An. Şt. Univ. Ovidius Constanţa
Vol. 11(1), 2003, 105–112
THE USE OF LAGUERRE-SONINE
POLYNOMIALS IN SOLVING
BOLTZMANN’S EQUATION (II)
Gheorghe Lupu
To Professor Silviu Sburlan, at his 60’s anniversary
Abstract
Generally, one can solve Botzmann’s equation only by using approximation methods, and the expansion of the distribution function in spherical harmonics. In this paper, we shall study Boltzmann’s equation for
a fully ionised inhomogeneous plasma with Laguerre-Sonine polynomials as coefficients of the spherical harmonics expansion. We establish
also the cross-coupling relations between Laguerre-Sonine polynomials
of different orders, useful relations in order to obtain the approximative
solutions of Boltzmann’s equation.
1. Introduction
We base our considerations on the following assumptions:
(i) The Laguerre-Sonine polynomials are of the form:
l+ 21
Lr
p2 v 2 .
(ii) The weight function coincides with the equilibrium distribution function of electrons.
For Boltzmann’s equation for electrons, we take the following form :
→
−
−
∂f →
e →
1 −
→
v × B ∇v f =
+−
v ∇r f +
E+
∂t
me
c
= Nv
Z
Z
0 0
0
→
−
2 0
f ( v )σ v , χ d Ω − N v f σ (v, χ) d2 Ω .
105
(1)
106
Gh. Lupu
In order to solve eq. (1) in agreement with (i) and (ii) , we assume for
distribution function the following expansion:
+∞ +l +∞
−
→
− β 3 (→
r , t) −β 2 v2 X X X l,m →
l+ 1
−
Rr ( −
r , t) Lr 2 β 2 v 2 Yl,m (0, ϕ)
e
×
f →
r ,vΩ,t =
3
π2
i=0 m=−l r=0
(2)
where:
r
m
β=
,
−
2kT (→
r , t)
l+ 1
Lr 2 β 2 v 2 are the Laguerre-Sonine polynomials:
l+ 1
Lr 2
2 2
β v
=
eβ
2 2
v
β 2 v2
r!
−(l+ 12 )
h
i
dr
2 2r+l+ 21 −β 2 v 2
β
v
e
d (β 2 v 2 )
(3)
and Yl,m (θ, ϕ) are the spherical harmonics.
A development of the distribution function only in spherical harmonics
permits to ensure that the Maxwellian character of the equilibrium
distribution function be not altered by the anisotropy due to the presence of
both electric and magnetic fields, even in the relativistic case. The assumed
double expansion (2) maintains this property. Indeed, for m = l = r = 0, one
obtains the maxwellian distribution function.
h 1
i
−
−
L02 = Y0,0 = 1
and
R00,0 (→
r , t) = n0 (→
r , t)
2. Derivation of the cross-coupling relations
Substituting (2) into (1), we obtain:
Al,m
t
=
1
(l + m + 1) (l + m + 2) ·
−
2 (2l + 3)
e
l l+1,m+1
l+1,m+1
l+1,m+1
l+1,m
· vAx
− iAy
−
−
(Ez − iEy )
C
− Dv
me
v
l l+1,m
c
1
(l + m + 1) vAl+1,m
−
Ez
C
− Dvl+1,m +
−
z
2l + 3
me
v
1
v Al+1,m−1
+ iAl + 1, m − 1y −
z
2 (2l + 3)
e
l l+1,n−1
l+1,m−1
−−
(Ez + iEy )
C
− Dv
+
me
v
+
107
The use of Laguerre-Sonine polynomials
1
(l − m − 1) (l − m) v Al−1,m+1
− iAl−1,m+1
+
z
y
2 (2l − 1)
e
l + 1 l−1,m+1
l−1,m+1
+
−
C
+ Dv
(Ez − iEy )
me
v
l + 1 l−1,m
e
1
l−1,m
l−1,m
−
(l − m) vAz
+
C
+ Dv
Ez
−
2l − 1
me
v
l
1
e
l−1,m−1
l−1,m−1
l−1,m−1
Cl−1,m−1 + Dv
(Ez + iEy )
+
−
v Ax
+ iAy
+
2 (2l − 1)
me
v
+
e
e i
(Bx − iBy ) C l,m + m
iBz C l,m −
me c 2
me c
e
(l + m)!
l,m−1
−
i (Bx − iBy ) C
+ 4πN v
σl − σ0 C l,m ,
me c
(l − m)!
+ (l + m + 1) (l − m)
(4)
where we have written:
Al,m
x
3 −β 2 v 2
−β e
2 2
= 3 − 2β v
∞
X
∂Rl,m
r
r=0
∂xi
l+ 1
Lr 2
∞
∂β −β 2 v2 X l,m l+ 21
R r Lr −
e
β
∂xi
r=0
2
4 2 −β 2 v 2
+ 2β v e
∞
X
l+ 1
Rrl,m
r=0
∂Lr 2 ∂β
∂ (β 2 v 2 ) ∂x1
(5)
(here i = 1, 2, 3, 4, so : x1 = x, x2 = y, x3 = z, x4 = l)
C l,m = β 3 e−β
2 2
v
∞
X
l+ 12
Rrl,m Lr
(6)
r=0
and
Dvl,m
5
= 2β ve
−β 2 v 2
∞
X
r=0
l+ 1
Rrl,m
∂Lr 2
l+ 1
− Lr 2
2
2
∂ (β v )
!
.
(7)
Taking into account the following recurrence relations for Laguerre polynomials:
l
2 2
3 2 ∂Lr β v
(8)
β v
= rLlr β 2 v 2 − (r + l) Llr−1 β 2 v 2
∂ (β 2 v 2 )
and
β 2 v 2 Llr β 2 v 2 = (2r + l + 1) Llr β 2 v 2 −
− (r + l) Llr−1 β 2 v 2 − (r + 1) Llr+1 β 2 v 2 ,
(9)
108
Gh. Lupu
the expressions (5) and (7) become:
2
Al,m
xi = 2β
∞
i
∂β −β 2 v2 X l,m h
l+ 1
l+ 1
(r + 1) Lr+12 − (r + l) Lr 2 +
Rr
e
∂xi
r=0
+β 3 e−β
2 2
v
∞
X
∂Rl,m
0
r=0
and
Dvl,m = 2β 3 e−β
2 2
v
∂x
l+ 21
0
(5 )
Lr
∞
X
1 l+ 1
1
l+ 1
l+ 1
Rrl,m − rLr 2 + β 3 vLr 2 + (r + l) Lr−12 .
v
v
r=0
0
0
(7 )
0
Substituting (5 ) and (7 ) into (4), we obtain a set of eqs. which contains
Laguerre-Sonine polynomials corresponding to the indices l + 21 , l + 23 , l − 21 .
In order to utilize the orthogonality relation for these polynomials
Z
l+ 1
2 2
l+ 1
e−β v Lr 2 β 2 v 2 Lk 2 d β 2 v 2 =
=
(
0,
f or
Γ(r+l+ 23 )
r!
k 6= 0
f or
(10)
k=r ,
0
they must have the upper index. Multiplying now (5’),(6) and (7 ) by v, then
applying relations:
β 2 v 2 Llr β 2 v 2 = (r + l) Ll−1
β 2 v 2 − Ll−1
β 2 v2
(11)
r
r
and
2 2
β 2 v 2 Llr β 2 v 2 = 3 (r + l + 2) Ll+1
−
β 2 v 2 − (3r + 2l + 1) Ll+1
r
r−1 β v
2 2
2 2
(12)
+ (r + l) Ll+2
− (r + 1) Ll+1
r+2 β v
r+1 β v
we obtain for (4) an expansion which contains Laguerre-Sonine polynomials of
l+ 1
l+ 1
the same upper index. Multiplying this result by β 2 v 2 2 d β 2 v 2 Lk 2 and
subsequently performing the integration over the interval (0, ∞) and summing
relatively to the index mk, we obtain the following infinite set of equations:
2β 2
∂Rrl,m l
∂β l,m (r + 1) Llr+1,k+1 − (r + l) Llr,k + β 3
Rt
Lr =
∂t
∂t
X + (l + m + 1) (l + m + 2) 1
2 (2l + r)
±
−1
109
The use of Laguerre-Sonine polynomials

 X

−
X
"
j
(−1) Bj β
j=0,1
(−1)
j
l+1,m±1
<j Rr+j
j=0,1,2

+
X
(−1)
j=−2,−1,0,1,2
j
X
"
(−1) D2+j β
j=−2,−1,0,1
∂β
∂β
∓i
∂x
∂y
l+1,m±1
l+1,m±1
∂Rr+j
∂Rr+j
∓
∂x
X
1
+
2 (2l − 1) ±


j
!
#

2e 3
l+1,m±1
+
β (Ez ∓ iEy ) Rr+j
+

me
+ (l − m − 1) (l − m)
−1
l−1,m±1
U2+j Rr+j
−
∂β
∂β
∓
∂x
∂y
l−1,m±1
l−1,m±1
∂Rr+j
∂Rr+j
∓i
∂x
∂y
!
·
+
#

e 3
l−1,m±1
β (Ex ∓ iEy ) Rr+j
+
−2

me

l−1,m
∂R
1
r+j
j
j
l−1,m ∂β
+
+
(−1) D2+j
(l − m) 
+
(−1) U2+j Rr+j
2l − 1
∂z
∂z
j=−2,−1,0,1
j=−2,−1,0,1,2

ieβ 3 l
I
+
me c r,k
+
X
±
X
X
)
(
X + (l + m + 1) (l − m) l,m
l,m±1
+ mBz Rr
· (Bz ∓ iBy ) Rr
+
−1
±
0 #
(l+m)! 4π
X
+ (l−m)!
N βσe v
j
l,m
2l+1
,
(−1) − U1+j Rr+j
+
−4πN σ0
j=−1,0,1
"
where we have set:

Γ(r+l+ 23 )


+ l + 1) r + l + 23

r!
 <0 = 2 (r
Γ(r+l+ 52 )
<1 = 2 (r + l + 1) + (r + 1) r + l + 52
(r+1)!


7

 <2 = 2 (r + 1) Γ(r+l+ 2 )
(r+2)!

 B = r + l + 3 Γ(r+l+ 32 )
0
2
r!
5
 B1 = Γ(r+l+ 2 )
(r+1)!
(13)
(14)
(15)
110
Gh. Lupu

Γ(r+l− 12 )


U0 = 2 (r + l − 1) r + l − 21

(r−2)!


Γ(r+l+ 1 )

1


U1 = 2 (r + 1) r + l + 2 + (r + l − 1) (3r + 2l) (r−2)!2


Γ(r+l− 21 )
U2 = 2 (r + 1) r + l + 12 + (r + l − 1) 3r + l + 32
(r−1)!


5


7 Γ(r+l+ 2 )

U3 = 2 (r + 1) r + l + 4

(r+1)!



 U = 2 (r + 1) (r + 2) Γ(r+l+ 72 )
4
(r+2)!

1

1 Γ(r+l− 2 )

D
=
r
+
l
−
0

2
(r−2)!



 D = (3r + 2l) Γ(r+l+ 21 )
1
(r−1)!
3

3 Γ(r+l+ 2 )

D
=
3r
+
l
+

2
2
r!


5

 D = U = (r + 1) Γ(r+l+ 2 )
3
2
(r+1)!

 U = r + l + 1 Γ(r+l+ 21 )
0
2
(r+1)!
 U = 2r + l + 3 Γ(r+l+ 23 )
1
and
l
Ir,k
=
Z
2
∞
ve−β
2 2
β 2 v2
v
0
l+ 12
(16)
(17)
(18)
r!
l+ 21
Lr
l+ 12
Lk
d β 2 v2 .
(19)
3. Application
In order to compute the integral (19)
Z ∞
l+ 1 l+ 1 l+ 1
2 2
l
ve−β v β 2 v 2 2 Lr 2 Lk 2 d β 2 v 2 ,
Ir,k
=
0
we use the following property of Laguerre polynomials:
Z ∞
0
α1 +α2
e−z(S+ 2 ) xµ+β Lµk (a1 x) Lµr (a2 x) dx =
0
=

0,


 Γ



where
k 6= r 
 
0
0
1+µ+β
µ+β
A2


F
,1+
,1+µ,
1+µ+β Γ(1+µk)
2
2
B2
dk 

× dh
1+µ 1+µ+β
h
k!k!Γ(1+µ)
(1−h)
B


f or
0
A2 =
4a1 a2 h
(1 − h)
2;B
=S+
a1 + a 2 1 + h
2
1−h
, f or k = r,
h=0
(A.1)
(A.2)
111
The use of Laguerre-Sonine polynomials
and
0
0
F α, β , γ, z
=
0
0
β +1
z2 +
γ (γ + 1) 1.2
0
0
0
α (α + 1) (α + 2) β β + 1 β + 2
1+
+
αβ
z+
γ1
α (α + 1) β
γ (γ + 1) (γ + 2) 1.2.3
The equality (A.1) is satisfied if and only if:
a1 + a 2
Re S +
> 0,
a1 > 0,
a2 > 0,
2
z 3 + ...
(A.3)
0
Re µ + β > −1.
With (A.1), we write:
Z
1 ∞ −β 2 v2 2 2 l+ 12 + 12 l+ 12 l+ 21
l
e
β v
Ir,k =
Lr Lk d β 2 v 2 .
β 0
(A.4)
(19’)
Let us note that:
s = 0; a1 = a2 = 1; µ = l +
0
1
1
4h
1+h
, β = ; A2 =
.
;B =
2
2
2
1−h
(1 − h)
(A.5)
If we substitute (A.5) in (A.4), then all the inequalities are verified. Taking
into consideration (A.5), we obtain for (A.1) the following expression:
Γr,k = 0
f or
k 6= r
and


 
3
4h
l+2 l+3

;
;
l
+
;
F
r
2
2
2
2 (1+h)
1 Γ (l + 2) Γ l + r +
d 
l

Irl = Ir,r
=
.
·
−1
l+2
 dhr

β
r!r!Γ l + 32
(1 − h) 2 (1 + h)
h=0
(A.6)
The hypergeometric function F in this case has the form :
3
2
F = F1 = 1 +
∞
X
l+p+1
2 p hp
.
[2 (l + p + 1) − 1] p! (1 + h)2p
p=1
(A.7)
For the particular case of Section 3, we have:
I00 =
1
β0
and
I01 =
5
.
β0
(A.8)
We will note also that, because of the form of z (and hence of A and
B), the integrals Irl , for all finite values of r and l, have a finite number of
terms, although the hypergeometric function has been given by a series with
an infinite number of terms.
112
Gh. Lupu
References
[1] LUPU Gh.: ”The hypergeometric function in the study of collision integral of the
Boltzmann’s equation”,- Proceedings of ICIAM-99, Edinburgh, p. 150.
[2] HASSANI S. ”Mathematical Physics - A modern introduction to its Foundations” Edit. Springer-Verlag, 1998.
[3] LUPU Gh. and LEAHU A.: ”The use of Laguerre-Sonine polynomials in solving
Boltzmann’s equation (I)”, Sixième colloque Franco-Roumain de Mathématiques Appliquées - 2002, Perpignan France, p. 125.
”Ovidius” University of Constanta,
Faculty of Mathematics and Informatics,
B-dul Mamaia 124,
8700 Constantza,
Romania
e-mail: ghlupu@univ-ovidius.ro