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Exact green function for the Lenard-Bernstein operator in the presence of a static magnetic field
PHYSICS
Volume 34A, number 7
EXACT
GREEN FUNCTION IN THE PRESENCE
19 April 1971
LETTERS
FOR THE LENARD-BERNSTEIN OF A STATIC MAGNETIC
OPERATOR FIELD
Y. FURUTANI Laboratoire
de Physique des Plasmas * , Facultk des Sciences CpOrsay,
Universitd
Paris XI, 91- Orsay, France
Received 12 March 1971
An exact Green function for the Lenard-Bernstein static magnetic field is obtained.
collision
operator in the presence
of an external
In the classical theory of the Brownian motion, Chandrasekhar [l] has found the joint probability p(x, v,tlxO, vo, 0) in the set of two dynamical random variables {x(t), v(t)). Apart from the purely academic interest of obtaining a complete description of the Brownian motion in the (x, v) space, one is required to know an exact joint probability or Green function for the equation
Lf, at
v.
3 + !? y ax
(1)
x B
7.i2
This Green function, useful in plasma kinetic theory, can adequately describe the propagation of “singular” initial conditions in velocity space, such as transverse velocity impulses, in the presence of small-angle collisions. Although the collision operator of the Ornstein-Uhlenbeck process, usually called the Lenard-Bernstein operator [2], does not account exactly for Coulomb collisions, it is frequently used in the litterature [2-51. In this letter we report a solution of eq. (1) in terms of the Green function fix, v, tl x0, vo, 0) f(x,
v, t) = j-ax0 dv,
P(x,
v,tlxO,
vo, 0) Ax,,
vo, 0) .
(2)
In eq. (l), fl and q are the effective collision frequency and the diffusion coefficient, both assumed to be independent of v. Other notations are customary. In eq. (2), f(x,, vo, 0) is an initial distribution function. as follows The solution of the form, slightly modified by inclusion of the B, -field, is expressed
4x9
Y,tlXo,vo,0) =2r(F
2(F,G,-H!-K2)’
where
S = v- exp(-Pt) h?= T-To-
with
* Laboratoire
F
112
I
’
R,,~.
v.
(S2+S2) w, x y -
Q, a ( 1 - ew (-Bt) Ruct)
eBO
wc =-,
F, S$ - 2 f&S.&
1 42) G 2 z
m
2(FzG,
(S&+SyRy)
1 h&w
+ GHz
(2a)2(F,_ G, -H! -K!)
-Hi)
+ 2K,(SyR,-S,Ry)
(3)
+G,i$+R;)~]
(4) (5)
- v.
Sl2 =-$-J
and512
=$.
C
associe’ au C.N.R.S. 357
Volume 34A, number 7
PHYSICS
LETTERS
19 April 1971
R, is the well-known rotation matrix given by: R,
= [iI:neO
tl
H] .
The quantities with indices z and _Lare referred to the B, - field which is directed to the z-axis. They are given as follows F,e = 4 {2Pt-(l-exP(-Pt))(3-exp(-pt))} P3 FL =-${2Pt-+
3p2402
P2 + 4exp(-Bt)p2,w2
C
H,
2
(coswct-?sinwcf)
,
Gz = $1~exp(-2Pt))
-exp(-2pt))
;
(6)
,
C
(l-2 exp (-Pt) co8 w,t+l)
=q
Hz = $(l-exp(-pt))2
,
K_I_ = pw,(1-2$cexp(-Pt)sinwct-exp(-2Bt)) P3
,
and
(7)
G, =$l+g)(1-exp(-2,%)). P2 The mean values are
(v(t)> = exp (-Bt) Ruct. v. ,
(8)
(x(t)> = x0+ Q,. (1 -exp (-Pt) Rwct) 1 v. .
(9)
The variances and the covariance are easily evaluates to be <{ v(t) - (v@))]~)
= s(l-exp(-2flt)) FL
<{x(f)
-
wt)>}2)= i
“,/P2)
0
0
0
Fu’U++~~)
0
0
0
Fz
/(l+w
I Hl
and ({x(t)
- (x(t))}{
(10)
1 = Gz 1
v(t) - (v(t))})
=
/(l+f.&P2) -K
(11)
/(l+w2/P2)
&1+&2)
KL /(l+wf/P2)
0
0
0 0
*
(12)
Hz The detailed resolution of eq. (1) and the application of eq. (3) to the problem of a linear response of a magnetoplasma to pulsed transverse excitations will be reported in a separate paper.
References [l] [2] [3] [4) [5]
S.Chandrasekhar, Rev. Mod.Phys. 15 (1943) 1. A. Lenard and Ira B. Bernstein, Phys. Rev. 112 (1958) 1456. V. I.Karpmsn, Soviet Phya. JETP (Engl. Transl. ) 24 (1967) 603. Y. H. Ichikawa and P. Suzuki, Progr. Theoret,Phys. 41 (1969) 313. Y. Furutani and J. Coste, J. Plasma Phys. 4 (1970) 843.
****
358
*
Volume 34A, number 7
EXACT
GREEN FUNCTION IN THE PRESENCE
19 April 1971
LETTERS
FOR THE LENARD-BERNSTEIN OF A STATIC MAGNETIC
OPERATOR FIELD
Y. FURUTANI Laboratoire
de Physique des Plasmas * , Facultk des Sciences CpOrsay,
Universitd
Paris XI, 91- Orsay, France
Received 12 March 1971
An exact Green function for the Lenard-Bernstein static magnetic field is obtained.
collision
operator in the presence
of an external
In the classical theory of the Brownian motion, Chandrasekhar [l] has found the joint probability p(x, v,tlxO, vo, 0) in the set of two dynamical random variables {x(t), v(t)). Apart from the purely academic interest of obtaining a complete description of the Brownian motion in the (x, v) space, one is required to know an exact joint probability or Green function for the equation
Lf, at
v.
3 + !? y ax
(1)
x B
7.i2
This Green function, useful in plasma kinetic theory, can adequately describe the propagation of “singular” initial conditions in velocity space, such as transverse velocity impulses, in the presence of small-angle collisions. Although the collision operator of the Ornstein-Uhlenbeck process, usually called the Lenard-Bernstein operator [2], does not account exactly for Coulomb collisions, it is frequently used in the litterature [2-51. In this letter we report a solution of eq. (1) in terms of the Green function fix, v, tl x0, vo, 0) f(x,
v, t) = j-ax0 dv,
P(x,
v,tlxO,
vo, 0) Ax,,
vo, 0) .
(2)
In eq. (l), fl and q are the effective collision frequency and the diffusion coefficient, both assumed to be independent of v. Other notations are customary. In eq. (2), f(x,, vo, 0) is an initial distribution function. as follows The solution of the form, slightly modified by inclusion of the B, -field, is expressed
4x9
Y,tlXo,vo,0) =2r(F
2(F,G,-H!-K2)’
where
S = v- exp(-Pt) h?= T-To-
with
* Laboratoire
F
112
I
’
R,,~.
v.
(S2+S2) w, x y -
Q, a ( 1 - ew (-Bt) Ruct)
eBO
wc =-,
F, S$ - 2 f&S.&
1 42) G 2 z
m
2(FzG,
(S&+SyRy)
1 h&w
+ GHz
(2a)2(F,_ G, -H! -K!)
-Hi)
+ 2K,(SyR,-S,Ry)
(3)
+G,i$+R;)~]
(4) (5)
- v.
Sl2 =-$-J
and512
=$.
C
associe’ au C.N.R.S. 357
Volume 34A, number 7
PHYSICS
LETTERS
19 April 1971
R, is the well-known rotation matrix given by: R,
= [iI:neO
tl
H] .
The quantities with indices z and _Lare referred to the B, - field which is directed to the z-axis. They are given as follows F,e = 4 {2Pt-(l-exP(-Pt))(3-exp(-pt))} P3 FL =-${2Pt-+
3p2402
P2 + 4exp(-Bt)p2,w2
C
H,
2
(coswct-?sinwcf)
,
Gz = $1~exp(-2Pt))
-exp(-2pt))
;
(6)
,
C
(l-2 exp (-Pt) co8 w,t+l)
=q
Hz = $(l-exp(-pt))2
,
K_I_ = pw,(1-2$cexp(-Pt)sinwct-exp(-2Bt)) P3
,
and
(7)
G, =$l+g)(1-exp(-2,%)). P2 The mean values are
(v(t)> = exp (-Bt) Ruct. v. ,
(8)
(x(t)> = x0+ Q,. (1 -exp (-Pt) Rwct) 1 v. .
(9)
The variances and the covariance are easily evaluates to be <{ v(t) - (v@))]~)
= s(l-exp(-2flt)) FL
<{x(f)
-
wt)>}2)= i
“,/P2)
0
0
0
Fu’U++~~)
0
0
0
Fz
/(l+w
I Hl
and ({x(t)
- (x(t))}{
(10)
1 = Gz 1
v(t) - (v(t))})
=
/(l+f.&P2) -K
(11)
/(l+w2/P2)
&1+&2)
KL /(l+wf/P2)
0
0
0 0
*
(12)
Hz The detailed resolution of eq. (1) and the application of eq. (3) to the problem of a linear response of a magnetoplasma to pulsed transverse excitations will be reported in a separate paper.
References [l] [2] [3] [4) [5]
S.Chandrasekhar, Rev. Mod.Phys. 15 (1943) 1. A. Lenard and Ira B. Bernstein, Phys. Rev. 112 (1958) 1456. V. I.Karpmsn, Soviet Phya. JETP (Engl. Transl. ) 24 (1967) 603. Y. H. Ichikawa and P. Suzuki, Progr. Theoret,Phys. 41 (1969) 313. Y. Furutani and J. Coste, J. Plasma Phys. 4 (1970) 843.
****
358
*