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A note on model kinetic equations
Volume 24A, number 3
PHYSICS
LETTERS
T h e s e c a s e s c o v e r i n g a l l t h e i n i t i a l l i n k s * of fig. 1, it i s n a t u r a l to a n t i c i p a t e t h a t a l i k e d e g r e e of a c c u r a c y w i l l be a t t a i n e d w h e n a p p l y i n g the f o r m u l a (1) to l i n k s of h i g h e r o r d e r (i.e., with N > 3). W o r k is b e i n g u n d e r t a k e n to v e r i f y this c o n j e c t u r e .
30 January 1967
T h e f u l l a c c o u n t of the p r e s e n t c o n s i d e r a t i o n s w i l l be p u b l i s h e d e l s e w h e r e .
* Other examples of Bopp (He -~ C 2+ and Be 2+ --. Mg8+), exhibiting an e r r o r of about 10%, are not covered by the present theorem (1) as they involve passing from N, = 2 to N = 4 (or, omitting one member of a chain, fig. 1.
References 1. F. Bopp, Z. Physik 156 (1959) 348. 2. A.J. Coleman, Rev. Mod. Phys. 35 (1963)668. 3. D. Ter Haar, Rept. Prog. Phys. 24 (1961)304.
* * * * *
A NOTE
ON MODEL
KINETIC
EQUATIONS
I. F I D O N E and G. G R A N A T A
Groupe de Recherches de l'Association Euratom-CEA sur la Fusion Fontenay-aux-Roses (Seine), France Received 13 December 1966
In this note an attempt is made to obtain an explicit expression for the collision constants appearing in the isotropic Fokker-Planck model kinetic equations.
T o t a k e into a c c o u n t c o l l i s i o n e f f e c t s in t h e study of s m a l l a m p l i t u d e o s c i l l a t i o n s in a p l a s m a , m o d e l k i n e t i c e q u a t i o n s h a v e b e e n u s e d which m a y be s u p p o s e d to g i v e q u a l i t a t i v e l y c o r r e c t r e s u l t s [1-3]. Both the m o d e l of B h a t n a q a r et al. [1] and the i s o t r o p i c F o k k e r - P l a n c k [2,3] m o d e l c o n t a i n a g r o u p of i n d e t e r m i n e d c o n s t a n t s , l o o s e l y c a l l e d e f f e c t i v e c o l l i s i o n f r e q u e n c i e s . It is v e r y u n l i k e l y that a m o d e l e q u a t i o n , in w h i c h the c o m p l i c a t e d c o l l i s i o n p r o c e s s i s d e s c r i b e d by v e l o c i t y i n d e p e n d e n t c o n s t a n t s , can be d e d u c e d in a s a t i s f a c t o r y way f r o m t h e B o l t z m a n n i n t e g r o - d i f f e r e n t i a l e q u a t i o n or, in the c a s e of a fully i o n i z e d g a s , f r o m the F o k k e r - P l a n c k e q u a t i o n . N e v e r t h e l e s s , in t h i s n o t e an a t t e m p t , to s o m e e x t e n t a r b i t r a r y , i s m a d e to o b t a i n an e x p l i c i t e x p r e s s i o n f o r the c o l l i s i o n c o n s t a n t s in t e r m s of u s u a l p a r a m e t e r s l i k e t e m p e r a t u r e , m a s s o r d e n s i t y of t h e p a r t i c l e s f o r m i n g t h e p l a s m a . L e t us c o n s i d e r a b i n a r y p l a s m a of i o n s (i) and e l e c t r o n s (e). T h e c o l l i s i o n t e r m in t h e F o k k e r - P l a n c k form is written.
I
(i)
where ~,fl m e a n ions, electrons. The diffusion and friction coefficients are given by
..
4,e'ae:
(mafiv2afl ~
_
"
"
(2)
,. eJ m
m
175
Volume 24A. number 3
PHYSICS
LETTERS
30 jtmuaLv 1967
w h e r e vail = Va - v/3, rna~ = m a m ~ / ( m a + m ~ ) a n d K D1 = D e b y e d i s t a n c e We now a s s u m e that the t e n s o r ( vaZAvaJ} m a y be r e p l a c e d by its v a l u e a v e r a g e d o v e r the d i s t r i b u t i o n f a and d e f i n e a n a v e r a g e d i f f u s i o n t e n s o r Dij by Dij = ] f a (AvaiAv33}dva" T o o b t a i n an e x p l i c i t f o r m for Dij we r e f e r to a p l a s m a with a v e l o c i t y d i s t r i b u t i o n for each c o m p o n e n t g i v e n a p p r o x i m a t e l y by a d i s p l a c e d M a x w e l l i a n , the m e a n v e l o c i t y and the t e m p e r a t u r e b e i n g d e f i n ed by
Ua : (I/nc*)./"v a f a
~-naTa : { - m a f
dva,
(va -
Ua)2fa dVa,
"a
=J'f~ d v a
M o r e o v e r , the m e a n v e l o c i t y u is s u p p o s e d s m a l l e r than the t h e r m a l v e l o c i t y . F o r Dij we have 41re 2
Dij ~
m2a
2
[ m~m/3_) "/3
" j'j
d v a dv~ In
\(2~) ~ :/~ 7~
P u t t i n g T a r = (m a
~,i vj \ (ma/3v2{~](5i: a~__,/3)ex-p . . . . . . . \eae/3KD~ \va3 va3 1
___ v2 - _fi ,2
Tfi+m/3 Ta )/(m a +m/3) the f i n a l r e s u l t is
2T~ :
Dij
ma
2
2
eae~9 J 2 ~ m a f i P'al~ : 3 3 m a T a~ 23 In d i s c u s s i n g the f r i c t i o n c o e f f i c i e n t
(3)
n/3 ~taflSij, ( T
~ -,
\e a e~ K D /
(Av a} we note that in g e n e r a l withffl =ffl(Iv - urt ) we have
( A v a }= 1 - (v a
-ufi) O(]v a - u~i) ,
(5)
w h e r e 0 is a g i v e n f u n c t i o n d e p e n d i n g on the s t r u c t u r e off~3. S i n c e a n a v e r a g i n g p r o c e s s on eq. (5) is difficult to d e f i n e we c h o o s e for ¢ a c o n s t a n t v a l u e given by 0 = n3 tzaB ~?aB w h e r e Ta B does not d e p e n d on v a and will be d e t e r m i n e d in s u c h a way to s a t i s f y the g e n e r a l p r o p e r t i e s of Sa/3. ' P u t t i n g eqs. (3) and (5) in eq. (1) we get the f o l l o w i n g m o d e l e q u a t i o n dr- =
nfl ~a/3 ~ a
" ~ ma~ Ova + r/a/3 (va
Let u s now r e c a l l the g e n e r a l p r o p e r t i e s of S^.o. ~p T h e i n t e g r a l m a v a , ~ m a v a 2 , s a t i s f i e s the f o l l o w i n g r e l a t i o n s : (a) Ia/3(1) = 0,
(b) I a (qa) = 0,
(c)
(6)
Ia/3=j d v a q a S a f i ,
Ia/3(qa) + I/3a (q~) = 0
w h e r e qa = 1,
a ~ ~?
C o n d i t i o n s (a), (b), and (c) can be s a t i s f i e d by (6) p r o v i d e d that 3T =
77a/3 = ~/3a
_
a/3
_
(7)
3 T a ~ + ma/3 (ua_03)2__
with ~?a~ g i v e n by (7) Sa/3 = 0 in e q u i l i b r i u m a s it m u s t be. We r e m a r k that 77a,~ ,~ 1 even when (u a - u . ) ¢ 0 b e c a u s e the c o n d i t i o n 3 T a / 3 >> ma/3 (u a -U•) 2, ~ " T h e c o l l i s i o n c o n s t a n t s tta/3 g i v e n by eq. (4J a r e of the s a m e f o r m we m e e t in r i g o r o u s c a l c u l a t i o n s of k i n e t i c t h e o r y . To show this we d i s c u s s two s i m p l e e x a m p l e s c o n c e r n i n g a h o m o g e n e o u s i s o t r o p i c p l a s m a . We f i r s t c o n s i d e r u e = u i = 0 a n d c a l c u l a t e d T a / d t due to c o l l i s i o n s with/3 p a r t i c l e s .
176
Volume 24A, number 3
P HY SI CS L E T T E R S
30 January 1967
F r o m (6) we get dT~ -
-
2% =
n~/~afl (Ta - T ~ )
-
dt
(8)
+%
%
w h i c h c o i n c i d e s with S p i t z e r ' s f o r m u l a [4]. N e x t we c a l c u l a t e the m e a n e l e c t r o n v e l o c i t y u e u n d e r the e f f e c t of an e l e c t r i c f i e l d E = E o e x p ( - i w t ) . A g a i n u s i n g eq. (6) we h a v e due eE dt - m e - n i ~ e i T / e i ( U e
-ui)
P u t t i n g ~/ei ~ 1 and u e >>u i we get e 4 In (T e / e 2 K D)
e E e - me ni/Zei - iw'
/Zei ~ ~-
(9) m½T3 e e
C o m p a r i n g (9) with t h e c o r r e s p o n d i n g e x p r e s s i o n o b t a i n e d u s i n g t h e c o r r e c t f o r m (1) of S a ~ [5] we c o n c l u d e t h a t n itz ei i s t h e c o l l i s i o n f r e q u e n c y f o r m o m e n t u m e x c h a n g e f o r a p r o c e s s v a r y i n g with a f r e q u e n c y w > > n i / ~ e i . T h i s l e a d s to the c o n c l u s i o n that eq. (6) with ~ f l f r o m (4) could be a p p l i e d to s m a l l a m p l i t u d e o s c i l l a t i o n s with f r e q u e n c y g r e a t e r than nfi ~c~fl" References 1. P . L . B h a t n a g a r , E . P . G r o s s and M. Krook, Phys. Rev. 94 (1954)511. 2. J . P . D o u g h e r t y , Phys. Fluids 7 (1964)1788. 3. A.Oppenheim, Phys. Fluids 8 (1965) 900. 4. L.Spitzer J r . , Physics of fully ionized gases, (Interscience Publishers, Inc. New York, 1956). 5. V.L.Ginzburg, Propagation of electromagnetic waves in plasma, (Gordon and Breach Science Publishers, Inc. New York, 1961).
A NEW
LAW
IN
ELECTRODYNAMICS
O. C O S T A DE B E A U R E G A R D Institut Henri Poincare , Paris, France Received 31 December 1966
As a consequence of the ,Laulace" force RE xl] applied to a moving magnetic charge R, a varying dipole of moment 34 undergoes a force E × d34/dt, a n d a slowly varying current of intensity i a force (di/dt)~ VSl, where V denotes the scalar potential.
Let us recall first that a straightforward argum e n t of r e l a t i v i s t i c c o v a r i a n c e s h o w s that a m a g n e t i c p o i n t c h a r g e R ( e . m . u . ) , w h i c h by d e f i n i t i o n u n d e r g o e s , w h e n at r e s t , t h e C o u l o m b f o r c e R / ' / m u s t , when in m o t i o n with v e l o c i t y cfl, u n d e r g o the "Lorentz" force
F=R(Exfl +H).
(1)
It f o l l o w s t h e n t h a t a m a g n e t i c d i p o l e m a d e of two
p o l e s +R and -R with s e p a r a t i o n a and m o m e n t M =- Ra, s i t u a t e d at a g i v e n p o i n t in a s t a t i c e l e c tric field E, undergoes when M changes a force F = E × dM/dt.
(2)
If we a s s u m e f o r s i m p l i c i t y t h a t t h e s o u r c e of E is a single point charge Q with separation r from t h e d i p o l e ' s c e n t e r , t h e n one can r e w r i t e eq. (2) in t h e f o l l o w i n g f o r m 177
PHYSICS
LETTERS
T h e s e c a s e s c o v e r i n g a l l t h e i n i t i a l l i n k s * of fig. 1, it i s n a t u r a l to a n t i c i p a t e t h a t a l i k e d e g r e e of a c c u r a c y w i l l be a t t a i n e d w h e n a p p l y i n g the f o r m u l a (1) to l i n k s of h i g h e r o r d e r (i.e., with N > 3). W o r k is b e i n g u n d e r t a k e n to v e r i f y this c o n j e c t u r e .
30 January 1967
T h e f u l l a c c o u n t of the p r e s e n t c o n s i d e r a t i o n s w i l l be p u b l i s h e d e l s e w h e r e .
* Other examples of Bopp (He -~ C 2+ and Be 2+ --. Mg8+), exhibiting an e r r o r of about 10%, are not covered by the present theorem (1) as they involve passing from N, = 2 to N = 4 (or, omitting one member of a chain, fig. 1.
References 1. F. Bopp, Z. Physik 156 (1959) 348. 2. A.J. Coleman, Rev. Mod. Phys. 35 (1963)668. 3. D. Ter Haar, Rept. Prog. Phys. 24 (1961)304.
* * * * *
A NOTE
ON MODEL
KINETIC
EQUATIONS
I. F I D O N E and G. G R A N A T A
Groupe de Recherches de l'Association Euratom-CEA sur la Fusion Fontenay-aux-Roses (Seine), France Received 13 December 1966
In this note an attempt is made to obtain an explicit expression for the collision constants appearing in the isotropic Fokker-Planck model kinetic equations.
T o t a k e into a c c o u n t c o l l i s i o n e f f e c t s in t h e study of s m a l l a m p l i t u d e o s c i l l a t i o n s in a p l a s m a , m o d e l k i n e t i c e q u a t i o n s h a v e b e e n u s e d which m a y be s u p p o s e d to g i v e q u a l i t a t i v e l y c o r r e c t r e s u l t s [1-3]. Both the m o d e l of B h a t n a q a r et al. [1] and the i s o t r o p i c F o k k e r - P l a n c k [2,3] m o d e l c o n t a i n a g r o u p of i n d e t e r m i n e d c o n s t a n t s , l o o s e l y c a l l e d e f f e c t i v e c o l l i s i o n f r e q u e n c i e s . It is v e r y u n l i k e l y that a m o d e l e q u a t i o n , in w h i c h the c o m p l i c a t e d c o l l i s i o n p r o c e s s i s d e s c r i b e d by v e l o c i t y i n d e p e n d e n t c o n s t a n t s , can be d e d u c e d in a s a t i s f a c t o r y way f r o m t h e B o l t z m a n n i n t e g r o - d i f f e r e n t i a l e q u a t i o n or, in the c a s e of a fully i o n i z e d g a s , f r o m the F o k k e r - P l a n c k e q u a t i o n . N e v e r t h e l e s s , in t h i s n o t e an a t t e m p t , to s o m e e x t e n t a r b i t r a r y , i s m a d e to o b t a i n an e x p l i c i t e x p r e s s i o n f o r the c o l l i s i o n c o n s t a n t s in t e r m s of u s u a l p a r a m e t e r s l i k e t e m p e r a t u r e , m a s s o r d e n s i t y of t h e p a r t i c l e s f o r m i n g t h e p l a s m a . L e t us c o n s i d e r a b i n a r y p l a s m a of i o n s (i) and e l e c t r o n s (e). T h e c o l l i s i o n t e r m in t h e F o k k e r - P l a n c k form is written.
I
(i)
where ~,fl m e a n ions, electrons. The diffusion and friction coefficients are given by
..
4,e'ae:
(mafiv2afl ~
_
"
"
(2)
,. eJ m
m
175
Volume 24A. number 3
PHYSICS
LETTERS
30 jtmuaLv 1967
w h e r e vail = Va - v/3, rna~ = m a m ~ / ( m a + m ~ ) a n d K D1 = D e b y e d i s t a n c e We now a s s u m e that the t e n s o r ( vaZAvaJ} m a y be r e p l a c e d by its v a l u e a v e r a g e d o v e r the d i s t r i b u t i o n f a and d e f i n e a n a v e r a g e d i f f u s i o n t e n s o r Dij by Dij = ] f a (AvaiAv33}dva" T o o b t a i n an e x p l i c i t f o r m for Dij we r e f e r to a p l a s m a with a v e l o c i t y d i s t r i b u t i o n for each c o m p o n e n t g i v e n a p p r o x i m a t e l y by a d i s p l a c e d M a x w e l l i a n , the m e a n v e l o c i t y and the t e m p e r a t u r e b e i n g d e f i n ed by
Ua : (I/nc*)./"v a f a
~-naTa : { - m a f
dva,
(va -
Ua)2fa dVa,
"a
=J'f~ d v a
M o r e o v e r , the m e a n v e l o c i t y u is s u p p o s e d s m a l l e r than the t h e r m a l v e l o c i t y . F o r Dij we have 41re 2
Dij ~
m2a
2
[ m~m/3_) "/3
" j'j
d v a dv~ In
\(2~) ~ :/~ 7~
P u t t i n g T a r = (m a
~,i vj \ (ma/3v2{~](5i: a~__,/3)ex-p . . . . . . . \eae/3KD~ \va3 va3 1
___ v2 - _fi ,2
Tfi+m/3 Ta )/(m a +m/3) the f i n a l r e s u l t is
2T~ :
Dij
ma
2
2
eae~9 J 2 ~ m a f i P'al~ : 3 3 m a T a~ 23 In d i s c u s s i n g the f r i c t i o n c o e f f i c i e n t
(3)
n/3 ~taflSij, ( T
~ -,
\e a e~ K D /
(Av a} we note that in g e n e r a l withffl =ffl(Iv - urt ) we have
( A v a }= 1 - (v a
-ufi) O(]v a - u~i) ,
(5)
w h e r e 0 is a g i v e n f u n c t i o n d e p e n d i n g on the s t r u c t u r e off~3. S i n c e a n a v e r a g i n g p r o c e s s on eq. (5) is difficult to d e f i n e we c h o o s e for ¢ a c o n s t a n t v a l u e given by 0 = n3 tzaB ~?aB w h e r e Ta B does not d e p e n d on v a and will be d e t e r m i n e d in s u c h a way to s a t i s f y the g e n e r a l p r o p e r t i e s of Sa/3. ' P u t t i n g eqs. (3) and (5) in eq. (1) we get the f o l l o w i n g m o d e l e q u a t i o n dr- =
nfl ~a/3 ~ a
" ~ ma~ Ova + r/a/3 (va
Let u s now r e c a l l the g e n e r a l p r o p e r t i e s of S^.o. ~p T h e i n t e g r a l m a v a , ~ m a v a 2 , s a t i s f i e s the f o l l o w i n g r e l a t i o n s : (a) Ia/3(1) = 0,
(b) I a (qa) = 0,
(c)
(6)
Ia/3=j d v a q a S a f i ,
Ia/3(qa) + I/3a (q~) = 0
w h e r e qa = 1,
a ~ ~?
C o n d i t i o n s (a), (b), and (c) can be s a t i s f i e d by (6) p r o v i d e d that 3T =
77a/3 = ~/3a
_
a/3
_
(7)
3 T a ~ + ma/3 (ua_03)2__
with ~?a~ g i v e n by (7) Sa/3 = 0 in e q u i l i b r i u m a s it m u s t be. We r e m a r k that 77a,~ ,~ 1 even when (u a - u . ) ¢ 0 b e c a u s e the c o n d i t i o n 3 T a / 3 >> ma/3 (u a -U•) 2, ~ " T h e c o l l i s i o n c o n s t a n t s tta/3 g i v e n by eq. (4J a r e of the s a m e f o r m we m e e t in r i g o r o u s c a l c u l a t i o n s of k i n e t i c t h e o r y . To show this we d i s c u s s two s i m p l e e x a m p l e s c o n c e r n i n g a h o m o g e n e o u s i s o t r o p i c p l a s m a . We f i r s t c o n s i d e r u e = u i = 0 a n d c a l c u l a t e d T a / d t due to c o l l i s i o n s with/3 p a r t i c l e s .
176
Volume 24A, number 3
P HY SI CS L E T T E R S
30 January 1967
F r o m (6) we get dT~ -
-
2% =
n~/~afl (Ta - T ~ )
-
dt
(8)
+%
%
w h i c h c o i n c i d e s with S p i t z e r ' s f o r m u l a [4]. N e x t we c a l c u l a t e the m e a n e l e c t r o n v e l o c i t y u e u n d e r the e f f e c t of an e l e c t r i c f i e l d E = E o e x p ( - i w t ) . A g a i n u s i n g eq. (6) we h a v e due eE dt - m e - n i ~ e i T / e i ( U e
-ui)
P u t t i n g ~/ei ~ 1 and u e >>u i we get e 4 In (T e / e 2 K D)
e E e - me ni/Zei - iw'
/Zei ~ ~-
(9) m½T3 e e
C o m p a r i n g (9) with t h e c o r r e s p o n d i n g e x p r e s s i o n o b t a i n e d u s i n g t h e c o r r e c t f o r m (1) of S a ~ [5] we c o n c l u d e t h a t n itz ei i s t h e c o l l i s i o n f r e q u e n c y f o r m o m e n t u m e x c h a n g e f o r a p r o c e s s v a r y i n g with a f r e q u e n c y w > > n i / ~ e i . T h i s l e a d s to the c o n c l u s i o n that eq. (6) with ~ f l f r o m (4) could be a p p l i e d to s m a l l a m p l i t u d e o s c i l l a t i o n s with f r e q u e n c y g r e a t e r than nfi ~c~fl" References 1. P . L . B h a t n a g a r , E . P . G r o s s and M. Krook, Phys. Rev. 94 (1954)511. 2. J . P . D o u g h e r t y , Phys. Fluids 7 (1964)1788. 3. A.Oppenheim, Phys. Fluids 8 (1965) 900. 4. L.Spitzer J r . , Physics of fully ionized gases, (Interscience Publishers, Inc. New York, 1956). 5. V.L.Ginzburg, Propagation of electromagnetic waves in plasma, (Gordon and Breach Science Publishers, Inc. New York, 1961).
A NEW
LAW
IN
ELECTRODYNAMICS
O. C O S T A DE B E A U R E G A R D Institut Henri Poincare , Paris, France Received 31 December 1966
As a consequence of the ,Laulace" force RE xl] applied to a moving magnetic charge R, a varying dipole of moment 34 undergoes a force E × d34/dt, a n d a slowly varying current of intensity i a force (di/dt)~ VSl, where V denotes the scalar potential.
Let us recall first that a straightforward argum e n t of r e l a t i v i s t i c c o v a r i a n c e s h o w s that a m a g n e t i c p o i n t c h a r g e R ( e . m . u . ) , w h i c h by d e f i n i t i o n u n d e r g o e s , w h e n at r e s t , t h e C o u l o m b f o r c e R / ' / m u s t , when in m o t i o n with v e l o c i t y cfl, u n d e r g o the "Lorentz" force
F=R(Exfl +H).
(1)
It f o l l o w s t h e n t h a t a m a g n e t i c d i p o l e m a d e of two
p o l e s +R and -R with s e p a r a t i o n a and m o m e n t M =- Ra, s i t u a t e d at a g i v e n p o i n t in a s t a t i c e l e c tric field E, undergoes when M changes a force F = E × dM/dt.
(2)
If we a s s u m e f o r s i m p l i c i t y t h a t t h e s o u r c e of E is a single point charge Q with separation r from t h e d i p o l e ' s c e n t e r , t h e n one can r e w r i t e eq. (2) in t h e f o l l o w i n g f o r m 177