Structure of BBGKY hierarchy for Boltzmann gas and particle distribution

Frieman, E. A.

Goldman, R.

English

NASA Technical Reports Server

ON THE STRUCTURE OF THF. BBGKY HIERARCHY FOR A BOLTZMANN GAS 
R, Goldman 
Xnst i tu te  f o r  F lu id  Dynamics and Applied Mathematics 
Universi ty  of Maryland, College Park,  Maryland 
and 
E. A, Frieman 
Department, of Astrophysical Sciences 
Prince ton Univers i t y  
Princeton,  New Jersey  
ABSTRACT 
We treat the  evolut ion in .  time of a s p a t i a l l y  uniform Boltzmann gas 
with no i n i t i a l  c o r r e l a t i o n s ,  For the  case o f  cut-off p o t e n t i a l s  and arbi-  
t r a r y  i n i t i a l  velocqty c l i s t r i l p t i o n  func t ions ,  on using the expansion 
parameter nr3$, wi th  n = p a r t i c l e  densi ty  and ro = range of binary poten- 
t i d ,  w e  show f o r  terms of the  order  (pr?)* and 1Qwer t h a t  t he  hierarchy 
is formally se l f -c los ing  even with the  inc lus ion  of mgny body e f f e c t s ;  i,e,, 
0 
a t  a given order in 
t he  s i n g l e  p a r t i c l e  d i s t r i b u t i o n  funct ion of the  same w d e r ,  t he  binary 
Correrat ion func t ion  which, deteymines the  k i n e t i c  behavior of the  s i n g l e  
(nr’], wi th  $he expeption of contr ibut ions l i n e a r  i n  
0 
p a r t i c l e  d i s t r i b u t i o n  func t ion  only depends on funct ions which themselves 
f u l l y  determinable wi th in  a prescr tbed i t e r a t l o n  procedure. 
vergence of $he var ious orders  of t h e  formal expansian is discussed f o r  
The a c t u a l  con- 
i v i t i a l l y  a r i b t r a r y  ve loc i ty  dlls t r i bu t ions  and f o r  l i n e a r i z a t i o n s  around 
Maxwellian ve loc i ty  q l s t r i b u t i o n s .  I 
https://ntrs.nasa.gov/search.jsp?R=19680018139 2020-03-12T10:42:52+00:00Z
a 
t he  BBCKY hierarchy i n  the  Boltzmann approximation' takes the  form: 
The equations are i n  dimensionless u n i t s  with digtances i n  u n i t s  of ro, the  
range of t he  binary i n t e r a c t i o n  and v e l o c i t i e s  i n  u n i t s  of 
mean square; p a r t i c l e  ve loc i ty ;  therefore  times are I n  un i f s  of 
t i m e  of a b h a r y  i n t e r a c t i o n ,  
vav t h e  r o o t  
ro/vav, the  
Equations (1) hqve been obtained from the  general  hierarchy equa- 
- t i ons  by choosing <O>/mviv n, l ,  wi th  (4) the  c h a r a c t e r i s t i c  s t r eng th  of 
the  p o t e n t i a l ,  and nro - E , The gs a re  defined i n  a recurs ive  
mapner from the reduced d i s t r i b u t i o n  funceions , 
3 
f s  , by the  r e l a t i o n s  : 
Expansion procedure 
scale procedure2 amended t o  allow f o r  add i t iona l  many space scale va r i a t ions .  
Our expansion procedure is a vers ion  of the  many Cipe 
We assume 
. . I  
I 
. .  
. .  
I - HQ . + 6HS 
5 3  s 
i 
For m T 0 pe yqe The above. For m = 
later) frooa.the m - 2 equation, whi.ph 
. : . 
. .  
. .  
itself next receives cpntribufigns 
4 
from the  m = 3 equation. Therefore i n  p l a w  o f  $he basic h ie rwchy  for 
m 3, 1 we qan write a mut i la ted  hierarchy af the form 
For 
wi th  A ' ( s , m )  
Determination- 
s f i l l  t o  be spec i f ied .  
& On using equation (I") fa r  m = 0 f o r  a t i m e  T 0 
order  un i ty  (T 2 collision durat ion)  we hwv for  ( X ~ , ~ + ~ I  F r0 
and 
O f  
(3) 0 0 rJ + 611 c ei ,s+l - - Hs+l * R* -b %+l 
then 410 and 611 are zero. There are no secular contributions involving 
terms in and 6Ll ,  since these are only non-zsro %or times o l  order 
un i ty  . 
5 
By means of (3) and (2), t he  r i g h t  hand s i d e  of (2") f o r  m = 1 
can be  put: i n  the: f o p  
with  
I 
. .  
sfl 
0 0 
-Hs+l= 0 
Q -%+IT - J e Hs+l k i p  S.t.1 
I 
(witb tbs addi t ion  of terms of order  E) a The terms l i n e a r  ie Hs and Hsal 
cont r ibu te  t o  a t o t a l  des iva t lve  with respec t  t o  t i m e  of quant i ty  of order  
unicy, The terns l i n e a r  i n  vs+l - v ( 5  v F 
can be put  i n  the  form 
.) have a cont r ibu t ion  which s+l,1 
with  
6 
with das+l denoting a surfgce element. As long as a l l  of t he  s p a r t i c l e s  
i n  the  c o l l e c t i o n  (1, 2 ,. . ,s) are f u r t h e r  apa r t  than d is tance  of order  
un i ty ,  61 = 0. It has  been shown earlier3 f o r  g1 t o  be 'bounded t h a t  ong 2 S 
m u s t  take: 
0 
ps. This completes the  d e f i n i t i o n  of 
Defeminat ion  of .& general Within the  m = 2 equations of (1") f o r  
I J x i  , s+1 
1 
r b  w e  have as i n  the  cage of t he  m = 1 equations: 
-t- 6 ( i , s + l ;  l ,* .* , s+l ,  t )  . 20 
For the  p a r t i c l e s  ( l , * . . , s )  f a r  apa r t  w e  assume t h a t  620 i s  small com- 
pared t o  the  terms lPnear i n  e '+lT. This has been v e r i f i e d  i n  some 
deqa i l  f o r  s = 1, 2 and 3 ,  
order  t o r  x-' compared t o  the  t e r m s  l i n e a r  i n  e 
0 -H 
where one f inds  t h a t  620 i s  c h a r a c t e r t s t i c a l l y  of 
0 
(Here and S + P  
-H -1 
7 
6: i=1 J 
+ E  
i=l 
afterward x i s  taken t o  be t h e  minimum i n t e r p a r t i c l e  separa t ion  of the  s 
p a r t i c l e s , )  
The terms, excluding A(s,m) + ~ ~ ( s , m + l ) ,  on the  right; hand side 
of (1") f o r  m = 2 can therefore  be wr i t t en :  
0 
a e -Hs+lT 
( V s + p i )  - 
axs+l h, s+l 0 
0 
e .  
i7-, 
-Hs+lT 0 
%+l 
0 
H6+lT 
x ( I 2 ~ i , S + 1 , l , . ~ . ~ S , t )  + e 620) dSls+l 
w i t h  
By coupling the  arguments following ( 4 )  with considerat ions which put  t he  
i n t e g r a l  term i n  Eq. (1") f o r  m = 1 of order  t o r  x-' compared with -1 
one f inds  chat t h e  dominant terms i n  ( 6 )  are l i n e a r  i n  gb ' 
The cont r ibu t ion  from (8) wi th in  ( 6 )  may be put  i n  the  form 
A(s, l )  + B ( s , l )  + C(S,l)  + 622 
8 
w i t h  
A(s,l) = - 
i= l 
0 1 I d3 "si-1 dasi-l IVsi-l,$ I 'gl(s+l> g,W 
s*li =ro 'xi 9 si-1 1 1 -v f y si-1 I 
0 
-Hs+lT 
d3 "si-1 dasi-l I"s+l,i I .  
I Xi, s+l I =ro $Xi, si-l 1 1  vi, 94-1 
9 
-1 The t e r m  i n  6- is 0;f O P & ~  t . o r  x-' compared t o  A + B. The term 
l inear  i n  
cor rec t ion  
We 
t i o n  with 
LL 
0 1  gs gl (s+l) evaluated 
t o  g: which w e  w t l l  
are f r e e  t o  "promote" 
m = 1; s ince  i t  leads 
a t  x i ,s+l 1 1  -?,s+l y ie lds  a higher  order  
neglec t  
t he  A ( s  ,1) term t o  the  lower order  equa- 
t o  exponential  damping and does no t  r e s u l t  
i n  ;M e s s e n t i a l l y  more complicated equat ion t o  so lve ,  w e  do so. 
w e  have f o r  (1") wi th  m 2 2,  s 2 2: 
Therefore 
A'(s ,2)  = - A ( s , l )  9 A"(s,2) 
The t e r m  i n  A" contains terms which i f  they were pot "promoted" would lead 
t o  divergent  behavior i n  
values obtained for g1 on the  omission of the term i n  E A ( s , ~ ) ~  
Such terms can be est imated by usage of the  gS 
s 
For s = 2 ,  B(s , l )  is  of order  1x121-l f o r  Ix121>:1. Consequently 
the  cont r ibu t ion  t o  g i  through (1") i s  logarithmic.  Therefore w e  write 
1 1 1 
g2 = (g2I1 + (g2I2 
1 
and'"promote" the  terms wi th in  B(2,l)  l i n e a r  in .  (g,),. For s = 3 the  con- 
t r i b u t i o n  t o  g2 i s  f i n i t e  ( a t  l e a s t  as concerns x, v arguments which 
u l t imate ly  cont r ibu te  t o  g f o r  Ix121 C ro) .  Hence w e  do no t  l'promQte" 
3 
2 
B(3, l ) .  For 
f o r  B ( 3 , l ) .  
Hence 
(subjec t  t o  a 
s 2 4 w e  assume t h a t  the  conclusion f o r  B(s , l )  i s  the  same as 
1 
1 1 
t he  so lu t ion  f o r  gs, s 2 3 
knowledge of g1 and g2 )o  
i s  a matter of simple i t e r a t i o n  
10 
Solut ion for g: & p a r t i c u l a r  For g1 we have from (1"): 2 
We w r i t e  
with: 
= B ( 2 , O )  
12 
The term [ R ( 2 , 0 )  - A(2,O) -B(2,P.)3 f a l l s  off  with increas ing  1 x12 I 
more rap id ly  than, \x12 }-2. Consequently (g2& 1 1 does not  copt r ibu te  t o  
2 s ecu la r  behavior i n  (g,). The term B(2,O) va r i e s  with increas ing  Ix121 
. ConsequeDtly the  r i g h t  hand s i d e  of (13) i s  of the  order 
and t h e  r i g h t  hand s i d e  of 
-1 1 -1 
(14) i s  of order  E an(61x121) f o r  
IxI21 E from which (g,) Y is of order  E f o r  1xl21 E 
1 and (g2)y are of the  form 1 For EX >> 1 one has t h a t  both (g2)pjl 
13 
with  a pos i t i ve ,  of order  un i ty ,  and bounded from below. - a E X  E;e 
Determination of gs _and g2 & p a r t i c u l a r  W e  now tu rn  t o  the  detennina- 2 2 
2 t i o n  of g The r i g h t  hand s i d e  of (1") f o r  m = 3 can be w r i t t e n ,  analog- 
ously t o  ( 6 )  and (7 ) :  
S 
0 0 
-Hs+l+ 0 
H,+l e 
-'sf' - ! J e I x i  p s + l  I <  ro f=l lxi,s+p E. is1 E J  
I 
-Hs+lT 
O I  axs+l -Hs+lT+ i=l J ] x  i , s + l l < f o  
0 
%1 e e 
a 
( V s + p i )  -+ ! J  
i=1 Ixi,s+ll<'o 
0 
Hs*lr 
'30) dns+l x (I ( i , s + l , l , a . , e , s s t )  + e 3 (15) 
wi th  
2 One may f i r s t  ca l cu la t e  t he  various g s ( t  - r) without the  contr i -  
but ions A'(s,3) and then determine the r e s u l t i n g  cor rec t ions  from the  
A' (s  ,3). For s = 3, at least as concerns which cont r ibu tes  d i r e c t l y  83 
0 
a e -H3T . g3(t-.r> 2 t o  g2 f o r  Ix12 I < to, the  term i n  v3% * 
2 does not cont r ibu te  t o  g i n  order  E o r  lower. W e  assume t h a t  t he  
i t e r a t i o n  with A ' ( S  b3) does not  decrease the  order of g2 and w e  take the  
2 
3' 
14 
s i z e  of t h i s  term as rep resen ta t ive  of t he  s i z e  of o the r  terms i volving 
g3 ( t  - T) wi th in  (15)- Also we take t h e  cont r ibu t ion  from s = 3 t o  be 
f 
2 
a bound on the  contr ibut ions from s 2 4 ,  s o  t h a t  these  too can be  neglected 
2 i n  f inding g2 t o  order  E f o r  Ix121 < r . 
0 
1 For A'(2,3) w e  note  t h a t  due t o  the  presence of (g2IY one expects 
t h a t  ope may w r i t e  
wi th  g 2 p  
a time scale of order  t he  mean f r e e  time. 
of order  un i ty  on a length scale of order  the  mean free path and 
Correspondingly w e  have 
and 
, 
w e  note  t h a t  the equation is  of t h e  form: For g 4 2  
S decreases s u f f i c i e n t l y  rap id ly  with increasing 
1 2 is obtained sub jec t  t o  a knowledge of g and g l i2 ,  1 the  so lu t ion  f o r  g 2 f z ,  
IxI2 1 - t h a t  t o  order  un i ty  
2 
Discuseion of Eq,  (1%) and comments on the  genera l  s t r u c t u r e  v
Equation (12’) treats the many body e f f e c t s  whose exis tence has been 
previously noted by o the r  authors 4’5y6y70 
e n t i r e l y  unexpected sinee the  binear-y co r re l a t ion  funct ion as d is tances  of 
order the  mean free path i n  general  is of order  
of the binary c o r r e l a t i o n  funct ion in order  h i @ e r  than 
The behavior Sn Eqo (1%) f s  no t  
2 E One expects t h a t  parts 
c2 w i l l  have equa- 
t i o n ~  whose homogeneous p a r t s  are s i m i l a r  in form t o  the  l e f t  hand s i d e  of 
( l a ) ,  Since the co r re l a t ion  func t ion  f o r  n bodies a t  d is tances  of order  
the mean free path from each o the r  i s  i n  general  of order  
higher ,  parr of t h e  
and 
n body co r re l a t ion  funct ion w i l l  s a t i s f y  an integro- 
d i f f e r e n t i a l  equation whose homogeneous terns are acted on by the operator  
with ~ ( i ~ n 4 - 1 )  exchanging the  l abe l s  i and n+l. 
__c- Form 5f s i n g l e  particle d i s t r i b u t i o n  funct ion 
The s i n g l e  p a r t i c l e  d i s t r i b u t i o n  funct ion g is  a f  t he  f o m  1 
0 2 1 2 2  
11 =+ E 81 g1 = g1 =+ Lgl =+ E Rn Eg 10 
and g: of order  un i tyo  Its behavior f o r  a11 t i m e s  fi, 0 1  with 81’ ggo’ g11 
given by the  equations: 
17 
He re 
S(x) = 1, x > 0 
S(x) = 0 ,  x < 0 
and the  operators  Pi 
t i ons  f d l l m i n g  them. 
denote the ex is tence  of cont r ibu t ions  from the  func- 
Equation (20) is  the  Boltzmann equation. Equations (22) and (24) are 
l inea r i zed  s p a t i a l l y  homogeneous Boltzmann. equations with known source terms, 
i s  given by 
2 I1 From Eq. (25) the  behavior due t o  g 
ag’l 11 - -  - 0  a t  (26) 
The form 111’ i s  independent of g 211 
Within (271, t he  t e r m  i n  g 
of (27) follows from the  f a c t  t h a t  the  inhomogeneous i n t e g r a l  terms i n  (17) 
are o f  order  E. 
2 Descr ipt ion of t he  asymptotic t i m e  behavior of &1 t o  order F- 
I f  t h e  dev ia t ion  of go1 from i t s  asymptotic value i n  t i m e  approaches 
and g l  - gill approach 1 1  10’ 811 zero exponent ia l ly  o r  more r ap id ly ,  then g 
t h e i r  asymptotic values exponent ia l ly .  However, a normal mode8” ana lys i s  o f  
(17) and (27) revea ls  t h a t  
a lgeb ra i ca l ly  as t m3I2 f o r  t >> I, 
g2 may i n  p r i n c i p l e  approach i t s  asymptotic value 
111 
I f  the  devia t ion  of go 1 from i t s  asymptotic value approaches zero 
less rap id ly  than exponent ia l ly ,  then i n  general  one cannot even conclude 
1 2 t h a t  gl0 and g1 approach l i m i t s .  
1 1  remain bounded However, provided g 
a s u f f i c i e n t l y  long t i m e ,  l i n e a r i z e  (20) i n  the  devia t ions  of 
10' g l l  and 81 11 one can, a f t e r  
go1 from its 
asymptotic Maxwellian v e l w i t y  form, 
cut-off potentials!' i t  appears t h a t  the devia t ion  of 
value decays exponent ia l ly  i n  t i m e  and t h e  corresponding conclusions as t o  
Then, f o r  cut-off .Maxwellian or "harder" 
from i t s  asymptotic go 1 
I 
follow. 1 1 2  2 g lo '  811' 81 - gill and gill 
F ina l ly  f o r  t he  po ten t i a l s  j u s t  mentioned, i f  go1 has a constant 
s m a l l  dev ia t ion  from t h e  Maxwellian ve loc i ty  d i s t r i b u t i o n  and one l i n e a r i z e s  
i n  t h i s  small devia t ion ,  the cont r ibu t ion  from those modes which y i e l d  pre- 
vious t -3'2 cont r ibu t ion  i s  f i n i t e  e 
Acknowledgment 
This work was supported by the  National Aeronautics and Space 
Administration under Grant N s G  220-62 and the  A i r  Force Off ice  of S c i e n t i f i c  
Research of the  Off ice  o f  Aerospace Research under Contract AF 49(638)-1555, 
19 
References 
1, See J. E. McCune, G. Sandri ,  and E, A, Frieman, Third Symposium on 
Rarefied Gas Dynamics (Academic P res s ,  No Y , ,  1963) Vol, I, 
pp. 102-114, f o r  f u r t h e r  discussion i n  terms of t h e  not ion o f  
temperature, 
2. E. A. Frieman, Journal  of Math, Phys, 4, 410 (1963) 
3, E.  A. Frieman and R,  Goldman ( t o  b e  published i n  Journal  of Math. 
Phys . ) 
4 .  H. Grad - Handbook of Physics - Vol. X I I ,  205 (1958), Springer 
Verlag - Berlin, 
5. 
6. K ,  Kawasaki and I. Oppenheim, Phys. Rev. 139A, 1763 (1965). 
J. R. Dorfman and E. G. D o  Cohen, Phys. Letters 16, 124 (1965). 
7. J. Weinstock, Phys. Rev. 140A, 460 (1965). 
8. L. Sirbvich,  Phys. Fluids  6, 10 (1963). 
9.  L. Sirovich,  Phys. Fluids  6,  218 (1963). 
IO. H e  Grad, Physics of Flu.ids 6 ,  147 (1963), 


On the structure of the BBGKY hierarchy for a Boltzmann gas

Abstract

Similar works

Full text

Available Versions

NASA Technical Reports Server