Particle flux associated with stochastic processe

Newman, C. E.

Sturrock, P. A.

Tademaru, E.

English

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https://ntrs.nasa.gov/search.jsp?R=19710002378 2020-03-12T00:49:16+00:00Z
PARTICLE FLUX ASSOCIATED 
WITH STOCHASTIC PROCESSES 
* 
C. E. Newman, P. A. S tur rock  
and E. Tademaru t 
August 1970 
SUIPR Report No. 386 
Prepared under 
A i t  Force Off ice of S c i e n t i f i c  Research 
Contract  F44620-69-C-0008 
and 
Nat ional  Aeronautics and Space Adminis t ra t ion 
Grant NGL-05-020-272 
I n s t i t u t e  for Plasma Research 
S tanford  E lec t ron ic s  Labora tor ies  
S tan fo rd  Univers i ty  S tanford ,  Ca l i fo rn ia  
* Also Department of Applied Physics ,  S tanford  Universi ty  
.t Now a t  t h e  Nat ional  Radio Astronomy Observatory,  
C h a r l o t t e s v i l l e ,  V i rg in i a  
Abst rac t  
The Lagrange expansion, which may be used t o  de r ive  t h e  Fokker- 
Planck equat ion,  i s  he re  used t o  de r ive  t h e  corresponding expression 
for t h e  f l u x  of p a r t i c l e s  sub jec t  t o  a s t o c h a s t i c  s c a t t e r i n g  process .  
The c o e f f i c i e n t s  which occur  i n  t h i s  expression a r e ,  i n  genera l ,  not 
t h e  same a s  t h e  c o e f f i c i e n t s  which occur  i n  t h e  Fokker-Planck equa- 
t i o n  i t s e l f .  In  t h e  s p e c i a l  c a s e  t h a t  t h e  p a r t i c l e  d i s t r i b u t i o n  
involves  only  one independent va r i ab le ,  t h e  p a r t i c l e  f l u x  i s  d e t e r -  
mined by the  f a m i l i a r  Fokker-Planck c o e f f i c i e n t s .  Evaluat ion of 
p a r t i c l e  f l u x  i s  of s p e c i a l  i n t e r e s t  i n  t h e  s tudy of s t o c h a s t i c  
a c c e l e r a t i o n .  
The behavior of p a r t i c l e s  moving under t h e  in f luence  of random 
f o r c e  f i e l d s  i s  of i n t e r e s t  i n  a s t rophys ic s .  One example i s  t h e  hea t ing  
and d i f f u s i o n  of charged p a r t i c l e s  i n  t he  s o l a r  wind (Barnes, 1967; 
J o k i p i i  1966), and another  i s  t h e  a c c e l e r a t i o n  of charged p a r t i c l e s  by 
t h e  Fermi-type process  of s t o c h a s t i c  a c c e l e r a t i o n  ( H a l l  and Stur rock ,  
1967; Melrose, 1969; Tsytovich, 1966). Under cond i t ions  which a r e  nor- 
mally accepted, such a p rocess  may be represented  by a Fokker-Planck 
equation, which g i v e s  t h e  formula f o r  t h e  t i m e  d e r i v a t i v e  of t h e  p a r t i c l e  
d i s t r i b u t i o n  func t ion .  However, i n  i n t e r p r e t i n g  va r ious  s o l u t i o n s  of 
t h i s  equat ion ,  i t  i s  h e l p f u l  t o  know t h e  f l u x  of p a r t i c l e s - - i n  r e a l  space, 
momentum space,  or i n  energy, a s  i s  appropr i a t e  t o  t h e  problem. The f o r -  
mula f o r  t h e  p a r t i c l e  f l u x  i s  simple, but appears no t  t o  be  widely known. 
Th i s  paper t h e r e f o r e  g i v e s  a b r i e f  d e r i v a t i o n  of t h i s  formula. 
I n  an e a r l i e r  p u b l i c a t i o n  (S tur rock ,  1960; h e r e a f t e r  c a l l e d  paper  I ) ,  
i t  was shown t h a t  t h e  Fokker-Planck equat ion  may convenient ly  be der ived  
from a g e n e r a l i z a t i o n  of t h e  Lagrange expansion. T h i s  pub l i ca t ion  a l s o  
presented  a formula f o r  p a r t i c l e  f l u x ,  so it  o f f e r s  a convenient s t a r t i n g  
po in t  f o r  t h i s  paper .  
W e  cons ide r  t h e  d i s t r i b u t i o n  of p a r t i c l e s  i n  a space enumerated by  
v a r i a b l e s  x r = 1, . . ., n .  These may be s p a t i a l  coord ina tes ,  compo- 
nen t s  of v e l o c i t y  or momentum, or any combination t h e r e o f .  The d i s t r i -  
bu t ion  f u n c t i o n  f ( x , t )  has  t h e  p rope r ty  t h a t  f ( x , t ) d n x  i s  t h e  number 
of p a r t i c l e s  i n  t h e  volume d x cen te red  on t h e  p o s i t i o n  x a t  t i m e  t .  
I f ,  wi th  t h e  requirement t h a t  dx  be small compared with t h e  s c a l e  d e t e r -  
mined by t h e  g r a d i e n t  of f ,  f d x i s  no t  a l a r g e  number, it i s  necessary  
t o  i n t e r p r e t  t h i s  number a s  an "expectation" va lue .  
r' 
n 
n 
If ,  when t changes t o  t + A t ,  t h e  p a r t i c l e  which was a t  p o s i t i o n  
x moves t o  p o s i t i o n  x + Ax, t h e  d i s t r i b u t i o n  func t ion  a t  t i m e  t + A t  
i s  r e l a t e d  t o  t h a t  a t  t i m e  t by t h e  Lagrange expansion. An a p p r o p r i a t e  
- 3- 
change of n o t a t i o n  of equat ion  (3.6) of paper I l e a d s ’ t o  
The Fokker-Planck equat ion  i s  der ived  from equat ion  (1) by assuming 
t h a t  one may choose t h e  t i m e  i n t e r v a l  A t  such t h a t  ( a )  t h e  d i s t r i b u t i o n  
f u n c t i o n  f changes onLy s l i g h t l y  i n  t h i s  t ime;  (b )  t h e  a s soc ia t ed  va lues  
of 
f ;  
Axr a r e  small  compared w i t h  t h e  s c a l e  determined by t h e  g r a d i e n t  of 
and (c )  t h a t  a l a r g e  number of random processes  occur  i n  t h i s  t i m e  
w i t h  t h e  r e s u l t  t h a t  
where angular  b racke t s  denote t h e  expec ta t ion  va lue  of a q u a n t i t y .  When 
t h e  t i m e  i n t e r v a l  A t  i s  chosen i n  t h i s  way, equat ion  (1) t a k e s  t h e  
usua l  form of the  Fokker-Planck equat ion:  
- af - - a (f())+ 1 /2  
a t  - axr 
Note t h a t  t h e  term Axr/At 
f o r c e  f i e l d  as w e l l  a s  t h a t  of a randomly f l u c t u a t i n g  f o r c e  f i e l d .  
may r ep resen t  t h e  e f f e c t  of a small s t eady  
W e  now wish t o  o b t a i n  an express ion  f o r  t h e  f l u x  Fr, t h e  d e r i -  
v a t i o n  of which i s  t o  be analogous t o  the  d e r i v a t i o n  of equat ion  ( 3 ) .  
The same change of n o t a t i o n  i n  equat ion  (3*9)  of paper I g i v e s  
t h e  fo l lowing  express ion  f o r  t h e  averaged f l u x :  
-4- 
W e  see t h a t ,  al though t h e  f irst  t e r m  on t h e  right-hand s i d e  of equat ion  
(4) a l s o  appears  I n  t h e  Fokker-Planck equat ion  (equat ion  3) , t h e  second 
term does n o t .  Nevertheless,  w e  may confirm t h a t  t h e  c o n t i n u i t y  equa- 
t i o n  
r 
r 
aF 
- 0  - + - -  af at ax 
i s  equ iva len t  t o  equat ion  (3 ) .  
For t h i s  purpose, i t  is  convenient t o  rewrite t h e  f l u x  a s  
where 
T h i s  may be  expressed a s  
r = r + rrs 
v rs -rs 
( 5 )  
where r and Trs a r e  symmetric and antisymmetric, r e spec t ive ly ,  i n  
t he i r  s u f f i x e s .  The symmetric component i s  e x p r e s s i b l e  a s  a t o t a l  d e r i -  
rs -
v a t i v e  and may t h e r e f o r e  be  r e l a t e d  t o  the second term of t h e  Fokker- 
Planck equation: 
AxrAxs 
r = 112 Axs + Ax = 1/2  = A t  ) (9)  
r rs -
The antisymmetric term, g iven  by 
dAx 
rs Axs - Ax r 2) d t  , 
-5- 
does not appear  i n  t h e  Fokker-Planck equat ion because t h e  term 
i s  divergence free. S u b s t i t u t i o n  of formula (4 )  i n t o  equa- a - axS (f ‘3) 
t i o n  ( 5 )  t h e r e f o r e  y i e l d s  equat ion ( 3 ) .  
W e  see t h a t ,  i n  genera l ,  t h e  p a r t i c l e  f l u x  cannot be expressed i n  
terms of t h e  c o e f f i c i e n t s  of t h e  Fokker-Planck equat ion alone but i n  
add i t ion  r e q u i r e s  c o e f f i c i e n t s  of t h e  form (x AX,) , which a r e  
r e a d i l y  c a l c u l a t e d  i n  t h e  same manner a s  t h e  usual  Fokker-Planck c o e f f i -  
d Axr 
c i e n t s .  An important except ion t o  t h i s  r u l e  i s  t h e  case  t h a t  t h e  system 
has only one independent coord ina te .  I f ,  f o r  i n s t ance ,  w e  a r e  concerned 
only wi th  t h e  energy E of p a r t i c l e s ,  t h e  Fokker-Planck equat ion has 
t h e  form 
and t h e  f l u x  ( i n  efiergy) i s  given by 
with no a d d i t i o n a l  terms. 
This  formula i s  of s p e c i a l  i n t e r e s t  for discuss ion  of s t o c h a s t i c  
acce le ra t ion ,  s i n c e  it is  important t o  know whether a p a r t i c u l a r  solu- 
t i o n  of t h e  Fokker-Planck equat ion r ep resen t s  t r a n s f e r  of p a r t i c l e s  
from low energy t o  high energy ( a c c e l e r a t i o n )  or from high energy t o  
low energy ( d e c e l e r a t i o n ) .  T h i s  cons ide ra t ion  has  s p e c i a l  re levance 
t o  a recent  a r t i c l e  by Melrose (1969) a s  w i l l  be discussed i n  more 
d e t a i l  i n  a s e p a r a t e  a r t i c l e  ( Tademaru, Newman and Jones,  1970) .  
-6 - 
ACKNOWLEDGEMENT 
The au tho r s  ( E .  T.  and C .  E. N . )  acknowledge an i n t e r e s t i n g  co r re s -  
pondence wi th  D r .  D.  B. Melrose. Th i s  work has  been supported by t h e  
A . i r  Force O f f i c e  of S c i e n t i f i c  Research under c o n t r a c t  F 44620-69-c 
0008 and t h e  Nat ional  Aeronaut ics  and Space Adminis t ra t ion  under g ran t  
NGL-05-020-272. 
-7- 
References 
Barnes, As: 1967, Phys. F l u i d s  10, 2427. 
Ha l l ,  D. E .  and S tu r rock ,  P. A.:  1967, Phys. F l u i d s  10, 2620. 
-
-
J o k i p i i ,  J .  R.: 1966, Astrophys.  J. 146, 480. 
Melrose, D. €3.: 1969, Astrophys.  Space S c i .  5 ,  181. 
Stur rock ,  P .  A . :  1960, J .  Math. Phys. 1, 403. 
-
- 
- 
Tademaru, E . ,  Newman, C .  E .  and Jones,  F. C.:  1970, Astrophys. Space S c i .  
( t o  be pub l i shed) .  
Tsytovich,  V .  N.: 1966, Sov ie t  Phys. Uspekhi - 9, 370. 
-8- 


Particle flux associated with stochastic processes

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