution (except in thetail), it does not change in time any more, i.e., it becomes universal (but stilldependent on the value of e). The universal distribution, being normalized, ischaracterized by e and its (shrinking) width alone. Define the inverse width,ξ, as follows: ξ2(t) ≡ 3
2〈v2;t〉 . In the universal range, dξ/dt has dimensions of
inverse length; the only available parameter having this dimension is nσT (theinverse mean free path). It follows that dξ/dt = CnσT , where C is a constant.Mean field calculations (see remark in Appendix B) suggest that C ∝ ε, inconformity with systematic studies (cf. e.g., Sec. (4)). This is easy to under-
stand since ξ does not change in time when ε = 0. Defining λ by C ≡ ελπ− 12 ,
one obtains:ξ(t) = ελnσTπ
− 12 (t− t0) + ξ(t0) , (10)
where t0 is a time beyond which the distribution is largely (except at thefar tail) universal. With this definition, one can express the bulk of the dis-tribution function as a scaling function: f(u, t) = nξ3(t)f(uξ(t)), where theprefactor is determined by dimensional considerations, and where f is univer-sal (its dependence on e has been suppressed).
Let u be a speed in the range corresponding to the universal (and scaling)form of f , at time t. Clearly, after a sufficient time elapses, u will be in the tailof the distribution, since the entire distribution shrinks. Let t(u) be a time atwhich 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 ofSec. (3.1): f(u, t) = exp [−nσTu(t− t(u))] f(u, t(u)) for t > t(u), hence, using
Eq. (10): f(u, t) = exp[−√πu ξ(t)−ξ(t(u))ελ
]f(u, t(u)). Substituting now the
scaling form of f , multiplying both sides of the equation by u3 and rearrangingterms, one obtains:
44 I. Goldhirsch, S. H. Noskowicz, and O. Bar-Lev
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 = 3e = 0.5 ; c = 3e = 0.7 ; c = 3
0.1 0.2 0.4 0.6 0.8 1 2 3 4 5 6
100
101
102
103
ξ u
Loss−−−−−−−−
Gain
e = 0.2e = 0.5e = 0.7
Fig. 1. Left: Development of the distribution function as a function of the numberof accumulated collisions per particle, c. Shown are the negatives of the secondderivative of the logarithm of the rescaled distributions versus the rescaled speedξu for three values of the coefficient of restitution, e. Notice the crossover from theMaxwellian value, 2, to the value 0, corresponding to the exponential decay. Alsonotice the fast convergence of the bulk versus the slow convergence of the tail. Theinitial distribution is Maxwellian. Right: The ratio of the loss term to the gainterm extracted from a converged numerical solution of the Boltzmann equation forthree values of e, versus the rescaled speed. For uξ < 0.1 the ratio is smaller thanunity, 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/√ε
the ratio is proportional to (ξu)γ(ε), with γ ≈ 3, in conformity with the theoreticalcalculation (Appendix C-3).
u3ξ3(t)f(uξ(t)) exp
[√πuξ(t)
ελ
]= exp
[uξ(t(u))
√π
ελ
]u3ξ3(t(u))f(uξ(t(u))) .
(11)Eq. (11) implies that there must be a constant, A, such that uξ(t(u)) = A.Clearly, AÀ 1 since u is in the tail of the distribution at t = t(u). It followsthat, for t > t(u) the distribution, f(u), is given by:
f(u, t) = n
(A3f(A) exp
[A√π
λε
])1
u3exp
[−ξ(t)u
√π
λε
]. (12)
Fig. (1) depicts the convergence of a Maxwellian initial condition to a uni-versal distribution for several values of e (a similar result is obtained fornon-Maxwellian initial conditions). The crossover between the Maxwellian-like range and the exponential range is visible. As the values of ε used hereare O(1) (for technical reasons) one cannot read the scaling of the crossover‘points’, in terms of ε, off Fig. (1).
To reiterate, the exponential tail is, strictly speaking, an intermediate tail(developing out of the scaling part of the distribution at an earlier time), sincethe true tail retains the memory of the initial condition.
The Homogeneous Cooling State revisited 45
Assume for the moment that the near-Maxwellian range of speeds crossesover smoothly to the exponential range. Equating exp
(−ξ2u2
)with the ex-
ponential form, one obtains that the crossover must occur at uξ ∝ 1/ε. How-ever, as explained above (see also Sec. (4)), the Maxwellian range is lim-ited to uξ < 1/
√ε. Therefore there must be a crossover region, ‘starting’
at uξ ∝ 1/√ε, and ending where the exponential decay ‘starts’. The crude
matching of the Maxwellian and exponential ranges may be taken to sug-gest that the exponentially decaying range starts at uξ ∝ 1/ε, a conclusioncorroborated below.
4 The near-Maxwellian range of speeds
This section is devoted to a study of the near-Maxwellian range of speeds. Tothis end it is convenient to perform a perturbative expansion of the pertinentBoltzmann equation in powers of the degree of inelasticity, ε = 1 − e2. Onlythe universal distribution is sought.
Denote the (universal) scaled distribution function by f(u), where u ≡ξ(t)u, f(u, t) = nξ3(t)f(u), and ξ(t) is given by Eq. (10). Notice thatthe parameter λ, cf. Eq. (11), is left to be determined by the perturba-tion theory. With this scaling, the Maxwellian solution transforms to fM ≡π−3/2 exp
(−u2
). Upon scaling the impact parameter, b, by its maximal value,
2a (b ≡ b2a ), the (rescaled) Boltzmann equation reads:
3f(v1) + v1df(v1)
dv1=
1
ελ√π
∫d~v2b db dφ
∣∣∣~v12∣∣∣(1
e2f(v′1)f(v
′2)− f(v1)f(v2)
)
≡ 1
ελ√πB(f , f ; e) , (13)
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 ε; itcan be shown that this is the only choice of a small parameter which is a func-tion of ε, up to a multiplicative constant, that yields a consistent perturbationtheory. Notice that when ε = 0, φ = 0. The zeroth order (Maxwellian) distri-bution is conveniently chosen to possess the correct kinetic energy (averageof u2). Therefore φ must satisfy:
∫∞0v4 exp
(−v2
)φ(v)dv = 0 (so that it does
not affect the average kinetic energy), as well as∫∞0v2 exp
(−v2
)φ(v)dv = 0
(so as to preserve the normalization of f). Define now the following expan-sions: φ(u, e) =
∑∞k=1 ε
kφk(u), and λ(e) =∑∞k=0 ε
kλk. It is also convenient
to define the expansion of the nonlinear (Boltzmann) operator B in powersof ε: B(f, g, e) =
∑∞k=0 ε
kBk(f, g). The last definition does not include anexpansion of f or g in powers of ε; the only sources of ε dependence in theexpansion of B are the explicit 1/e2 term in the definition of this operatorand the ε dependence of ~v ′1, ~v ′2 on ~v1, ~v2. Therefore the full expansion of B
46 I. Goldhirsch, S. H. Noskowicz, and O. Bar-Lev
in powers of ε, when fM (1 +∑k ε
kφk) is substituted for f , requires a furtherexpansion. This, slightly cumbersome definition, is calculationally convenient,as seen below. The tilde superscripts are removed below for sake of notationalsimplicity. The zeroth order in ε, now including the expansion of f in powersof ε, of the Boltzmann equation, reads:
1
λ0√π[B0(fMφ1, fM ) +B0(fM , fMφ1)] = 3fM + u
dfMdu− 1
λ0√πB1(fM , fM ).
(14)It is convenient to multiply both sides of the above equation by uλ0/4fM ,thus obtaining:
L ∗ φ1(u) ≡u
4fM√π[B0(fMφ1, fM ) +B0(fM , fMφ1)]
=uλ04fM
(3fM + u
dfMdu− 1
λ0√πB1(fM , fM )
). (15)
The left hand side of Eq. (15) defines the action of the linearized Boltzmann
operator, L, and the right hand side is a source term, s1, whose functionaldependence on u is easy to calculate, cf. Eq. (55), as fM is given and B1 iseasily 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 straightforwardlycarried out, yielding:
L ∗ φ =√π
∫ u
0
dvvφ(v) erf(v) +√πeu
2
erf(u)
∫ ∞
u
dvvφ(v)e−v2
−(u
4e−u
2
+
√π
4
(1
2+ u2
)erf(u)
)φ(u)−
∫ u
0
dv
(v2u2 +
v4
3
)e−v
2
φ(v)
−∫ ∞
u
dv
(uv3 +
vu3
3
)e−v
2
φ(v) . (16)
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) ≡∫∞0vfM (v)a(v)b(v)dv. Also,
L ∗ vk = 0 for k = 0, 2 [14]: the functions 1 and v2 are the only normalizable
zero eigenfunctions of L; they clearly correspond to the conservation of particlenumber and energy (in the elastic limit). Hence:
(vk, L ∗ φn
)=(vk, sn
)=
∫ ∞
0
dvvk+1fM (v)sn(v) = 0 k = 0, 2. (17)
These are the two solubility conditions at order n. In addition, the two nor-malization conditions that φn must satisfy, read:
The Homogeneous Cooling State revisited 47
(vk, φn) =
∫ ∞
0
dvvk+1fM (v)φn(v) = 0 k = 1, 3. (18)
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 inhomogeneoussolution.
All in all φn must satisfy two conditions, Eq. (18), and these determinethe parameters an and bn. The source term, sn, which depends on one freeparameter, λn−1 (e.g., s1 depends on λ0), must also satisfy two conditions,Eq. (17). This does not lead to an overdetermination of λn−1, because, asshown in Appendix C-2, the condition corresponding to k = 0 in Eq. (17)is identically satisfied by sn. In the corresponding elastic case the conditionk = 2 is also identically satisfied by the source term, in conformity with thefact that there is no additional free parameter in this case.The condition k = 2, applied to sn, reads:
(v2, s1
)= π−3/2
∫ ∞
0
dvv3e−v2
[λ04
(3v − 2v3
)− I2(v)
]= 0, (19)
where I2 is given by Eq. (54). This integral can be carried out analytically
yielding: λ0 =√23 , in conformity with the mean field result, Eq. ( 50). The
correction φ1 can be straightforwardly found from the equation L ∗ φ1 = s1in terms of a polynomial expansion: φ1(u) = a1 + b1u
2 + A4u4 + A6u
6 + ...,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 perturbativesolution agrees well with the numerical solution of the Boltzmann equation upto u = O (1/
√ε) and slightly beyond (for values of ε which are O(1)). Notice
the rather rapid decrease of these coefficients. A straightforward asymptoticanalysis of the equation L ∗ φ1 = s1 (using Eq. (16) for uÀ 1), reveals that:φ1(u) ∼ −u2ln(u)−2
√2u/√π+3.9538u2, i.e., the ‘asymptotic’ solution of the
Boltzmann equation for large u is, to linear order in ε: π−3/2 exp(−u2
)(1 −
εu2ln u); this limit makes sense only for ε ¿ 1 and u < O (1/√ε). This,
and the above results imply that the perturbation expansion is valid for u <1/√ε, which is also the range of the near-Maxwellian distribution. The reduced
equation presented in the next section shares this property.
5 Reduction of the Boltzmann equation for the HCS
The goal of the present section is to present a derivation of a reduced equationfor the HCS, on the basis of the Boltzmann equation. The reduction is achievedby an asymptotic analysis of the Boltzmann equation at high speeds, u À 1(in the rescaled variables). The error allowed in the non-linear Boltzmann op-erator is algebraically subdominant to the loss term at large speeds. It turns
48 I. Goldhirsch, S. H. Noskowicz, and O. Bar-Lev
out that the error of the resulting equation at low speeds is formally O(ε). Thesolutions of the resulting equation, much like those of other equations derivedby asymptotic methods, correctly reproduce the HCS distribution function,far beyond the nominal limits of its derivation. As a matter of fact, the solu-tions of the reduced equation are practically indistinguishable from those ofthe full Boltzmann equation for all speeds, even at values of the coefficientof restitution which are as low as 0.4. A formal perturbative analysis of thereduced equation in powers of ε yields a leading order correction which isprecisely that obtained in the previous section for 1 À u À 1/
√ε. Possible
explanations for these features are proposed. Needless to say, the three uni-versal speed ranges are fully reproduced by the reduced equation (analyticallyand numerically). This section ends with a second reduction of the Boltzmannequation, which is of the form of a transcendental equation. The latter yieldssolutions which are in excellent agreement with the numerical solutions (and,of course, with the first reduced equation). Thus this section provides a nearlyanalytic and uniform solution for the HCS. Finally, we wish to stress that themethods presented below are non-perturbative.
0 1 2 3 4 50
0.02
0.04
0.06
0.08
0.1
0.12
0.14
0.16
0.18
ξ u
f
NumericalPerturbation
0 1 2 3 4 50
5
10
15
20
25
30
ξ u
−lo
g[ f(
ξ u
) ]
Simplified EquationFull Equation
Fig. 2. Left: A comparison of the first order perturbation theory result for the HCSversus the numerical solution of the Boltzmann equation for e = 0.8. The (rescaled)distributions are shown versus the rescaled speed, ξu. Right: A comparison of thenumerical solution of the Boltzmann equation for e = 0.8 with the numerical solutionof the reduced equation (29). Shown are the logarithms of the distribution functions.Notice the close correspondence, even for small speeds and not-so-small ε.
5.1 Derivation of the reduced equation
The goal of the present subsection is to derive a simplification of the Boltz-mann equation, which is formally valid at large values of the speed.
As in the above, it is convenient to employ a non-dimensional representa-tion with u = ξu, f(u, t) = nξ3f(u) and all lengths rescaled by the particle
The Homogeneous Cooling State revisited 49
radius. The integrals, Gi, are rescaled as follows: Gi ≡ Gi/(4πσTn
2ξ2). It
follows that the gain term is given by:
(∂f
∂t
)
g
=
9∑
i=1
Gi = 4πσTn2ξ2 ·
9∑
i=1
Gi,
and the loss term is:
(∂f
∂t
)
l
= 4πσTn2ξ2 · f (u)
[ ∫ u
0
(uv2 +
v4
3u
)f (v) dv
+
∫ ∞
u
(v3 +
u2v
3
)f (v) dv
]≡ 4πσTn
2ξ2(∂f
∂t
)
l
.
so that
nξξ2
(3f + u
df
du
)= 4πσTn
2ξ2 ·[
9∑
i=1
Gi −(∂f
∂t
)
l
].
Recalling that ξ = ελnσTπ−1/2 (following Eq. (19): λ =
√2/3 + O(ε)), one
obtains
ελπ−
32
4
[3f + u
df
du
]=
[9∑
i=1
Gi −(∂f
∂t
)
l
], (20)
In the sequel all entities are assumed dimensionless and the tilde signs aredropped for notational simplicity.
As shown in Appendix A-2, only G2 is non-negligible with respect to theloss term in the large-u limit. It is convenient to define (the dimensionless)G∗2 as follows:
G∗2(ε, u) ≡8
(1 + e)32 (3− e)u
∫ u
0
dv1v1f(v1)
×∫ ∞
√(u2−v21)(3−e)/(1+e)
dv2
√(3− e)(v21 − u2) + (1 + e)v22 v2 f(v2), (21)
where the dependence of G∗2 on ε has been spelled out. The difference betweenG∗2 and G2 is in the ranges of integration. Denote this difference by: G2−G∗2 =−δ1G2 − δ2G2, where:
δ1G2(ε, u) ≡8
(1 + e)32 (3− e)u
∫ u(1−e)/2
0
dv1v1f(v1)
×∫ [2u−(1−e)v1]/(1+e)
√(u2−v21)(3−e)/(1+e)
dv2
√(3− e)(v21 − u2) + (1 + e)v22 v2 f(v2), (22)
and
50 I. Goldhirsch, S. H. Noskowicz, and O. Bar-Lev
δ2G2(ε, u) ≡8
(1 + e)32 (3− e)u
∫ u
0
dv1v1f(v1)
×∫ ∞
[2u−(1−e)v1]/(1+e)dv2
√(3− e)(v21 − u2) + (1 + e)v22 v2 f(v2). (23)
It is shown in Appendix A-2-2 that δG1 and δG2 are algebraically subdomi-nant to the loss term for uÀ 1. It follows that the gain term can be substitutedfor by G∗2, the error being subdominant to the loss term. Consider the latternext. The large-u form of the loss term is: uf(u)
∫∞0dv v2f(v) = 1
4πuf(u),where the normalization of the dimensionless form of the distribution functionhas been used. Therefore, for large u, the Boltzmann equation for the HCSreduces to:
ελ
4π32
[3f + u
df
du
]= G∗2 −
1
4πuf(u), (24)
Interestingly, Eq. (24) becomes an identity for ε = 0 andf(u) = π−3/2 exp
(−u2
), as can be checked by direct substitution. Thus,
Eq. (24) can be regarded to be a uniform approximation of the Boltzmannequation, whose error is algebraically small (with respect to the loss term)for u À 1, yf
The Homogeneous Cooling State revisited 51
Further straightforward considerations (taking into account that the distribu-tion crosses over from a near-Maxwellian to an exponential for u > O( 1ε ) andthat h(µz) < h(z), since µ > 1), reveal (in agreement with numerical tests)that the integration over x in Eq. (26) is dominated by the ‘small’ x range,
where the distribution is near-Maxwellian, hence h is near exponential andψ′(µx) = 1 in this range. One can therefore replace ψ′ in Eq. (26) by unity,obtaining:
G∗2 (ε, u) '√π
(3− e) (1 + e)√z
∫ z
0
h (z − x) h (µx) dx (27)
Next, define: h (z) ≡ (3−e)(1+e)4 π(−3/2)h (z). Substituting this definition into
Eq. (27) and using Eq. (24), one obtains:
ε2λ√πz
32h
′
(z) +
(z +
3ελ√π
√z
)h (z) =
∫ z
0
h (z − x)h (µx) dx (28)
As a final step, we neglect ε√z with respect to z since this contribution is only
important when z < ε2, i.e., for values of z at which the asymptotic evaluationis not valid. Also the equation is valid only to within O(ε). Therefore the finalform of the reduced equation is:
ε2λ√πz
32h
′
(z) + zh (z) =
∫ z
0
h (z − x)h (µx) dx (29)
One can easily check that the reduced equation admits a Maxwellian solution(i.e., h(z) = e−z) when ε = 0, and that the leading term (in z = u2) of itsperturbative expansion to first power in ε reads: h(z) = e−z(1 − 1
2εzlnz), inagreement with the large speed limit (u À 1 and u
√ε < 1, for ε ¿ 1) of
the perturbative solution of the full equation. Thus, although the error of theequation is O(ε) at low speeds, it is correct to order εu2. As the numericalresults (and some further analysis not presented here) show, cf. Fig. (2,right),the equation faithfully reproduces the distribution function for the HCS.
5.2 Second reduction of the Boltzmann equation
Although Eq. (29) represents an enormous simplification of the full Boltz-mann equation it still requires a certain numerical effort to solve it. A furtherreduction is therefore useful.Let φ(z) ≡ − lnh. With this definition, Eq. (29) transforms to:
z − ε 2λ√πz
32φ
′
(z) =
∫ z
0
eφ(z)−φ(µx)−φ(z−x)dx (30)
The arguments presented in the previous subsection, as well as numericalstudies (cf. Fig. (3)), show that the integral in Eq. (30), like the integral in
52 I. Goldhirsch, S. H. Noskowicz, and O. Bar-Lev
0 1 2 3 4 5
−0.8
−0.6
−0.4
−0.2
0
x
φ(z)
− φ
(µ x
) −
φ(z
−x)
0 10 20 30 40 50
−8
−6
−4
−2
0
x
φ(z)
− φ
(µ x
) −
φ(z
−x)
Fig. 3. Plot of the argument of the exponent in Eq. (30) versus x. The value ofε 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. (29).
Eq. (26), is dominated by the small values of x when z > O(1/ε), hence oncecan affect the following (approximate) replacement for z > O(1/ε):
where the second approximate equality follows from the fact that the distribu-tion 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/ε). In this range the (approximate) linearityof φ in its argument, renders Eq. (31) valid as well. Finally, it is straight-forward to deduce from Eq. (29) that h(0) is close to unity (to within O(ε),hence one can substitute φ(0) = 0. With these approximations, the integralon the right hand side of Eq. (30) (i.e., the integral of an exponential) can beimmediately performed and Eq. (30) reduces to:
z − ε 2λ√πz
32φ
′
(z) =1− e−q(z)q (z)
z, (32)
where q(z) ≡ z (µ − φ′(z)). Substituting φ′(z) = µ − q(z)/z, which followsfrom the definition of q, into Eq. (32), one obtains the following transcendentalequation for q(z):
2ελ√π
(µ√z − q(z)√
z
)− 1 +
1− e−q(z)q(z)
= 0 (33)
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. (33) shows thatfor z < O(1/ε), q(z) is approximately given by:
The Homogeneous Cooling State revisited 53
q(z) ≈4ελµ√πz
√z + 4ελ√
π
(34)
It follows that in this range φ(z) = µz + O(√z), i.e. the distribution isnear-Maxwellian. When z > O(1/ε2), the solution of Eq. (33) is given by:q(z) ≈ µz − √πz/ (2ελ), hence φ(z) ≈ √πz/ (ελ), which corresponds (as ex-pected) to an exponential distribution. A full numerical evaluation of q(z)from Eq. (33) and the calculation of φ(z) on the basis of this result yield adistribution function (i.e., exp [−φ(z)]), which is essentially indistinguishablefrom the numerical solution of the Boltzmann equation. Therefore there isno point in presenting a graph showing this correspondence. It is somewhatsurprising that the second reduction still yields such good agreement with thenumerical solution.
6 Concluding remarks
A study of the properties of the HCS has been presented. Among the surpris-ing features is the fact that the reduced descriptions so faithfully agree withthe numerical solution (even in terms of perturbative corrections), in spiteof the fact that the asymptotic analyses leading to the reductions are basedon the large speed properties of the Boltzmann equation. It is possible thatthis good correspondence is due to the fact that the bulk of the distributionfunction is not significantly removed from a Maxwellian (except at values ofe which are near zero) and that the asymptotics sets in (as numerical resultsshow) at values of u which are basically of the thermal speed (for e not closeto 1); when e is close to unity the approximate equation is a faithful approx-imation of the Boltzmann equation anyway. Another interesting feature isthe nontrivial structure of the transition region, between the near-Maxwellianand the exponential ranges. This region is prominent for near-elastic valuesof the coefficient of restitution but it does exist (for O(1/√ε) < u < O(1/ε)at every value of e. It is perhaps important to mention that the dominanceof 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 uin the transition region, cf. Appendix C-3 and Fig. (1)). We believe that ourdetailed studies of the properties of the non-linear Boltzmann operator for theHCS should be useful to other, more complex granular flows, both in terms ofresults (e.g. concerning the linearized Boltzmann operator) and techniques. Itshould be noted that granular hydrodynamics depends on low order momentsof the distribution function, and, as is well known, the latter are provided,to a good degree of approximation, by perturbation theory (except for coef-ficients of restitution which are smaller than about 0.7). Therefore the abovestudy is not essential for the derivation of hydrodynamic descriptions (thisdoes not necessarily imply that a thorough study of the distribution function
54 I. Goldhirsch, S. H. Noskowicz, and O. Bar-Lev
pertaining to an important system, such as the HCS, is not warranted; distri-bution functions are actually measured experimentally these days and theirnon-Gaussian nature is of great interest). It may however be the starting pointfor the study of very inelastic systems or systems with large gradients, such asencountered in shocks, where the Chapman-Enskog theory is no longer valid.Whether this is indeed the case remains to be seen.
Acknowledgment: This work has been partially supported by the the US- Israel Binational Science Foundation (BSF), the Israel Science Foundation(ISF) and INTAS (the EU).
Appendix A: The gain term for the HCS
A-1: Reduction of the gain term
Since the sought distribution function is isotropic, one has:
(∂f(u, t)
∂t
)
g
=1
4π
∫dΩ~u
(∂f(~u, t)
∂t
)
g
(35)
Using the identity 14π
∫dΩ~u δ(~v1 − ~u) = 1
2πu δ(v21 − u2), one obtains fromEq. (35):
∫d~v1
∫dΩ~u4π
δ(~v1 − ~u)(∂f(v1, t)
∂t
)
g
=1
2πu
∫d~v1 δ(v
21 − u2)
(∂f(v1, t)
∂t
)
g
. (36)
Substituting this result into Eq. (5) and transforming the integration to in-coming velocities, one obtains:
(∂f(~u, t)
∂t
)
g
=1
2πu
∫v ′12f(v
′1)f(v
′2)δ(v
21 − u2)d~v ′1d~v ′2b ′db ′dφ ′ , (37)
where v1 is the outgoing velocity of particle ‘1’ and it represents a functionof the incoming velocities, ~v ′1 and ~v ′2. The impact parameter, b′, and the az-imuthal angle, φ′, relate to the incoming (precollisional) variables. In Eq. (37)the only φ′ and b′ dependent part of the integrand is the δ-function. The in-tegration over the angles in the above expression is tedious. The result is that(∂f(~u,t)∂t
)g=∑9i=1Gi(u), where:
G1 =2(3− e)1 + e
∫ ∞
u
dv1
∫ ∞
(2u+(1−e)v1)/(1+e)H(v1, v2)dv2 (38)
The Homogeneous Cooling State revisited 55
G2 =2√
1 + eu
∫ u
u(1−e)/2dv1
∫ (2u−(1−e)v1)/(1+e)
√(u2−v21)(3−e)/(1+e)
√w2 − (3− e)u2 ×
×H(v1, v2)dv2 (39)
G3 =2(3− e)u(1 + e)
∫ u
0
dv1
∫ ∞
(2u+(1−e)v1)/(1+e)v1H(v1, v2)dv2 (40)
G4 =2
u
∫ 2u/(1−e)
u
dv1
∫ (2u−(1−e)v1)/(1+e)
0
v2H(v1, v2)dv2 (41)
G5 =1
u√1 + e
∫ 2u/(1−e)
0
dv1
∫ (2u+(1−e)v1)/(1+e)
(2u−(1−e)v1)/(1+e)
[√1 + e v2
+√w2 − (3− e)u2
]H(v1, v2)dv2 (42)
G6 =1
u√1 + e
∫ ∞
2u/(1−e)dv1
∫ ((1−e)v1+2u)(1+e)
((1−e)v1−2u)/(1+e)
[√1 + e v2
+√w2 − (3− e)u2
]H(v1, v2)dv2 (43)
G7 =3− e
u(1 + e)
∫ 2u/(1−e)
u
dv1
∫ (2u+(1−e)v1)/(1+e)
(2u−(1−e)v1)/(1+e)(u− v1)H(v1, v2)dv2 (44)
G8 =3− e
u(1 + e)
∫ ∞
2u/(1−e)dv1
∫ ((1−e)v1+2u)/(1+e)
((1−e)v1−2u)/(1+e)(u− v1)H(v1, v2)dv2 (45)
G9 =3− e
u(1 + e)
∫ u
0
dv1
∫ (2u+(1−e)v1)/(1+e)
(2u−(1−e)v1)/(1+e)(v1 − u)H(v1, v2)dv2 (46)
where w2 ≡ (3− e)v21 + (1 + e)v22 and H(v1, v2) ≡ 16πσT(3−e)(1+e)v1v2f(v1)f(v2).
A-2: The gain term for large speeds
Below we present an analysis of the gain term for values of the speed thatare far larger than the typical (thermal) speed. In this limit it turns outthat Gi ; i 6= 2 are algebraically small with respect to the loss term. Morespecifically: [Gi(u)] / [nσTuf(u)] < C/u, for i 6= 2 and u À 1, irrespective ofthe value of e. Here C is an O(1) constant. Numerical results actually showthat the ratio of the gain to the loss term for large u is smaller than the abovebound; however this bound suffices for the purposes of our analysis.
A-2-1: Analysis of the Gi ; i 6= 2It is convenient to study the dimensionless Gi and compare to the dimension-less loss term. For notational simplicity we shall omit below the tilde signsand assume that all velocities are scaled by ξ, Gi are scaled by 4πσTn
2ξ2,
56 I. Goldhirsch, S. H. Noskowicz, and O. Bar-Lev
and that the same scaling is applied to the loss term (see Section 5.1). Inaddition, f(u) denotes below the non-dimensionalized universal distributionas a function of the non-dimensionalized speed.
The bounds below are based on two assumed properties. The first is thatasymptotically any small power of u times f(u) can be bounded by a constant.This will be referred to as ‘property 1’. The second property is developed next.Define In ≡
∫∞uvnf(v)dv. Let ψ(u) ≡ − ln f(u). Since f(u) is a decreasing
function of u, ψ′(u) is positive (the prime superscript denotes a derivativewith respect to u). For small u: ψ′(u) ∝ u and ψ′′(u) ∝ 1, whereas for largeu: ψ′(u) ∝ 1 and ψ′′(u) → 0. We therefore assume that ψ′ is bounded frombelow for u > uc > 0, where we conveniently choose uc to be larger than unityso that 1/ψ′ is bounded by an O(1) value, denoted by C0. From the propertiesof ψ′′ we conjecture that ψ′′ > 0 for all u. Using straightforward integrationby parts, applied to the definition of In, one obtains:
In =un
ψ′ (u)f (u) +
∫ ∞
u
nvn−1
ψ′ (v)f(v)dv −
∫ ∞
u
vnψ′′(v)
ψ′2(v)f(v)dv.
It follows that In ≤ C0unf(u)+C0nIn−1. Consequently, for small values of nand u > uc there is an O(1) constant, C, such that: In < Cunf(u). We referto the latter inequality as ‘property 2’. Below we use the symbol C to denoteall O(1) bounding prefactors, irrespective of their values.
Consider first G1. It is easy to check that G1 < CI21 , hence using property2: G1 < Cu2f2(u). For u À 1 it follows from property ‘1’ that G1 < Cf(u),hence it is subdominant to the loss term (the latter being Cuf(u) for largeu).
Consider next G3. Using its definition and the definition of In one obtains:G3 <
Cu
∫ u0dv1v
21f(v1)I1 ([2u+ (1− e)v1] /(1 + e)). Since I1 is a decreasing
function of its argument, I1 in the latter integral can be bounded by I1(u).Next, since the integral
∫ u0vnf(v)dv is bounded by a constant for any positive
value of n, it follows that: G3 <Cu I1(u). Now, using property 2, it follows
that: G3 < Cf(u). A similar analysis yields G4 < Cf(u), hence G3 and G4
are subdominant to the loss term for uÀ 1.It is convenient to analyze G5 and G9 together. Define Ga5 as the part of
G5 in which the integration over v1 ranges from 0 to u and Gb5 as the partof G5 in which the integration over v1 ranges from u to 2u/(1− e). Clearly:G5 = Ga5+G
b5. Consider next the sum Ga5+G9. It follows from their definitions
that:
Ga5 +G9 =C
u
∫ u
0
dv1v1f(v1)
∫ [2u+(1−e)v1]/(1+e)
[2u−(1−e)v1]/(1+e)dv2v2f (v2)
×[3− e1 + e
(v1 − u) + v2 +
√w2
1 + e− 3− e
1 + eu2
].
The term in the square brackets increases with v2, hence it can be boundedfrom above by substituting the maximal value of v2 in the range of integration.
The Homogeneous Cooling State revisited 57
Since f(v2) is a decreasing function of v2 one can bound it by substituting theminimal value of v2 in the range of integration for its argument. In addition,one can substitute for the explicit factor v2, its maximal value in the range ofintegration. It follows that:
Ga5 +G9 ≤ (1− e)Cu
∫ u
0
dv1v31f (v1) [2u+ (1− e) v1] f
(2u− (1− e) v1
1 + e
)
By replacing v1 in the argument of f ([2u− (1− e)v1] /(1 + e)) by u one fur-ther increases the result, hence Ga5+G9 ≤ C(1− e)f(u) and this sum is againsubdominant to the loss term.
Next consider Gb5. One can subdivide the range of integration over v1to [u,Λu], where Λ is a constant satisfying : 1 < Λ < 2/(1− e), and[Λu, 2u/(1− e)]. The contribution of the second range is proportional (us-
ing property ‘2’) to (1 − e)u3f(2−(1−e)Λ
1+e u)f(Λu); using property ‘1’ one
concludes that this sum is asymptotically smaller than and clearly sub-dominant to (1 − e)Cf(Λu), hence it is smaller than (1 − e)Cf(u). In thefirst range on can now use property ‘2’, obtaining by straightforward anal-
ysis that Gb5 ≤ (1 − e)Λu3f(2−(1−e)Λ
1+e u)f(u). Next, using property ‘1’,
this contribution is bounded by C(1 − e)f(u). Thus, for u À 1 it fol-lows G5 + G9 < C(1 − e)f(u). An analysis similar to that for Gb5 yields:G7 ≤ (1− e)Cf(u). Since in the expressions for G6 and G8, the lower boundfor v1 is 2u/(1− e), the corresponding integrals are clearly subdominant tothe loss term. A straightforward analysis shows that they too can be boundedfrom above by (1 − e)Cf(u). Note that we have not absorbed the constant(1 − e) in C for Gi ; 5 ≤ i ≤ 9, as we wish to emphasize that these contri-butions to the gain term vanish in the elastic limit.
A-2-2: Large speed bounds on δ1G2 (ε, u) and δ2G2 (ε, u)
The goal here is to prove that for u À 1, δ1G2 (ε, u) and δ2G2 (ε, u), cf.Eqs. (22,23), are algebraically (in u) subdominant to the loss term (which is∝ u f(u)) for large u. As before, C represents a generic prefactor denotingO(1) numbers. Below, all functions and speeds are dimensionless.
Consider first δ1G2, Eq. (22). The lowest value of v1 in the range of in-tegration is zero and the lowest value of v2 is 3−e
2 u =(1 + 1−e
2
)u. There-
fore, replacing f(v1)f(v2) in the integrand by f(0)f((1 + 1−e
2 )u)increases
the value of the integral. With this substitution, the integral over v1 andv2 can be straightforwardly performed, yielding the following upper bound:
δ1G2 <Cu
(((1− e)u)5
)f((1 + 1−e
2 )u). When (1 − e)u is of order unity or
less, the bound is clearly subdominant to the loss term. When (1 − e)u islarger than order unity, the value of f
((1 + 1−e
2 )u)is (at least) exponentially
subdominant to f(u) and again the bound on δ1G2 is subdominant to the lossterm.
58 I. Goldhirsch, S. H. Noskowicz, and O. Bar-Lev
Consider next δ2G2, cf. Eq. (23). Using (1 + e)/(3− e) ≤ 1 and replacingthe v1-dependent lower limit of the integration over v2 by its smallest value,u, one obtains the following bound:
δ2G2 ≤C
u
∫ u
0
dv1 v1 f(v1)
∫ ∞
u
dv2
√v21 + v22 − u2 v2 f(v2). (47)
Let η(u) be defined by f(η(u)) = 1/u2. Clearly, η(u) is proportional to a powerof log(u) for large u. Divide now the range of integration over v1 in Eq. (47)into a ‘range A’, in which 0 < v1 < η(u), and let ‘range B’ be defined by:η(u) < v1 < u. Denote the corresponding values of the integrals by δ2G
A2 and
δ2GB2 , respectively. Now, in range B, replace f(v1) by its largest value (recall
that f is a decreasing function of its argument), i.e. by f(η(u)) = 1/u2, and re-place the argument in the square root by 2v22 , thereby increasing it. Next, per-form the integration over v1. One obtains: δ2G
B2 ≤ C
u f(η(u))∫∞udv2v
22f(v2).
Using property 2 one obtains: δ2GB2 ≤ C
u f(η(u))u2f(u), and using the defi-
nition of f(η(u)): δ2GB2 ≤ C
u f(u), hence this contribution is subdominant tothe loss term. Next, consider δ2G
A2 . In this range it is convenient to majorize
f(v1) by f(0) and perform the integration over v1. One obtains:
δ2GA2 ≤
C
u
∫ ∞
u
dv2 v2 f(v2)[(η2(u) + v22 − u2
) 32 −
(v22 − u2
) 32
]. (48)
Next divide the range of integration in Eq. (48) into u < v2 < u1 (rangeA1) and u1 < v2 < ∞ (range A2), where u1 satisfies u21 − u2 = αη2(u),with α = 10, for example. In the range A1 the argument in the squareroot in Eq. (48) is majorized by an O(1) number times η2(u). It followsthat δ2G
remains Maxwellian. In this case, a substitution of the Maxwellian distributionin the r.h.s of the Boltzmann equation, followed by a multiplication of theresult by v21 and integration over v1, yields:
·ξ = ε
√2
3nσTπ
− 12 , (50)
i.e., λ =√2/3 in the mean field approximation.
Appendix C: Perturbation theory
C-1: Calculation of 1/(ελπ1/2
)B(fM , fM ; e)(v1)
The following relation results from the kinematics of the collisions, Eqs. (1)and (2), and the definition of b (the outgoing rescaled impact parameter):v ′21 + v ′22 − v21 − v22 = (v21)
2(1 − b2)(1− e2
)/(2e2). Substituting this result
in Eq. (13) and performing the integrations involved in the calculation of theright hand side of Eq. (13), one obtains:
1
ελπ12
B(fM , fM ; e)(v1) =4fM (v1)
v1ε2λI(v1), (51)
where the function I is defined as:
I(u, ε) ≡∫ ∞
0
dvv exp(−v2
) [(u+ v)− |u− v| − ε
6(u+ v)3 − |u− v|3
− 1
2
∫ (u+v)2
(u−v)2
dy√yexp
(− ε
2(1− ε)y)]
, (52)
Eq. (52) can be further simplified to:
I(u) =
∫ u
0
e−t2
dt
(1− εu2
2− ε
4
)− εu
4e−u
2
−exp
(−u2 εK
1+εK
)
1 + εK
∫ u
0
exp
(− t2
1 + εK
)dt, (53)
where K ≡ (2(1− ε))−1. For u < O(1/√ε), one is justified in expanding I(u)in powers of ε: I(u) = I0(u)+ εI1(u)+ ε
2I2(u)+O(ε3). One obtains, to secondorder in ε: I0 = I1 = 0 and
I2(u) =
√π
2
(5
32+u2
8− u4
8
)erf(u) +
(3u
32− u3
16
)e−u
2
. (54)
Since B(fM , fM ; 1) = 0, it follows that:
60 I. Goldhirsch, S. H. Noskowicz, and O. Bar-Lev
limε→0
1
ελπ1/2B(fM , fM , e) =
1
λ0π12
B1(fM , fM ),
hence 1/(λ0π
1/2)B1(fM , fM ) = I2(u). It follows that the source term, s1,
i.e., the right hand side of Eq. (15), is given by:
s1 =λ04(3u− 2u3)− I2(u), (55)
with I2(u) given by Eq. (54). Notice that, unlike in this Appendix, the tildesdenoting dimensionless variables and functions are omitted in Eq. (15).
C-2: Proof of the first solubility condition
Recall that the general form of the n-th order in perturbation theory is L ∗φn = sn, where sn is a ‘source’ term, which depends on previous orders andon λn−1. Following Eq. (17), sn is required to satisfy both (1, sn) = 0 and(u2, sn) = 0, where we recall that the scalar product, (f, g), is defined by(f, g) ≡
∫∞0vfM (v)f(v)g(v)dv, with fM (v) = π−3/2 exp
(−v2
). It is shown
below that the condition (1, sn) = 0 is identically satisfied to all orders inperturbation theory (in powers of ε). All entities in this Appendix are assumedto be dimensionless (and the tilde signs are omitted).
It follows from Eq. (13) that: B(f, f ; e) = ελ√π(3f + u dfdu
), where u is
the speed variable. Formally expanding B(f, f ; e) in powers of ε (as explainedin Sec. (4), i.e., the function f is not expanded), one obtains:
B0(f, f) = ελ√π
(3f + u
df
du
)−
∞∑
n=1
εnBn(f, f). (56)
Let f = fM (1+φ) be the sought solution of the Boltzmann equation. Clearly,since B (cf. Eq. (13)) is a bilinear operator, one has:
B(f, f ; e) = B(fM , fM ; e)+B(fMφ, fM ; e)+B(fM , fMφ; e)+B(fMφ, fMφ; e).(57)
It therefore follows from Eq. (56) that:
B0(fMφ, fM ) +B0(fM , fMφ) = −B0(fM , fM )−B0(fMφ, fMφ)
+ ελ√π
(3f + u
df
du
)−
∞∑
n=1
εnBn(f, f). (58)
The left hand side of Eq. (58), multiplied by u/ (4fM√π) is the definition of
L ∗ φ, cf. Eq. (15). Therefore:
L ∗ φ =u
4fM√π
(−B0(fM , fM )−B0(fMφ, fMφ) + ελ
√π
(3f + u
df
du
)
−∞∑
n=1
εnBn(f, f)
). (59)
References 61
Eq. (59) can clearly be written as L∗φ = s, where s =∑∞n=1 ε
nsn. It thereforesuffices to show that (1, s) = 0. Noting that B0(fM , fM ) = 0, and using thedefinition of the scalar product in Sec. (4), one obtains:
(1, s) =
∫ ∞
0
duu2
4√π
(−B0(fMφ, fMφ)+ελ
√π
(3f + u
df
du
)−∞∑
n=1
εnBn(f, f)
).
(60)
First, notice that∫∞0du(3u2f(u) + u3 df(u)du
)= 0, as a simple integration by
parts reveals. Next, since u2du = 1/(4π)d~u (all functions depend on the speed,u, here) and since
∫d~uB0(f, f) = 0 for any function, f , it also follows that∫
duu2B0(fMφ, fMφ) = 0. Finally, notice that∫d~uB(g, g; e) = 0 for any func-
tion g. Expanding the last relation in powers of ε one obtains∫d~uBn(g, g) = 0.
It thus follows that∫duu2Bn(f, f) = 0. This completes the proof.
C-3: The asymptotic ratio of the gain to the loss term
Consider G∗ as given by Eq. (25). Since, as explained in the main text, this
integral is dominated by the small x domain, one can replace h(t + µx) by
a Maxwellian transformed to the new variables, h(t + µx) ≈ π−3/2e−(t+µx).
Next, replace h(z − x) in this integral by its asymptotic form (an exponen-
tial distribution in u): h(z − x) ≈ Cz−η exp(−α√z − x/ε
), where η, α and
C are O(1) constants (they are actually known, e.g., η = 3, but this is notimportant for the present derivation). With this substitution, a straightfor-ward asymptotic analysis shows that for z > O
(1/ε2
)(i.e. for u > O (1/ε)),
G∗2(u) ∝ u−2f(u), hence the ratio of the gain to the loss term is propor-tional to u−3, in conformity with the numerical findings. When 1¿ u¿ 1/ε,the gain term can be shown to be ∝ Cf , and its ratio to the loss term isproportional to 1/u, again in conformity with the numerical results.
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