STEADY STATES OF A BOLTZMANN EQUATION FOR DRIVEN

GRANULAR MEDIA

J. A. CARRILLO∗, C. CERCIGNANI†, I. M. GAMBA‡

Abstract. We study a driven granular media in which particles perform inelastic col-lisions among themselves while they follow brownian dynamics in between collisions.We consider a Boltzmann-type kinetic equation as a model and we analyze the homo-geneous regime showing that existence of homogeneous steady states is plausible. Weapproximate the steady states by a small inelasticity expansion showing they are givenby a maxwellian distribution corrected by a Sonine polynomial of fourth degree up tothird order in the expansion parameter.

1. Introduction

In this work, we perform an analysis from a statistical mechanics perspective of thedynamics of perfect spheres of diameter σ > 0 colliding inelasticaly in a thermal bathof infinite temperature. Because of inelastic collisions we are constantly loosing energy,then we have to include some sort of energy input mechanism to achieve a steady state.Here, we consider that the motion of the particles between collisions is Brownian. Theequation of motion can be written down as the Langevin equation

x′′ = Γ(t)

where Γ(t) is a white noise stochastic force with independent, identically Gaussian dis-tributed processes of variance F , that is, < Γ(t),Γ(t′) >= 2Fδ(t− t′).

Regarding the collision mechanism, if (x, v) and (x − σn,w) are the states of twoparticles before a collision, where n ∈ S2 is the unit vector along the center of bothspheres, the postcollisional velocities are found assuming that the total momentum ispreserved but they loose part of the normal relative velocity, that is,

n · (v′ − w′) = −e((v − w) · n)

where 0 < e ≤ 1 is called the restitution coefficient. Using both information we constructthe postcollisional velocities as

v′ =1

2(v + w) +

V ′

2(1.1)

w′ =1

2(v + w) − V ′

2(1.2)

∗ Department of Mathematics, University of Texas at Austin, Austin, TX 78712, USA; on leave fromDepartamento de Matematica Aplicada, Universidad de Granada, 18071 Granada, SPAIN.

† Politecnico di Milano, Milano, ITALY.‡ Department of Mathematics, University of Texas at Austin, Austin, TX 78712, USA.

1

2 J.A. CARRILLO, C. CERCIGNANI AND I.M. GAMBA

where V ′ = V − (1 + e)(V · n)n, V = v − w and V ′ = v′ − w′. Let us denote by v∗ andw∗ the precollisional velocities corresponding to v and w.

Therefore, following the standard procedures of kinetic theory [8, 9, 11, 13], we canfind a Boltzmann-Enskog equation for inelastic hard spheres in a thermal bath. Thisequation reads as

∂f

∂t= Qhs(f, f) + LFPf(1.3)

where Qhs is the collision operator for inelastic hard-spheres [9, 11] and LFP is the Fokker-Planck operator. This operator takes into account the white noise interaction betweencollisions and using Ito’s stochastic calculus is given by

LFPf = F∆vf .(1.4)

In a previous work [7], a simplification of the Boltzmann-Enskog collision operator wasintroduced as an analog to the case of maxwellian molecules in classical elastic Boltzmannequation. Thus, assuming the pseudo-Maxwellian approximation [7], we have that thecollision frequency does not depend on the relative velocities but is proportional to thesquare-root of the kinetic temperature by a constant S. This constant S will be fixed insuch a way that the energy loss coincides with the one from the hard-spheres collisionoperator Qhs.

The existence of such a steady state was proved in the one dimensional case for adifferent collision operator in [1] and discussed in [14]; see also [3, 2]. The main aim ofthis work is to show the possibility of steady states in the homogeneous regime and to findan approximate expression of these states by expanding around a suitable maxwelliandistribution for small temperature dissipation rate. In the 3D hard-spheres case, wewere strongly motivated by the numerical simulation in [4] in which the shape of thehomogeneous steady state obtained by particle dynamics simulation has been plotted.

The homogeneous Boltzmann equation for inelastic particles under the pseudo-maxwellianapproximation in a heat bath is given by

∂f

∂t= B(ρ, t)Qε(f, f) + LFPf(1.5)

where

Qε(f, f) =1

4π

∫R3

∫

S2

[

f(t, v∗)f(t, w∗)1

e− f(t, v)f(t, w)

]

dn dw(1.6)

with

B(ρ, t) = πSσ2G(ρ)√

θ(t) = B(ρ)√

θ(t)(1.7)

and the temperature dissipation rate ε = 1−e2

4. Here, v∗, w∗ are the precollisional veloci-

ties associated to the collision mechanism

v′ =1

2(v + w) +

1 − e

4(v − w) +

1 + e

4|v − w|n(1.8)

w′ =1

2(v + w) − 1 − e

4(v − w) − 1 + e

4|v − w|n(1.9)

STEADY STATES OF A BOLTZMANN EQUATION FOR DRIVEN GRANULAR MEDIA 3

Also, ρ, u and θ(t) are the density, the mean velocity and temperature of the distributionf . Since both Qε(f, f) and LFPf preserve density and mean velocity they are justconstants. The function G(ρ) takes into account dense gas effects [4, 11].

The corresponding weak form of Qε(f, f) is given by the integral

〈Qε(f, f), ψ〉 =1

4π

∫R3×R3×S2

f(t, w)f(t, v)[ψ(v′) − ψ(v)] dv dw dn,

where ψ ∈ C∞0 (R3) and v′ is computed by

v′ =1

2(v + w) +

1 − e

4(v − w) +

1 + e

4|v − w|n.(1.10)

We show first that the possibility of existence of stationary solutions with positivetemperature. This is a remarkable difference with respect to the case of inelastic hardspheres without a thermal background bath in which the only stationary solutions areDirac deltas, concentrated on their mean velocity. The last section is devoted to find anapproximation to the steady state expanding in the temperature dissipation rate ε abouta maxwellian distribution with the temperature of the steady state. We find that up toorder on in ε3 the steady state is given by the maxwellian distribution multiplied by aquartic polynomial related to the second eigenvalue of the linearized classical Boltzmannelastic operator.

2. Temperature dissipation

We first find the equation for the evolution of the second moment of the distributionfunction. Let us consider ρ, u and θ(t) the density, mean velocity and temperature off(t, v). Thus, f must satisfy

∫R3

f dv = ρ ,

∫R3

v f dv = ρu ,

∫R3

|v − u|2f dv = 3ρθ.(2.1)

Let us remark that the computation of the second moment without the Fokker-Planckoperator was done for isotropic solutions in [7] using Fourier transform techniques. Weinclude here this computation in a different way which is valid for general distributions.

Evaluating the weak formulation of the equation for the test function |v−u|2 we deduce

3ρθ′(t) =d

dt

∫R3

|v − u|2f(v) dv

=B(ρ, t)

4π

∫R3×R3×S2

f(v)f(w)(|v′ − u|2 − |v − u|2) dv dw dn+ 6Fρ

(2.2)

We change variables in the collision integral finding∫R3×R3×S2

f(v)f(w)(|v′ − u|2 − |v − u|2) dv dw dn

=

∫R3×R3×S2

f(v + u)f(w + u)(|v′|2 − |v|2) dv dw dn(2.3)

4 J.A. CARRILLO, C. CERCIGNANI AND I.M. GAMBA

Now, we need to compute |v′|2. In order to simplify this computation we use the unitarylinear change of variables given by the velocity of the center of mass - relative velocitysystem

(v, V ) =

(

v + w

2, v − w

)

,

so that

v′ = v +1 − e

4V +

1 + e

4|V | · n(2.4)

then

|v′|2 = |v|2 +1 + e2

8|V |2 +

1 − e

2v · V +

1 + e

2|V |v · n+

1 − e2

8|V |V · n .

Since the integral on the unitary vector n goes through in (2.3), the integrals containing|V |v · n and |V |V · n will vanish. On the other hand, due to v · V = 1

2(|v|2 − |w|2) and

(2.1) the integral containing v · V vanishes too. In addition, rewriting v and V in theoriginal variables yields

|v|2 +1 + e2

8|V |2 =

3 + e2

8

(

|v|2 + |w|2)

+1 − e2

4v · w(2.5)

therefore the collision integral of (2.2) can be computed as

B(ρ)θ1/2

∫R3×R3

f(v + u)f(w + u)

[

3 + e2

8

(

|v|2 + |w|2)

− |v|2]

dw dv.(2.6)

since the term with v · w vanishes too due to (2.1). Again, using properties (2.1), (2.6)results in

= B(ρ)θ1/2

∫R3×R3

f(v + u)f(w + u)

[

(−5 + e2)

8|v|2 +

3 + e2

8|w|2

]

dv dw

= −3B(ρ)ρ2 1 − e2

4θ3/2 = −3B(ρ)ρ2εθ3/2

(2.7)

Therefore combining (2.2) and (2.7) yields the equation

ρθ′(t) = −B(ρ)ρ2εθ3/2 + 2Fρ(2.8)

or equivalently the temperature dissipation equation

θ′ = −B(ρ)ρεθ3/2 + 2F .(2.9)

Now, we fix the value of the constant S. The dissipation term in temperature arisingfrom the collision operator Qε is given by

γε = πSσ2G(ρ)ρε = B(ρ)ρε

while the dissipation term arising from the hard-spheres original operator Qhs is givenby [4, 11]

γhs = 8√πσ2G(ρ)ρε.

STEADY STATES OF A BOLTZMANN EQUATION FOR DRIVEN GRANULAR MEDIA 5

We set S by γε = γhs and then S = 8√π. Thus, finally the temperature equation can be

written as

θ′ = −γεθ3/2 + 2F .(2.10)

The equilibrium point corresponds to a possible steady state f sρ,u, whose temperature

is θ∞(ρ), is explicitly given by

θ∞(ρ) =

(

2F

B(ρ)ρε

)2/3

=

(

2

γε

F

)2/3

.(2.11)

This temperature is asymptotically stable for the ODE (2.10). Moreover, it is easy tosee that a steady state f s

ρ,u can be obtained from a normalized steady state f s1,0 = f s by

the self-similar relation

f sρ,u(v) = ρA(ρ)−3/2f s

(

A(ρ)−1/2(v − u))

(2.12)

where

A(ρ) = ρ−2/3

(

G(1)

G(ρ)

)2/3

=θ∞(ρ)

θ∞=

√θ∞G(1)

ρ√

θ∞(ρ)G(ρ)

and θ∞ = θ∞(1). In this sense we can say that the steady states are self-similar solutions.Let us also finally point out that the dependency on ρ of θ∞(ρ) (2.11) and the relation

(2.12) for G = 1 coincides with the one dimensional granular media models developed in[1, 2]. In fact, the pressure P = ρθ∞(ρ) ≃ ρ1/3 for G = 1 was also obtained in [1, 2, 14].

3. An approximation formula for the steady state

Assuming the existence of a steady state solution f sρ,u we shall perform a small in-

elasticity expansion and a linearization of the operator to compute a small inelasticityapproximation of this steady state. Without loss of generality, we take unit density andzero mean velocity and we focus on an expansion of f s since using (2.12) we produce anexpansion for f s

ρ,u. Let us remember that f s has temperature

θ∞ =

(

2F

Bε

)2/3

(3.1)

with B = 8√πσ2G(1) and we assume that satisfies

B√

θ∞Qε(fs, f s) + LFPf

s = 0.(3.2)

The parameter of expansion is given by ε = 1−e2

4. Therefore, we can approximate the

restitution coefficient by e =√

1 − 4ε ≈ 1−2ε−2ε2. The reason for this choice is twofold:on one hand ε is the parameter involved in the dissipation term for the temperatureequation; on the other hand ε appears as a small eigenvalue of the linearized operatorQε about the delta Dirac distribution which has a lot of important consequences in theasymptotic behavior of the distribution for small inelasticity in the unheated case θb = 0,see [7].

6 J.A. CARRILLO, C. CERCIGNANI AND I.M. GAMBA

We perform the expansion on the weak formulation of the operator Qε since it is easierto evaluate it in this form. The precollisional velocity v′ is given by

v′ =1

2(v + w) +

1 − e

4(v − w) +

1 + e

4|v − w|n ≈

=1

2(v + w) +

1

2|v − w|n+

1

2(ε+ ε2)(v − w) − 1

2(ε+ ε2)|v − w|n

= v0 + (ε+ ε2)v1 .

(3.3)

Here v0 = 12(v + w) + 1

2|v − w|n is the precollisional velocity corresponding to an elastic

collision and v1 = 12(V − |V |n) the dissipated part of the post collisional velocity, which

depend only on the relative velocity and collision angle.We will look for the steady state as a ε2-perturbation of a maxwellian distribution with

temperature θ∞, therefore

f s(v) = Mθ∞(1 + ε2g) = Mθ∞ + g

with∫

gdv = 0. Linearizing Qε(f, f) about Mθ∞ on (3.2) yields an equation for g

B√

θ∞Qε(Mθ∞ ,Mθ∞) + 2B√

θ∞Qε(Mθ∞, g) + LFP (Mθ∞) + LFP (g) = 0.(3.4)

On the other hand, since Qε depends on the restitution coefficient e ≈ 1 − 2ε − 2ε2 weexpand Qε in the weak form by making Taylor series in ε of ψ(v′), that is, for ψ ∈ C∞

0 (R)and v = v+w

2,

ψ(v′) = ψ(v0 + (ε+ ε2)v1)

= ψ(v0) + (ε+ ε2)(v1 · (∇v)ψ)(v0)) +ε2

2v©∨ 21 · (Hvψ)(v0) + O(ε3)

(3.5)

where Hvψ(v0) denotes the Hessian of ψ with respect to v at v0. Then replacing (3.5) inthe weak formulation of the collision integral and integrating by parts we have

〈Qε(f, f), ψ〉 = 〈Q0(f, f), ψ〉 +(

ε+ ε2)

〈R(f, f), ψ〉 +ε2

2〈S(f, f), ψ〉(3.6)

whereQ0 is the the classical elastic Boltzmann collision operator for maxwellian molecules,

〈R(f, f), ψ〉 = − 1

8π

∫R3×R3×S2

(V − |V |n) · ∇v [f(v)f(w)]ψ(v0) dv dw dn(3.7)

and

〈S(f, f), ψ〉 =1

16π

∫R3×R3×S2

(V − |V |n)©∨ 2 ·Hv [f(v)f(w)]ψ(v0) dv dw dn .(3.8)

STEADY STATES OF A BOLTZMANN EQUATION FOR DRIVEN GRANULAR MEDIA 7

Since all the maxwellians distributions are in the kernel of the classical operator Q0

we have Q0(Mθ∞,Mθ∞) = 0. Therefore, g must satisfy the equation

0 = B√

θ∞⟨

(ε+ ε2)[

R(Mθ∞,Mθ∞) + 2R(Mθ∞ , g)]

+(3.9)

+ε2

2

[

S(Mθ∞ ,Mθ∞) + 2S(Mθ∞, g)]

, ψ

⟩

+ 2B√

θ∞ 〈Q0(Mθ∞ , g), ψ〉 + 〈LFPMθ∞ , ψ〉 + 〈LFP g, ψ〉 + O(ε3) .

It is straightforward to find

LFPMθ∞(v) =F

θ∞

(

−3 +|v|2θ∞

)

Mθ∞(v) = −ε2B

√

θ∞Mθ∞

[

3 − |v|2θ∞

]

.(3.10)

Next, replacing (3.10) into (3.9) and taking into account that g = ε2Mθ∞g the followingequation is valid for g

0 = ε−1B√

θ∞

[

〈R(Mθ∞,Mθ∞), ψ〉 −⟨

1

2Mθ∞

(

3 − |v|2θ∞

)

, ψ

⟩]

(3.11)

+B√

θ∞

⟨

(R+1

2S)(Mθ∞,Mθ∞), ψ

⟩

+ 2B√

θ∞ 〈Q0(Mθ∞, gMθ∞), ψ〉

+ 〈LFP (Mθ∞g), ψ〉+ O(ε) .

Next, we shall see that the terms R and S evaluated on (Mθ∞ ,Mθ∞) can be computedexactly. Hence, we must calculate (3.7) and (3.8) in order to replace in (3.11). First from(3.7), using (v, V ) coordinates

〈R(Mθ∞,Mθ∞), ψ〉

= − 1

8π

∫R3×R3×S2

(V − |V |n) · ∇v

[

Mθ∞(√

2v)Mθ∞

(

V√2

)]

ψ

(

v +|V |n

2

)

dv dV dn(3.12)

Since ∇vMθ∞(√

2v ) = − 2vθ∞Mθ∞(

√2v ) we have

1

4πθ∞

∫R3×R3×S2

Mθ∞(√

2v )Mθ∞

(

V√2

)

(V · v − |V |v · n)ψ

(

v +|V |n

2

)

dv dV dn.

(3.13)

Now, considering spherical coordinates V = |V |k with k ∈ S2, swaping the roles of k andn, interchanging the integrals over dn by the one over dk and rewriting V = |V |n, (3.13)becomes

1

4πθ∞

∫R3×R3×S2

Mθ∞(√

2v)Mθ∞

(

V√2

)

(|V |v · k − v · V )ψ(v) dv dV dk .(3.14)

Since the integration with respect to k on S2 comes through then the integral of theterm v · k vanishes. Returning to (v, w) coordinates and taking into account v · V =

8 J.A. CARRILLO, C. CERCIGNANI AND I.M. GAMBA

|v|2−|w|22

, (3.14) becomes

− 1

2θ∞

∫R3×R3

Mθ∞(v)Mθ∞(w)(

|v|2 − |w|2)

ψ(v) dv dw .(3.15)

Therefore the operator R(Mθ∞,Mθ∞) can be expressed as

−1

2

∫R3

1

θ∞Mθ∞(v)Mθ∞(w)(|v|2 − |w|2) dw = − 1

2θ∞Mθ∞(v)(|v|2 − 3θ∞)(3.16)

or equivalently

〈R(Mθ∞ ,Mθ∞), ψ〉 =

⟨

1

2Mθ∞(v)

[

3 − |v|2θ∞

]

, ψ

⟩

,(3.17)

This computation shows that the term of order ε−1 vanishes in (3.11).We proceed similarly in order to compute S(Mθ∞ ,Mθ∞). Expressing the first integral

on S in (v, V ) variables, computing

Hv

(

Mθ∞(√

2v ))

= Mθ∞(√

2v )

(

− 2

θ∞I +

4

θ2∞v©∨ 2

)

(3.18)

and swaping the role of n and k as done for R we have

1

8π

∫R3×R3×S2

Mθ∞

( V√2

)

Mθ∞(√

2v )

(

− 2

θ∞I +

2

θ2∞v©∨ 2

)

(

|V |2k©∨ 2 − 2|V |V ⊗ k + V ©∨ 2)

ψ(v) dk dv dV .

Now, performing the integration with respect to k ∈ S2, recalling∫

S2 k©∨ 2 dk = 4π

3I,

∫

S2 V ⊗ k dk = 0 and∫

S2 dk = 4π and reducing the matrices, (3.18) becomes

1

2

∫R6

Mθ∞

( V√2

)

Mθ∞(√

2v )

[

−|V |2θ∞

+2

3θ2∞|v|2|V |2 − |V |2

θ∞+

2

θ2∞v©∨ 2 · V©∨ 2

]

ψ(v) dv dV ,

thus, rewriting this integral in the original coordinates,

∫R6

Mθ∞(v)Mθ∞(w)

[

− |v − w|22θ∞

+|v − w|2|v + w|2

12θ∞− |v − w|2

2θ∞

+1

4θ2∞

(v + w)©∨ 2 · (v − w)©∨ 2

]

ψ(v) dv dw

=

∫R6

Mθ∞(v)Mθ∞(w)

[

− |v|2 + |w|2 − 2v · wθ∞

+(|v|2 + |w|2)2 − 4(v · w)2

12θ2∞

(3.19)

+(|v|2 − |w|2)2

4θ2∞

]

ψ(v) dv dw .

STEADY STATES OF A BOLTZMANN EQUATION FOR DRIVEN GRANULAR MEDIA 9

Now, we have to integrate with respect to w, for doing that we make use of the integral∫R3

Mθ∞(w)(v · w)2 dw =

∫ ∞

0

Mθ∞(w)|v|2|w|4d|w|∫

S2

(n1 · n2)2 dn2

=4π

3

∫R3

Mθ∞(w)|v|2|w|4 d|w| =1

3

∫R3

Mθ∞(w)|v|2|w|2 dw ,

and of the fact that the integral in w of the term v ·w vanishes. Therefore, the operatorS(Mθ∞,Mθ∞) becomes

Mθ∞(v)

∫R3

Mθ∞(w)

(

−|v|2 + |w|2θ∞

+|v|4 + |w|4

3θ2∞

− 4

9θ2∞|v|2|w|2

)

dw.(3.20)

Taking moments of Mθ∞(w) yields

S(Mθ∞ ,Mθ∞) = Mθ∞(v)

[

−|v|2θ∞

− 3 +|v|43θ2

∞+ 5 − 4

3

|v|2θ∞

]

(3.21)

= Mθ∞(v)

[ |v|43θ2

∞− 7

3

|v|2θ∞

+ 2

]

.

In particular∫

R(Mθ∞,Mθ∞) dv = 0 =

∫

S(Mθ∞,Mθ∞) dv

then we have checked our computations in the sense that the first two order terms inthe expansion of Qε preserve mass. Next, we can replace (3.17) and (3.21) in the weakformulation of the equation for g, given by (3.11) obtaining the following operator actingon g,

− 2B√

θ∞Q0(Mθ∞, gMθ∞) − LFP (gMθ∞) =

= B√

θ∞Mθ∞

[

1

2

(

3 − |v|2θ∞

)

+|v|46θ2

∞− 7

6

|v|2θ∞

+ 1

]

=1

6B

√

θ∞Mθ∞

[ |v|4θ2∞

− 10|v|2θ∞

+ 15

]

= −h(v) .

(3.22)

Next, let us consider the linear operator acting on g

L(gMθ∞) = 2B√

θ∞Q0(Mθ∞, gMθ∞) + LFP (gMθ∞) .(3.23)

We need to solve the problem

L(gMθ∞) = h(v) .(3.24)

We first discuss the eigenvalue problem associated to the operator L(gMθ∞), namely,we look for isotropic eigenfunctions g, that is,

L(gMθ∞) = −λgMθ∞ .(3.25)

10 J.A. CARRILLO, C. CERCIGNANI AND I.M. GAMBA

We recall from [8], [10] that the isotropic eigenfunctions of the operators Q0(Mθ∞, gMθ∞)are given by Sonine polynomials multiplied by Mθ∞(v). Looking at the second orderSonine polynomial [10] we find

S(2)1/2(ξ) =

3

24

(

15 − 20ξ + 4ξ2)

(3.26)

for any ξ ∈ R, and if we define the function

P (v) = S(2)1/2

( |v|22θ∞

)

=3

24

(

15 − 10|v|2θ∞

+|v|4θ2∞

)

(3.27)

then, the function h(v) is given by

h(v) = −4

3B

√

θ∞Mθ∞P(3.28)

Looking at the eigenvalues and eigenvectors of the operator Q0(Mθ∞ , gMθ∞) in [5, 6]and following its notation, we have

Q0

(

Mθ∞ , S(2)1/2

( |v|22θ∞

)

Mθ∞

)

= −λ20S(2)1/2

( |v|22θ∞

)

Mθ∞(v)(3.29)

with λ20 = 23.

Next, we show that the function Mθ∞P = Mθ∞S(2)1/2(

|v|22θ∞

) is an eigenvector with zero

eigenvalue for the operator LFP = F∆v up to an order of ε2/3. We first compute

LFP (Mθ∞P ) = FMθ∞

(

P( |v|2θ∞

− 3) 1

θ∞− v

θ∞· ∇vP + ∆vP

)

=1

θ∞FMθ∞

3

24

(

−105 + 85|v|2θ∞

− 17|v|4θ2∞

+|v|6θ3∞

)

.

(3.30)

Now, by (3.1) θ−1∞ = O(ε2/3) and thus, LFP (Mθ∞P ) = O(ε2/3).

To finish we come back to (3.24), take into account (3.28) and the previous results forthe eigenvalues of the operators to find out that

g(v) = S(2)1/2

( |v|22θ∞

)

Mθ∞(v)

is a solution of the equation (3.24) up to order ε. We conclude by writing the expansionof the stationary solution

f s(v) = Mθ∞(v)

[

1 + ε2S(2)1/2

( |v|22θ∞

)]

+ O(ε3).

Now, we use the self-similar relation (2.12) to obtain the main result of this paper.

Theorem 3.1. The steady state f sρ,u for (1.5), up to order ε3, is given by

f sρ,u(v) = ρMθ∞(ρ)(v − u)

[

1 + ε2S(2)1/2

( |v − u|22θ∞(ρ)

)]

+ O(ε3)

STEADY STATES OF A BOLTZMANN EQUATION FOR DRIVEN GRANULAR MEDIA 11

with

θ∞(ρ) =

(

2F

B(ρ)ρε

)2/3

=

(

2

γεF

)2/3

and S(2)1/2 given by formula (3.26).

4. Heat bath with friction

We may include also friction over the particles between collisions asuming that theparticles are in some sort of surrounding heat bath with a fixed finite temperature θb. Inthis case, the paths of the particles are governed by the Langevin equation

x′′ +1

τx′ = Γ(t)

where Γ(t) is a white noise stochastic force with independent, identically Gaussian dis-tributed processes of variance F , that is, < Γ(t),Γ(t′) >= 2Fδ(t − t′) with F = θb

τand τ > 0 the relaxation time corresponding to the damping force. The Fokker-Planckoperator reads now as

L2FPf =

1

τdivv(vf + θb∇vf) .

In this case, we can perform the same computations as before and the temperaturedissipation equation becomes

θ′ +2

τθ = −γεθ

3/2 +2

τθb .(4.1)

Also, the mean velocity is no longer preserved and is dissipated as u(t) = e−t

τ u(0).The equilibrium point corresponds to a possible steady state f s

ρ with zero mean velocitywhere θ∞(ρ) is given by the unique positive solution of the equation

θb = θ +τ

2γεθ

3/2.(4.2)

This temperature is globally asymptotically stable for the evolution of the ODE (4.1).In this case we do not have a simple relation between f s

ρ and the normalized f s.The same expansion procedure can be done for this operator. The main differences

are that we do not have a relation as (2.12) to reduce the computation to f s and thecomputation of L2

FP (Mθ∞(ρ)P ) with P given by

P (v) = S(2)1/2

( |v|22θ∞(ρ)

)

=3

24

(

15 − 10|v|2θ∞(ρ)

+|v|4

θ∞(ρ)2

)

The first one is easily solve by making the expansion directly for f sρ . Now, we compute

L2FP (Mθ∞(ρ)P ) =

Mθ∞(ρ)

τ

(

P(

3 − |v|22

)(

1 − θb

θ∞(ρ)

)

+ v(

1 − 2θb

θ∞(ρ)

)

∇vP + θb∆vP

)

(4.3)

12 J.A. CARRILLO, C. CERCIGNANI AND I.M. GAMBA

Now, by (4.2) we have θ∞(ρ) = θb + O(ε) and therefore, we can estimate the righthand side of (4.3) as

Mθ∞(ρ)

τ(O(ε) − v∇vP + θb∆vP ) = −4

Mθ∞(ρ)

τP + O(ε)

thus,

L2FP (Mθ∞(ρ)P ) = −4

τMθ∞(ρ)P + O(ε).

As a consequence, the expansion for the steady state in this case is

f sρ(v) = ρMθ∞(ρ)(v)

[

1 + ε2AS(2)1/2

( |v|22θ∞(ρ)

)]

+ O(ε3)

with

A =

(

τρB(ρ)

τρB(ρ) + 3

)

,

θ∞(ρ) the unique solution of

θb = θ +τ

2γεθ

3/2

and S(2)1/2 given by formula (3.26). Formally, we recover theorem 3.1 by taking the limit

τ → ∞, θb → ∞ in such a way that F = θb

τremains constant.

Remark 4.1. Let us finally remark that a detailed study of the eigenfunctions andeigenvalues of the linear operator L may eventually give, by expanding in this set ofeigenfunctions, information about the linear asymptotic stability of the steady state withrespect to small perturbations.

Acknowledgments. The first author is supported by the DGESIC MEC-Spain Per-feccionamiento de Doctores en el Extranjero fellowship. Partially supported by TMREuropean Union ERB FMBX-CT97-0157 and DGES (Spain) PB98-1281. The third au-thor is supported by NSF under grant DMS 9971779. We would like to thank H. L.Swinney, M. D. Shattuck and C. Bizon for helpful conversations.

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