CONSERVATIVE FORMS OF BOLTZMANN'S COLLISION OPERATOR

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Niveau: Supérieur, Doctorat, Bac+8
CONSERVATIVE FORMS OF BOLTZMANN'S COLLISION OPERATOR : LANDAU REVISITED C. VILLANI Abstract. We show that Boltzmann's collision operator can be written explicitly in divergence and double divergence forms. These conservative formulations may be of interest for both theoretical and numerical purposes. We give an application to the asymptotics of grazing collisions. Contents 1. Introduction 1 2. Basic identities and main results 5 3. First conservative form 8 4. How to go further 13 5. Computing in ?-representation 15 6. Landau's collision operator 21 7. The asymptotics of grazing collisions 23 References 25 1. Introduction Boltzmann's collision operator is the main model in statistical physics for describing the interaction between colliding particles. It reads (1) Q(f, f) = 12 ∫ R N dv? ∫ SN?1 d? B(v ? v?, ?) (f ?f ?? ? ff? ), where f : RN ?? R+ is a nonnegative L1 function, standing for the distribution of the velocities of the particles, (2) { v? = v ? (v ? v?, ?)?, v?? = v? + (v ? v?, ?)?, and f ? = f(v?), and so on. As ? varies in the unit sphere SN?1, v? and v?? describe all the possible postcollisional velocities of two particles 1

• collision operator


• dv? d?


• dv dv? d?


• double divergence


• ?? ?


• formally performed


• write formally


• between particles

Informations

• Publié par: mijec
• Nombre de lectures: 9
• Langue: English

Extrait

CONSERVATIVE FORMS OF BOLTZMANN’S
COLLISION OPERATOR :
LANDAU REVISITED
C. VILLANI
Abstract. We show that Boltzmann’s collision operator can be
writtenexplicitlyindivergenceanddoubledivergenceforms. These
conservative formulations may be of interest for both theoretical
andnumericalpurposes. Wegiveanapplicationtotheasymptotics
of grazing collisions.
Contents
1. Introduction 1
2. Basic identities and main results 5
3. First conservative form 8
4. How to go further 13
5. Computing in ?-representation 15
6. Landau’s collision operator 21
7. The asymptotics of grazing collisions 23
References 25
1. Introduction
Boltzmann’scollisionoperatoristhemainmodelinstatisticalphysics
for describing the interaction between colliding particles. It reads
Z Z
¡ ¢1 0 0(1) Q(f;f)= dv d!B(v¡v ;!) f f ¡ff ;⁄ ⁄ ⁄⁄
2 N N¡1S
N 1where f :R ¡!R is a nonnegative L function, standing for the+
distribution of the velocities of the particles,
(
0v =v¡(v¡v ;!)!;⁄
(2)
0v =v +(v¡v ;!)!;⁄ ⁄⁄
0 0 N¡1 0and f =f(v), and so on. As ! varies in the unit sphere S , v and
0v describe all the possible postcollisional velocities of two particles⁄
1
R2 C. VILLANI
colliding with respective velocity v and v , taking into account that⁄
the collisions are assumed to be elastic, i.e.
(
0 0v +v =v+v ;⁄⁄(3) 2 20 0 2 2jvj +jv j =jvj +jv j :⁄⁄
Moreover, B(z;!) is a nonnegative weight function for all possible di-
rections of the parameter !. It is always assumed on physical grounds
that B depends only onjzj andj(z;!)j. A factor 1=2 stands in front of
thecollisionalintegralbecauseintherepresentation(2)theunitsphere
is covered twice : alternatively, we could restrict the integration to the
set of (v;v ;!) satisfying (v¡v ;!)>0.⁄ ⁄
Q(f;f) at point v gives the variation of the “number” of particles
with velocity v, in a unit of time, due to collisions. It is often split
+ ¡into its positive and negative parts, Q = Q ¡Q , which are called
respectively the gain and loss terms of the collision operator.
Another useful representation of Q is obtained by using a local sys-
tem of spherical coordinates with axis v¡v and spherical angle ? :⁄
Z Z
¡ ¢
0 0e(4) Q(f;f)= dv d B(v¡v ;?) f f ¡ff ;⁄ ⁄ ⁄⁄
N N¡1S
with
8
v+v jv¡v j⁄ ⁄> 0v = + ?>< 2 2
(5)
> v+v jv¡v j> ⁄ ⁄0:v = ¡ ?;⁄
2 2
and
B(z;!)eB(z;?)= :‡ ·N¡2
zN¡12 ;!
jzj
If the interaction between two particles is given by an inverse-power
sforce 1=r (r being the distance between particles) with s ‚ 2, then
(Cf. [1])
?eB(z;?)=jzj ‡(?)
where ? 2 [0;…] is the angle between z and ?. If N = 3, then ? =
¡(s+1)=(s¡1)(s¡5)=(s¡1) and ‡ is smooth on (0;…=2), but ‡(?)» C?
with C > 0 as ? ! 0. This singularity corresponds to the so-called
grazing collisions, i.e. with a very small amount of momentum transfer
(equivalently,(v¡v ;!)’0). Thoughitisnonintegrable,itispossible⁄
RCONSERVATIVE FORMS OF BOLTZMANN’S OPERATOR 3
togiveadistributionalsensetoQ(f;f)underratherweakassumptions,
provided that
Z … ?2(6) Λ· d ‡ (?)sin <1;
20
i.e. thatthetotalcross-sectionformomentumtransferbefinite. See[6]
and the references therein for a detailed study.
For Coulomb interactions, s = 2 and the integral (6) diverges loga-
rithmically, due to the effect of long-range interactions. To circumvent
this difficulty, Landau [4] formally performed asymptotics in which the
grazing collisions become preponderant, and obtained a new collision
operator,
‰ ? ¶ Z
@ @f @f⁄
(7) Q (f;f)= dv a (v¡v ) f ¡f ;L ⁄ ij ⁄ ⁄
@v @v @vNi j ⁄j
with
zzi j
a (z)=ΛΨ(jzj)Π (z); Π (z)=– ¡ ;ij ij ij ij 2jzj
¡ ¢1?+2Ψ(jzj)=jzj = for Coulomb potential
jzj
Here we use Eisntein’s convention of implicit summation. The case
? =¡3 is formally treated in [2] for instance.
Assuming of course Λ to be finite, the operator (7) is well-defined
and is believed to give a satisfactory description of the collisions in a
dilute plasma. The basis of Landau’s analysis was to write formally
Boltzmann’s collision operator in a conservative form,
(8) Q(f;f)=¡r ¢sv
thand the j component of s is given by
(9) s (v)=j
Z Z
vj £ ⁄
¡ d!dv dw B(w¡w ;!) f(w+q)f(w ¡q)¡f(w)f(w )⁄ j ⁄ ⁄ ⁄
q >0 v ¡qj j j
where q = (v¡ v ;!)! is the transferred momentum; see [4], para-⁄
graph 41.
It is easily seen that this expression is a crude approximation of (1),
sincetheflux(9)countsparticlesgoingthroughahyperplane. Butboth
expressions (1) and (8) coincide in the limit when all the collisions be-
comegrazing,because,duetothesmallamountofmomentumtransfer,
the velocity of a particle changes continuously – and a small surface
element, viewed from very close, looks like a hyperplane. In fact, it is
R64 C. VILLANI
sometimes stated in physics textbooks that the expression (8) is mean-
ingless in the general case, because, contrary to a diffusion process,
the velocity of particles does not change continuously (hence particles
N“jump” in the velocity spaceR , and the flux is not well-defined).
However,weshallshowthat,atleastfromthemathematicalpointof
view, it is perfectly possible to write the Boltzmann collision operator
in a conservative form, even if particles undergo sudden changes of
velocity. For instance, one possible expression for the flux of particles
is
(10) J(f;f)=
Z Z
(v¡v ;!)⁄
¡ dv d!B(v¡v ;!) drf(v+r!)f(v +r!)!:⁄ ⁄ ⁄
(v¡v ;!)>0 0⁄
What is more, we shall prove that Q can be written as a double
divergence, that is,
X
(11) Q(f;f)= @ A (f;f):ij ij
ij
These formulas could be interesting for the mathematical study of the
Boltzmann equation, especially the singularity in B. In particular, we
note that in the formula (10) ay of order 1 in B has been
formally reported on the divergence operation. Indeed, if f is smooth,
Z (v¡v ;!)⁄ ¡ ¢
drf(v+r!)f(v +r!)=O j(v¡v ;!)j :⁄ ⁄
0
On the other hand, the representation (10) gives the flux as an inte-
graloperatorwithonemultiplicitymorethanthecollisionoperator(1),
i.e. 2N insteadof2N¡1. Infact,onecanalsowriteJ asanexpression
of the form
Z
(12) J(f;f)= dv dv f(v )f(v )D(v;v ;v ):⁄ – ⁄ – ⁄ –
2N
Suchaformulacouldbeusefulfordevisingnewdeterministicnumerical
schemes for the Boltzmann equation. Indeed, the fact that ! runs
throughtheunitspherein(1)appearstooftencausedelicateproblems
for discretization [3].
Theorganizationofthepaperisasfollows. Insection2,werecallthe
basicidentitiesinvolvingQ;thenwestateseveralformsofQ(f;f),con-
sideredeitherasadivergenceorasadoubledivergence. Wealsogivea
double divergence form for Landau’s collision operator (7). These ex-
pressionsareestablishedinsections3to6. Finally,insection7,wegive
an application of the results of section 3 : a simplified proof of a result
RCONSERVATIVE FORMS OF BOLTZMANN’S OPERATOR 5
by Desvillettes [2] that if a fixed function f is smooth enough, then in
a suitable grazing asymptotics process, Q(f;f) reduces to Q (f;f).L
Acknowledgement : The author thanks F. Golse for asking him
the question which is at the basis of this work.
2. Basic identities and main results
We first recall some well-known facts about Boltzmann’s collision
operator. Let ’ be a smooth test function; we consider
Z Z
¡ ¢
0 0eQ(f;f)’= dvdv d B(v¡v ;?) f f ¡ff ’(v):⁄ ⁄ ⁄⁄
Bythechangeofvariables(involutiveandwithunitJacobian)(v;v )!⁄
0 0 0 0e e(v;v ), noting that B(v ¡v ;?)=B(v¡v ;?), we obtain⁄⁄ ⁄
Z Z
0eQ(f;f)’= dvdv d Bff (’ ¡’);⁄ ⁄
ewhere we have omitted for simplicity the arguments of B. This is
Maxwell’s form [5] of Boltzmann’s collision operator. Now, by the
change of variables (v;v )!(v ;v),⁄ ⁄
Z Z
1 0 0eQ(f;f)’= dvdv d Bff (’ +’ ¡’¡’ ):⁄ ⁄ ⁄⁄
2
This formula can actually be taken as a definition of Q(f;f) in theR
2+?senseofdistributions[6],assoonas,say, f(v)(1+jvj )dv <1and
R
? 2eB(z;?)=Φ(jzj)‡(?) with Φ(jzj)•C(1+jzj ) and ? ‡(?)d < 1. It
is also clear that analogous formulas hold for the representation (1).
Similarly (Cf. [6]), for Landau’s collision operator
Z Z
1
Q (f;f)’= dvdv ff a (v¡v )(@ ’+(@ ’) )L ⁄ ⁄ ij ⁄ ij ij ⁄
2
Z
+ dvdv ff b (v