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A New Renormalized Form of the Boltzmann Equation
V. L. Saveliev1,2 and K. Nanbu2
1
Institute of Ionosphere, Almaty 480020, Kazakhstan
Institute of Fluid Science, Tohoku University, Sendai 980-8577, Japan
2
Abstract. On the basis of a recently discovered collision group, the Boltzmann collision integral is exactly rewritten in
two parts. The first part describes the scattering of particles with small angles. In this part the infinity due to the infinite
cross sections is extracted from the Boltzmann collision integral. Moreover, the Boltzmann collision integral is exactly
represented as a divergence of the flow in velocity space. Owing to this, the role of collisions in the kinetic equation can
be interpreted in terms of the non-local friction force that depends on the distribution function.
I. PARAMETERIZATION OF COLLISIONS IN THE BOLTZMANN
EQUATION BY A ROTATION MATRIX
Applications of the group theory remain underutilized in kinetic theory although they deserve to be utilized [1-3].
Recently[2], it has been shown that if we parameterize the two-particle collision by a matrix R̂ belonging to the
group of rotations O3+ then the transformation of particle’s velocities due to collision becomes linear one. In that
case, if a particle of mass m1 and velocity v collides with a particle of mass m2 and velocity u , the postcollisional velocities are given by the following equations:
v′ =
u′ =
mv + u + Rˆ(v − u )
,
1+m
mv + u − mRˆ(v − u )
1+m
(1)
.
The velocities v ′ and u ′ are determined by a partitioned scattering matrix ( 2 × 2 cells). The size of each cell is
obviously ( 3 × 3 ):
ˆ

1 − Rˆ 
 m + R

 1 + m
1 + m 
′
ˆ
ˆ
ξ = Sξ ,
(2)
S = 
.
 m(1 − Rˆ) 1 + mRˆ 

 1 + m
1 + m 
Here, ξ is a 6-dimensional bivector consisting of the components v and u . The scattering matrixes
Sˆ(Rˆ) ( Rˆ ∈ O3 ) constitute a group that is isomorphic to the group of orthogonal matrixes O3 (the group of
 v 
ξ =  u  ,
 
rotations including improper rotations):
Sˆ(Rˆ1 ) ⋅ Sˆ(Rˆ2 ) = Sˆ(Rˆ1 ⋅ Rˆ2 ),
ˆ) = Sˆ(Rˆ−1 ) .
Sˆ−1 (R
(3)
Here, it should be noted that parameterization [4] of a collision by the direction n normal to the plane of reflection
of the relative velocity ( v ′ = v − 2n n ⋅ v ) also provides a linear transformation of the particles’ velocities due
CP663, Rarefied Gas Dynamics: 23rd International Symposium, edited by A. D. Ketsdever and E. P. Muntz
© 2003 American Institute of Physics 0-7354-0124-1/03/$20.00
27
to collision. In this case, however, scattering matrixes do not constitute a group, which is crucial for our further
ˆ) satisfy the following conditions:
consideration. The matrixes Sˆ(R
 m 0  ˆ  m 0 
Sˆ 
S = 
,
 0 1 
 0 1 
det Sˆ = 1 ,
(4)
where Sˆ is the transpose of Sˆ . It provides the equality of velocity volumes:
dvdu = dv ′du ′ .
(5)
This relation is much simpler than dvdud Ω ′ = dv ′du ′d Ω which appears in the case of the conventional [5,6]
parameterization of a collision.
II. COLLISION INTEGRAL
To rewrite the collision integral in the conventional form[6] for the case when a collision is parameterized by
rotation matrix, integration over directions of the relative velocity vector v ′ should be replaced by integration on
dΩ
dRˆ
):
→
4π
8π 2
ˆ ˆ = dRˆ−1,
dRˆ = dRˆ0Rˆ = dRR
0
the invariant measure[7,8] over the group O3+ (
∫
(6)
dRˆ = 8π 2 .
The collision integral now takes the following form:
I ( f , ψ) = I (F ) =
dRˆ
∫ b ( v, µ )[ f (v ′)ψ(u ′) − f (v)ψ(u ) ] 2π du
=
(7)
dRˆ
du ,
= ∫ b ( v, µ )  F ( Sˆ(Rˆ)ξ ) − F (ξ) 
2π
ˆ
ˆ, v ) = v ⋅ R v is the cosine of the scattering
where b ( v, µ ) = v σθ ( v, µ ) is a scattering indicatrix, µ ( R
v2
angle, and F( ξ )=f D ψ ( ξ ) = f( v ) ψ( u ) is the two-particle velocity distribution function.
The fact (eq.(3)) that collision matrixes constitute a group gives us essentially new opportunities for investigating
the Boltzmann equation. In a paper [2], a new method to directly construct a class of discrete velocity models for
mixtures from the Boltzmann equation was proposed by replacing the integration of eq. (7) over the full group of
rotations by summation over the discrete subgroups of this group. In the following sections we make use of the
group property (3) of the collision matrixes (2), and present a new exponential form [1] of the collision operator and
a method of renormalization of collision integral (7).
A long-standing problem in the kinetic theory is how to effectively describe, within the framework of the
Boltzmann equation, the evolution of a system of particles interacting with long range forces, especially with
Coulomb forces. Many interesting methods (see [9, 10] and cited references therein) have been developed for
practical calculations of the velocity distribution function of Coulomb particles. It is expected that the exact
renormalization of the Boltzmann collision integral presented in our paper will be an important step in solving these
problems.
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REPRESENTATION OF THE GROUP OF SCATTERING MATRIXES IN THE
HILBERT SPACE AND THE BOLTZMANN COLLISION INTEGRAL
There are many equivalent ways to parameterize rotation matrixes, each of which has its own advantage. We will
here use a parameterization of proper rotations (reflections are not included) by the angle ( 0 ≤ φ ≤ π ) of
rotation and by the direction n of the axis of rotation, where n is the unit vector. Namaly, rotation of the angle φ
around the axis n is given by the following formula [7,8]:
Rˆ = e φnˆ = ( 1 − cos φ ) nˆ2 + ( sin φ ) nˆ + 1 ,
where the operator (matrix) n̂ is defined by the following expression:
df
nˆ = n × ,
(8)
df
nˆv = n × v .
(9)
The infinitesimal rotations take the form
Rˆ = e φnˆ 1 + φnˆ .
(10)
φ→0
Substituting eq. (10) into eq. (2) for the scattering matrix, we obtain the following equation for the infinitesimal
scattering matrix:
ˆ

1 − Rˆ 
 m + R

 1 + m
1 + m 
ˆ
S = 
 1 + φcˆ, ( φ → 0 ) ,
 m(1 − Rˆ) 1 + mRˆ 

 1 + m
1 + m 
From this we can see, that the generator ĉ for the scattering matrix Sˆ is
−nˆ 
1  nˆ


cˆ =
.
1 + m  −mnˆ mnˆ 
(11)
(12)
Making use of eq. (12) for the generator ĉ , we can obtain an exponential representation for the scattering matrix:
Sˆ = e φcˆ .
(13)
In accordance with the usual rules [7,8] of the theory of Lie groups, we can also easily construct a representation of
the scattering group in the Hilbert space of functions on bivector variable ξ :
e φσˆ F ( ξ ) = F ( Sˆ−1ξ ) .
(14)
To obtain the infinitesimal transformation in the Hilbert space, we substitute eq. (11) into eq. (14):
e φσˆ F ( ξ ) = F ( Sˆ−1 ξ ) F ( ( 1 − φcˆ) ξ ) =
φ→0


∂ 
∂ 
=  1 − φci, k ξk
 F ( ξ ) =  1 − φξcˆ  F ( ξ ) .


∂ξi 
∂ξ 
As a result, for the generator σ̂ we obtain the following expression:
ˆ,
σˆ = n ⋅ σ
where
ˆ =−
σ
(
)
)
(
(16)
1
∂
∂
v×
.
−m
1+m
∂v
∂u
The square of the vector operator σ̂ is as follows:
ˆ2 =
σ
(
(15)
1
∂
∂
∂
∂
−m
⋅ ( v2 − vv ⋅ )
−m
2 ∂v
∂u
∂v
∂u
(1 + m )
29
(17)
)
To better understand the algebraic properties of the operator σ̂ , we should change variables. Introducing the
relative velocity v = v − u and the center-of-mass velocity w =
(
)
mv + u
, we have
1+m
1
∂
∂
∂
.
(18)
−m
=
1 + m ∂v
∂u
∂v
It is seen from eq. (18) that the generator σ̂ depends only on relative velocity v and that its form is very simple:
∂
∂
ˆ = −v ×
σ
(19)
=
× v.
∂v
∂v
The operator (19), accurate up to constant, is the well-known quantum mechanical operator of angular momentum
with commutative relations:
[ σˆi , σˆk ] = eikl σˆl , [ σˆi , σˆ2 ] = 0 ,

∂ 
∂
σ
 = eikl
ˆi ,
,
 ∂ vk 
∂ vl
ˆi vi = vi σ
ˆi = 0 .
σ
[ σˆi , vk ] = eikl vl
,
(20)
The equation for e φσˆ acting on the product of two functions obviously is
e φσˆF1 ( ξ ) F2 ( ξ ) = F1 ( Sˆ−1ξ ) F2 ( Sˆ−1ξ ) =  e φσˆF1 ( ξ )   e φσˆF2 ( ξ )  .
(21)
Equation (14) allows us to rewrite the collision operator in a new form in which the group structure of a process of
particle scattering is explicitly used. The invariant measure for the group of proper rotations is as follows when
rotations are parameterized by the angle of rotation φ and the direction n of the rotation axis [7,8]:
dRˆ = 2(1 − cos φ)d φd Ωn ,
where
d Ωn = sin θd θd ϕ ,
(22)
( 0 ≤ φ, θ ≤ π; 0 ≤ ϕ ≤ 2π ) .
Using eq. (8), the equation for scattering cosine µ is as follows:
( 1 − cos φ ) v ⋅ nˆ2 v + ( sin φ ) v ⋅ nˆv + v2
v ⋅ Rˆ v
µ=
=
=
v2
v2
( 1 − cos φ ) [ n × v ]2
= 1−
= 1 − ( 1 − cos φ ) sin2 θ .
2
v
(23)
Substituting eq. (14) into eq. (7) and keeping eqs. (22) and (23) in mind, we obtain the final form for the Boltzmann
collision integral:
I ( f , ψ) =
=
=
∫
∫
∫
b ( v, µ ) [ f (v ′)ψ(u ′) − f (v)ψ(u ) ]
dRˆ
du =
2π
dRˆ
b ( v, µ )  e −φσˆ − 1  f( v )ψ( u ) =
2π
dRˆ  −φσˆ
du
e
− 1  b ( v, µ ) f( v )ψ( u ) .
2π 
du
(24)
The form of the collision integral given by eq. (24) provides new opportunities for its consideration. A rather simple
generator, σ̂ , of the scattering group determines all general properties of the Boltzmann collision operator. We will
not further discuss this question here, but only point out that the product of Maxwellian velocity distributions is an
eigenfunction of operator σ̂ with zero eigenvalue:
σˆ e
−
m1v 2
2kT
e
−
m2u 2
2kT
30
=0
(25)
and that σ̂ commutates with the three invariants:
[ σˆ, v2 ] = 0,
[ σˆ, µ ] = 0,
[ σˆ, ( mv + u ) ] = 0 .
(26)
Also, from eqs. (20) and (16) we have useful formulas:
[ σˆ, n ⋅ v ] = 0 ,
[ σˆ, v ] = −n × v ,
[ σˆ, n × v ] = −n × n × v .
(27)
(28)
III. RENORMALIZATION OF THE BOLTZMANN COLLISION OPERATOR
In this section, we present an exact method of rewriting the Boltzmann collision integral in the divergence form
based on eq. (24). Within the framework of this method, we show how to reduce or extract the singularities
connected with infinite total collision cross sections related to small-angle scatterings. We consider the Coulomb
collisions in detail.
To separate the singular part of the collision integral from the regular one, we do not use the usual Taylor series
but rather the Taylor series with a residual term. For any function ϕ ( α ) with derivatives we can write
ϕ′ ( 0)
ϕ( n −1 ) ( 0 ) n −1
α +"+
α
+ Oˆn ϕ ( α ) ,
(n − 1)!
1!
α
n ∂ n

ˆ
On ϕ ( α ) =  ∫ d α 
ϕ (α ) .
 0
 ∂α
ϕ (α ) = ϕ (0) +
( )
where
(29)
(30)
From this, we can obtain the Taylor series of the operator e −φσˆ with the residual term
1
e
−φσˆ
( −1 )n −1 n −1 n −1
( −1 )n n n
1
φ σˆ
φ σˆ ∫ d αqn ( α ) e −αφσˆ , (31)
= 1 − φσˆ + φ2 σˆ2 + ... +
+
(n − 1)!
2!
n!
0
where n = 1, 2, " and
1
n −1
qn ( α ) = n ( 1 − α )
∫ d αqn
,
(
α ) = 1.
(32)
0
ˆ in the collision integral of eq. (24) is invariant under the replacement Rˆ → Rˆ−1 .
The measure b ( v, µ ) dR
Therefore, we can symmetrize in respect with this replacement the factor e −φσˆ in the integral:
∫ dRˆ b ( v, µ )e−φσ
ˆ
=
1
dRˆ b ( v, µ ) (e−φσˆ + e φσˆ )
2∫
(33)
Substituting eq.(31),(33) into eq. (24), we obtain renormalized divergent form of the Boltzmann collision integral:
I ( f , ψ) = −
∂
⋅ J ( f , ψ ) , (n > 0 ) ,
∂v
where the flow in the velocity space is given by
Ji ( f , ψ ) = −
+
Here the brackets
1
e
du vi
0
1 + m ii0i1 ∫
(34)
 2k <n 1

ˆi ...σ
ˆi +
φ2k ni ...ni σ
1
2
2k
2k
(
)
 k∑
2
k
!
=1
1
φn ni σˆn −1 (e αφσˆ + ( −1 )n e−αφσˆ )
1
2⋅n!
}f
(
v ) ψ (u ) .
(35)
denote averaging over the parameters of collision and the new parameter α .
... =
∫ dα
dφ
d Ωn qn ( α ) (1 − cos φ)b ( v, µ ) [ ... ] .
π
31
(36)
IV. EXAMPLES
Here we consider the general renormalized expression (34) (35) for the Boltzmann collision integral for the most
important cases of n = 0,1, 2 and 4, where n is the series termination number. The same letter n for the order
of the residual term and the unit vector n may not cause confusion.
CASE n = 0
I ( f , ψ) =
∫
du
 e φσˆ + e −φσˆ

dφ
d Ωn ( 1 − cos φ )b ( v, µ ) 
− 1 f ( v ) ψ ( u )

π
2

(37)
Equation (37) is the usual form of the Boltzmann collision integral that is expressed through the generator of the
scattering group, where the principle of detailed equilibrium is explicitly satisfied.
CASE n = 1 . Inserting the value n = 1 into general equations (34) (35), we have a renormalized equation
for the Boltzmann collision integral:
I ( f , ψ) = −
and for the flow in the velocity space:
J ( f, ψ) =
=−
1
1+m
1
1+m
∫ du
∫ du
[
∂
⋅ J ( f , ψ ),
∂v
{ 21 φ (e
αφσˆ
− e −αφσˆ )
(38)
}
f ( v )ψ ( u ) =
(39)
φ
n × v ] [ f ( v+′ α ) ψ ( u+′ α ) − f ( v−′ α ) ψ ( u−′ α ) ] .
2
(40)
n×v
where new ”post-collisional” velocities can be expressed as
v±′ α = v +
u ±′ α
[ ( 1 − cos αφ ) nˆ2 ± ( sin αφ ) nˆ ](v − u )
Recall that
df
,
1+m
m [ ( 1 − cos αφ ) nˆ2 ± ( sin αφ ) nˆ ] (v − u )
.
=u−
1+m
nˆv = n × v
... =
and
∫ dα
(41)
(42)
dφ
d Ωn (1 − cos φ)b ( v, µ ) [ ... ]
π
The Boltzmann equation can be rewritten in the form of the Liouville equation with the help of the general
divergence form of the collision integral (34):
e
e
1
∂f
∂f
∂
F )f = 0 .
+v⋅
+
⋅( 1 E + 1 v ×B + g +
m1c
m1 coll
∂t
∂r
∂v m1
(43)
Note that in addition to the usual electromagnetic and gravity forces, we have non-local friction force Fcoll , which
depends on the distribution functions f (v ) and ψ(v ) . In the case of n = 1 this force has the following
expression:
Fcoll =
−1
J
=
(
)
m1 f v
m1 + m2
∫
du
[
n × v]
φ [ f ( v+′ α ) ψ ( u +′ α ) − f ( v−′ α ) ψ ( u−′ α ) ]
2
f (v )
(44)
For the Coulomb collisions, the probability of collision is given by the measure
d φd Ωn
 e e 2
dRˆ
b ( v, µ ) = ( 1 + m )2  1 2 
,
 m1  π v 3 ( 1 − cos φ ) sin 4 θ
2π
for n = 1 we have q1 ( α ) = 1 , and the flow J takes the form
d φd Ωn d α

2
φ [n × v ]
2  e1e2 
J ( f , ψ ) = − (1 + m ) 
d
u
[ f+′ ψ+′ + f−′ ψ−′ ] .

∫
3
4

 m1 
π v ( 1 − cos φ ) sin θ 2 ( 1 + m )
32
(45)
(46)
Now we have an essentially new form of the collision integral. The collision integral is represented by the
divergence of the flow in the velocity space. A new parameter, 0 ≤ α ≤ 1 , that reduces the strength of collision
is introduced. The obtained equation may be convenient for Monte-Carlo simulations.
CASE n = 2
The flow in the velocity space is given by the following equation:
1 2
(47)
φ ( n × v ) σˆ (e αφσˆ + e −αφσˆ ) f ( v )ψ ( u ) =
4
φ2
−1
∂
∂
(n × v )(n × v ) ⋅
d
=
−m
u
[ f ′ ψ ′ + f−′ ψ−′ ] .
2 ∫
4
∂v
∂u + +
(1 + m )
For n = 2 , the probability function qn ( α ) for the reducing parameter α is q1 ( α ) = 1 − α . If we change
the probability function q1 ( α ) into the delta function δ ( α ) , what is far from reality, then we have the collision
J ( f, ψ) =
1
1+m
∫ du
(
)
integral in the Landau-Fokker-Planck form[11,12]:
I ( f, ψ) =
φ2 ( n × v ) ( n × v )
∂
∂
∂
f (v )ψ(u )
⋅ ∫ du
⋅
−m
2
∂v
∂v
∂u
2 (1 + m )
(
)
(48)
CASE n = 4
The collision integral becomes
I ( f , ψ) =
∫
 φ2 σˆ2
φ 4 σˆ4 αφσˆ
du 
e
+
+ e −αφσˆ )
(
 2 !
2 ⋅ 4!

 f ( v )ψ ( u ) ,

(49)
and the flow in velocity space is
Ji ( f , ψ ) = −
eii i
01
1+m
For the averages φ2ni ni
1
2
∫
 φ2n n
φ 4 ni

i1 i2
1
ˆsi +
d u vi 
σˆ3 (e αφσˆ + e −αφσˆ )
0 
2
2
!
2
4
!
⋅



 f ψ.


(50)
it is easily to obtain the following equations:
1 2
ˆi ,
φ sin2 θ σ
2
1 2
ˆi σ
ˆk =
ˆi σ
ˆi =
φ 2 ni nk σ
φ sin2 θ σ
2
φ2 sin2 θ  ∂
 ∂
∂  2
∂


m
v
δ
v
v
=
−
−
−m
(
)

i
ik
k
2  ∂v


∂ui 
∂uk
 ∂vk
2 (1 + m )  i
ˆk =
φ 2 ni nk σ
(51)

.

(52)
Now the Boltzmann collision integral is represented by the sum of the two terms:
φ2 sin2 θ

1
∂
2
 ∂ − m ∂
I ( f , ψ) =
du
v
v
v
δ
−
(
)
i k 
ik
2
∂uk
 ∂vk
2 ( 1 + m )2 ∂vi ∫
1
du φ 4 σ 4 (e αφσˆ + e −αφσˆ ) f ( v ) ψ ( u ) .
+
2 ⋅ 4! ∫

 f (v ) ψ (u ) +

(53)
For the Coulomb collisions, we have
φ2d φ d cos θ
φ2 sin2 θ

2
2  e1e2 
(
)
.
(54)
= 1+m 
 m1  ∫ v 3 ( 1 − cos φ ) sin2 θ
2
The integral in eq. (54) diverges logarithmically as θ → 0 or π . To avoid this, we introduce the cut-off
angle θmin and consider the range θmin < θ < π − θmin , which is usual for Coulomb collisions.
33
ˆi = 0 and [ σˆ, sin θ ] = 0 , we have:
Due to the transversability condition vi σ
 n⊥
4

ˆ ) = sin θ 
ˆ  ,
σˆ = ( n ⊥ ⋅ σ
⋅σ
 n ⊥

4
4
4
(55)
Different from the first term, the second term of eq. (53) does not diverge as θ → 0 or π . The first term is in
essence the Landau-Fokker- Plank collision integral [11]. The only difference is that in the present formulation we
have the new effective collision frequency
φ2 sin2 θ
2
instead of the transport collision frequency
1 − µ = ( 1 − cos φ ) sin2 θ introduced in ref. [11]. At small φ , which always means small scattering
cosine µ but not vice versa, the two frequencies coincide. For the present effective collision frequency we can
easily prove the inequality
1−µ <
φ2 sin2 θ
π2
1−µ .
<
2
4
(56)
The small difference between the present and previous [11], frequencies is not very important for practical physical
problems, but it is important that the lack of coincidence of these two frequencies proves that the Landau series[11,
12] is divergent.
In conclusion, it is worth emphasizing that the renormalized equations for the Boltzmann collision integral
obtained in the present work are not approximate expansions but exact ones that are especially useful when a total
collision cross section diverges and the term collision is badly determined.
REFERENCES
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