PHYSICAL REVIEW AVOLUME 36, NUMBER 10 NOVEMBER 15, 1987
Fokke -Planck and Lange in equa ions o a bi a y slip eloci ies
R. Fe nandez-Fe ia and P. Riesco-Chueca
Depa men o Mechanical Enginee ing, Yale Uniue si y, Pos O hce Box 2159 Yale S a ion,
Xew Hauen, Connec icu 06520-2159
(Recei ed 26 June 1987)
An exp ession o he Fokke -Planck equa ion go e ning he eloci y dis ibu ion unc ion o
pa icles o hea y molecules imme sed in ahos ligh gas alid o a bi a y mean eloci ies o he
hea y componen is gi en. This exp ession gene alizes p e ious esul s which we e limi ed o
small di e ences be ween he mean eloci ies o he hea y and ligh componen s compa ed wi h
he he mal eloci y o he ligh gas. The de i a ion assumes aMaxwellian eloci y dis ibu ion
unc ion o he ligh gas, elas ic hea y-ligh collisions, and makes use o in eg als compu ed by
Riesco-Chueca, Fe nandez-Fe ia, and Fe nandez de la Mo a in Re . 1. The s ochas ic Lange in
equa ion associa ed wi h his Fokke -Planck collision ope a o is also ob ained. Mo e in gene al,
we de i e he Lange in equa ion co esponding o he gene al o m o he Fokke -Planck collision
ope a o , and pa icula ize i o he p esen case.
I. INTRODUCTION
As is well known, he Fokke -Planck kine ic equa ion
go e ning he eloci y dis ibu ion unc ion o pa icles
o hea y molecules dilu ed in ahos ligh gas can be de-
i ed om wo di e en poin s o iew. ' ' The mo e
adi ional one is based on he heo y o s ochas ic p o-
cesses and he Lange in equa ion, while he o he ap-
p oach makes use o he Bol zmann equa ion o he
hea y componen and expands he c oss-collision in-
eg als in powe s o he small mass a io. ''Based on
his second p ocedu e, we gi e in Sec. II an exp ession
o he Fokke -Planck equa ion no es ic ed o small
di e ences be ween he mean eloci ies o he hea y and
ligh componen s compa ed o he he mal eloci y o
he ligh gas, bu assuming aMaxwellian eloci y dis i-
bu ion unc ion o he ligh gas. In o de o connec
his kine ically de i ed Fokke -Planck equa ion wi h he
s ochas ic app oach, he co esponding s ochas ic
Lange in equa ion is de i ed in Sec. III.
whe e Up and up a e he molecula eloci ies o he
hea y componen be o e and a e he collision wi h a
ligh molecule and 4g=g' —
g, g=Up UgUp
To i s o de in he mass a io M, he c oss-collision in-
eg al may be w i en as
1=V„IB( ;u ) +—,V„[II( ;u) ]]+
(2)
whe e and a e he eloci y dis ibu ion unc ions o
he ligh and hea y componen s. —
mBand mII
ep esen , espec i ely, he a e o momen um and ene -
gy enso ans e om he ca ie gas o apa icle mo -
ing wi h agi en eloci y U; o plas ic collisions Band
II can be exp essed as [see Eq. (1)]
B( ;u )—
= du dQo(g, g)gbg (u)
m+m
dggg '"g up —
g
p
2
II. THE FOKKER-PLANCK EQUATION
FOR ARBITRARY SLIP VELOCITIES
11( ;u,)—= m+m
2
dudQu g,L9 gAgA u
The c oss-collision in eg al appea ing in he
Bol zmann equa ion co esponding o he hea y com-
ponen (deno ed by he subsc ip p) o abina y mix u e
whose cons i uen s ha e e y di e en molecula masses
can be simpli ied o aFokke -Planck o m, 'a e an
expansion in he a io o molecula weigh s M
—
:mjmp «1. This expansion is based on he small
ecoil eloci y o he hea y molecule o pa icle upon
collision wi h amuch ligh e molecule: F om he
momen um conse a ion i ollows ha , o elas ic col-
lisions,
m+m d'g g[-,
'(g'I —
3gg)Q"'(g)
+2ggQ"'(g)]
X (u~ —
g) .
In he abo e exp essions,
Q"(g)—
=2~ dg(1 —
cos'8)o(g, g) sing,
0
(4)
Up —
Up =—
m+mp c is he di e en ial sca e ing c oss sec ion o hea y-
ligh collisions, dQ= singdgdg, and (g, g, p) a e he
36 4940
FOKKER-PLANCK AND LANGEVIN EQUATIONS FOR. . . 4941
iU~ —
U[
(2kT/m)'" '(6)
whe e Uand U~ a e he mean eloci ies o he ligh and
hea y componen s, Tis he empe a u e o he ligh gas,
and kis Bol zmann's cons an . The assump ion «1
[mo e p ecisely, U=O(M' )] has been used, o ou
knowledge, in all p e ious wo ks on he subjec (Re s.
6—
10). No ice ha he assump ion u~/u =O(M' )
used in Re s. 6—
8is equi alen o U=O(M' ), since a
ame in which U=0 was used, and since
c~/c=O(M' ), whe e c—
:u—
Uand c~—=u~ —
U~ a e
he he mal eloci ies o he ligh and hea y componen s
[mo e accu a ely, c~ /c=O{(mT~/m~T)'i ), bu we as-
sume ha T~/T=O(1)]. To compu e he in eg als (3)
and (4) one needs o speci y he ligh -gas dis ibu ion
unc ion . In Re s. 6—
8 was aken o be aMaxwelli-
an dis ibu ion,
(u) = ()(u) —
=n2mkT
3/2 miu —
U[2
2kT
sphe ical coo dina es o g' in a e e ence ame in which
gis along he pola axis.
In his sec ion we shall gi e agene al exp ession o
he c oss-collision ope a o (2) no es ic ed o small
alues o he slip eloci y pa ame e U, de ined as
e n). Howe e , his assump ion is app op ia e in mos
p ac ical si ua ions and, mo eo e , i does no cons ain
he hea y-gas dis ibu ion unc ion since, as shown in
Re . 10, he Bol zmann equa ion o he ligh gas is kine -
ically uncoupled om he kine ic equa ion o he hea y
gas. The dis ibu ion unc ion o he ligh gas used in
Re s. 9and 10 con ains addi ional e ms p opo ional o
Kn, bu he assump ion U«1is used.
When is gi en by Eq. (7), he in eg als (3) and (4)
ha e been compu ed in Re . 1 o a bi a y alues o U.
In ha e e ence he ans e o momen um and ene gy
be ween species, in ol ing in eg als o Band II in
cspace, we e e alua ed by expansion o B( 0;u)and
II( o;u )in powe s o c/c a ound B( o,U)and
II( o;U ). Howe e , since no in eg a ion o Band II
a e equi ed he e, hese expansions need no be made, so
ha he pa ame e Uappea ing in he exp ession o B
and II gi en in Re . 1mus be subs i u ed he e by
(2kT/m )' (8)
[No ice ha u~ en e s in o he in eg als (3) and (4) as a
pa ame e , so ha he subs i u ion o uz by U~, and
he e o e o 6by U, does no change a all he o m o
hese in eg als. ]Using aLenna d-Jones po en ial o de-
sc ibe he in e ac ion be ween ligh and hea y com-
ponen s,
12 6
whe e nis he numbe densi y o he ligh gas. Mo e
gene al exp essions o we e used in Re s. 9and 10. In
pa icula , Re . 9used he i s o de o he Chapman-
Enskog expansion o (conside ing he ligh gas as a
pu e gas), while in Re . 10 he e ec o he hea y species
on he ligh -gas dis ibu ion unc ion was aken in o
accoun o i s o de in he Knudsen numbe o he
ligh gas. He e we shall assume ha is he Maxwellian
dis ibu ion (7) bu a bi a y alues o will be allowed
in he e alua ion o (3) and (4). The assump ion (7) will
es ic ou esul s (as in p e ious wo ks) o si ua ions in
which he ligh gas is in nea -equilib ium condi ions (i.e.,
Kn «1, whe e Kn is he Knudsen numbe o he ligh
gas, de ined as he a io be ween he equency o ligh -
ligh collisions and acha ac e is ic equency o he sys-
p( ) =4e (9)
one ob ains' "
u—
U
P
B= ~ (10)
=n, + , ,
2kT [ 2) (I—
eses )+ n) eses ],
7m
2kT
II2—
—5G n2(I —
3e&e&),
'Dmin
(1la)
(1lb)
(1lc)
whe e Iis he uni enso , e~ is he uni ec o along he
di ec ion o u~ —
U,
3
Vg =dxx exp[ —
(x +5 )]Q'" (xT*' )
2Sn" "(T*) asinha
CK
(12a)
5
& 2 =dx xexp[ —
(x +6 )]Q' '*(xT"' )
450'' '(T )cz
3cosho. 3sinha
+
Q 2Q3 (12b)
8ln g
~1 —
—
~ 1+25 +6 kTmp a=2xs,
e(m+m~) '3m' 2
16n( II""(T*)
1/2
(12c)
2II('2) (T+)
G=,II" '(Z')= dx x+exp( —)xQ2(i) (xTel/2)
5II(1,1) (T» )(j+1) (12d)
4942 R. FERNANDEZ-FERIA AND P. RIESCO-CHUECA 36
The dimensionless quan i ies g" a e ela ed o he in-
eg als de ined by Eq. (5) h ough Eq. (8.2-7) o Re . 12.
On he o he hand, he cha ac e is ic collision ime ~ be-
ween hea y and ligh componen s is ela ed o he i s
app oxima ion o he bina y di usion coe icien D ia
(e.g.,Re . 9)
m~D (n +np)
7=kT n
whe e n~ is he numbe densi y o he hea y gas.
Then, o a bi a y alues o U, he Fokke -Planck
equa ion which esul s om neglec ing he sel -collision
e m in he kine ic equa ion o he hea y componen
may be w i en as
+up 'V p
a 'V„(u~ U) s ~—
P
kT
+V„ ~ [ ii (I—
eses )+ i, eses
+5 G „z(I—
3eses)]
o a - om-equilib ium dispa a e-mass mix u es unde
condi ions o conside able p ac ical in e es .
III. LANGEVIN EQUATION
In he s ochas ic ea men o he B ownian mo ion o
pa icles o hea y molecules, he Fokke -Planck equa-
ion go e ning he eloci y dis ibu ion unc ion o hese
pa icles is ob ained om he s ochas ic Lange in equa-
ion, applying he heo y o Ma ko p ocesses. 'In his
sec ion we shall p oceed in e sely: Gi en he Fokke -
Planck equa ion (13), which has been de i ed om he
kine ic Bol z nann equa ion, we shall ob ain he
Lange in equa ion go e ning he mo ion o he indi idu-
al pa icles o hea y molecules whose dis ibu ion unc-
ion sa is ies ha equa ion. Mo e in gene al, we will
de i e he Lange in equa ion associa ed wi h a bi a y
momen um and ene gy ans e unc ions B( ;u)and
II( ;u~ )appea ing in he Fokke -Planck collision ope a-
o (2), o which he igh -hand side o Eq. (13) is apa -
icula case [when he dis ibu ion unc ion o he ligh
gas is he Maxwellian (7)].
Le us w i e he equa ion o mo ion o he indi idual
pa icles (Lange in equa ion) as
(13) de =F(u~; , )+ A( , ),
d (15)
In he pa icula limi whe e 6«1, he coe icien s ap-
p oach uni y and he abo e equa ion educes o he s an-
da d o m o he Fokke -Planck equa ion. 'In e ec ,
he in eg al exp essions (12) can be expanded o show
ha o 6«1 and T*))1 p 1+ &g6 gl, 6+
+—,
', 6—
—,
'„6...,while o 6«1 and T* «1,
mus be no iced ha , in he limi U«1,
5=u +O(M, M' ), so ha he abo e expansions
show he equi alence be ween Eq. (13) and he s anda d
o m o he Fokke -Planck equa ion o small slip eloci-
y (u «1). Fi ing exp essions o he coe icien s co -
e ing he ull ange o alues o 6in he high- and low-
empe a u e limi s a e'" Au~ =FA +I (16)
whe e he o al accele a ion o he pa icles due o col-
lisions wi h he ligh molecules has been di ided in wo
e ms: amean accele a ion F, and a luc ua ing o s o-
chas ic accele a ion A, which by de ini ion has ze o
mean. As in Eq. (13), he hea y componen is assumed
so dilu ed ha hea y-hea y collisions can be neglec ed.
In addi ion, we ha e assumed ha he e a e no ex e nal
o ces [ he inclusion o ex e nal o ces in bo h Eq. (13)
and Eq. (15) is as aigh o wa d ma e ]. Fo an in e -
al o ime A long compa ed o he pe iod o luc ua ion
o he accele a ion Abu sho compa ed o he in e -
als du ing which he mean accele a ion Fchanges ap-
p eciably, we can w i e Eq. (15) as
~ —
—
(1+0.45965 )', 1&& T* o 1&&5T*'~
z —
—
(1+0.47605 )'i, 1»T*o 1»5T*'i (14a) whe e
(17)
n2 —
(1+0.31252)'~3, 1&& T* o 1&&5T*'
ugg —
—
(1+0.3225 )'~, 1&&T* o 1&&5T*'~ (14b)
Mo e gene al in e pola ed o mulas a e gi en in Re . 1
o a bi a y alues o T* and . Al hough Eqs. (14a)
and (14b) o he coe icien s a e compu ed o a
Lenna d- Jones po en ial, exp essions (12a)—
(12c) a e
gene al ( o elas ic collisions) and can be pa icula ized
o any po en ial o in e ac ion. The limi T' »1is
ele an in mos a - om-equilib ium physical si ua ions
(shock wa es, impingemen o a low in apla e, e c.).
The opposi e limi is mo e uncommon; i can, howe e ,
be used o desc ibe he inal s ages o he expansion o a
je in o a acuum. The Fokke -Planck equa ion gi en in
Eq. (13) hus p o ides ab oad desc ip ion o he kine ics
(bu, 4u~) (18)
whe e (b,u~ )lb and (bu~ b.u~ )lb. a e, espec i ely,
he a e age ime a e o change o Au& and hu~ Au„.
Since Ihas ze o mean, om Eq. (16)
Using he heo y o Ma ko p ocesses, i can be shown
(e.g.,Re . 3) ha , neglec ing e ms O(b, )and
0(b,u/b, ), he Fokke -Planck equa ion go e ning he
eloci y dis ibu ion unc ion o he hea y componen is
FOKKER-PLANCK AND LANGEVIN EQUATIONS FOR. ..4943
(b,u~ )=Fb, ,
(b,ubu )=FF(h ) +( &, (19a)
(19b)
so ha iden i ying he igh -hand side o Eq. (18) wi h
he Fokke -Planck collision ope a o (2) one ob ains
F=—
B,
( &=a 11,
(20a)
(20b)
de g
d (u —
U)+ A, (21)
whe e (I I)=b, II, wi h II gi en by Eqs. (11). Fu -
he , i can be shown ha , unde he assump ion ha
he in e al o ime A is long compa ed o he pe iods
I
whe e highe -o de e ms in A ha e been neglec ed in
Eq. (20b). [Clea ly, he ela ions be ween (hu )and
(b,ubu )and he collision in eg als Band II could
ha e been ob ained di ec ly om he de ini ions (3) and
(4); see, e.g.,Re . 13.]The e o e, in he pa icula case
in which he dis ibu ion unc ion o he ligh gas is he
Maxwellian (7), he Lange in Eq. (15) becomes
o luc ua ions o he s ochas ic accele a ion A(in o he
wo ds, i he numbe Xo ligh molecules ha collide
wi h he pa icle o hea y molecule du ing A is ala ge
numbe ), he p obabili y dis ibu ion unc ion o 1is he
Gaussian
p( )=
, ,
2x II,„
(2~) (A de II )' (22)
whe e epea ed subsc ip s a e summed. In e ec , w i ing
N
mpbu =gp;, (23)
whe e p; is he momen um in e changed by asingle
hea y-ligh collision, and assuming ha he p; a e in-
dependen andom a iables, o Nla ge one can apply
he cen al limi heo em (see, e.g.,Re . 14; we assume
ha he p obabili y dis ibu ion unc ions o he a i-
ables p; a e well beha ed so ha his heo em applies) o
ob ain he p obabili y dis ibu ion unc ion o mz Au~ as
11
P(m b,u)= exp
(277) '—
de y (p p)'2
m, b,u„—
g(p,„)'m, au,,—g(p,,)'
(p ~pI )
+O(1/N) .(24)
In his exp ession (p; )and (p;p; )a e he i s wo mo-
men s o he p obabili y unc ion o p;. Thus g; &(p; )
and g+ &(p;p; )a e, espec i ely, he a e age o al
momen um and he a e age o al enso pp deli e ed by
he ligh molecules o apa icle o hea y molecule wi h
eloci y uz du ing b, . F om Eqs. (3) and (4) we ha e
N
g(p, &= —
m, Ba , (25)
and
de =—
—
(u —
U)+ A( , )
d (27a)
p( )=
p2
4kT A /mp~
(47 kT b, /m~ )
exp
(27b)
N
g(p,p, )=m'lis .
i=1
The e o e, subs i u ing Eqs. (16), (25), and (26) in o Eq.
(24), and making use o he ac ha he p obabili y dis-
ibu ion unc ion o Ihas ze o mean, one eadily ge s
F=—
B[Eq. (20a)] and he Gaussian dis ibu ion (22)
o P( I)[wi h e o s 0( 1/N )].
The Lange in equa ion de i ed abo e [Eq. (15) wi h
Eqs. (20a) and (22)] con ains p e ious esul s on he sub-
jec . Fo ins ance, when he dis ibu ion unc ion o he
ligh gas is he Maxwellian (7) and, in addi ion, he slip
eloci y pa ame e is small, in i s app oxima ion in
we ha e B=(u —
U)/ and II=(2kT/ m )I, so ha
(Re . 2, pp. 22—
24; in his e e ence he hos gas is a
es , U=0). In he case in which he i s app oxima ion
o he Chapman-Enskog expansion o he eloci y dis i-
bu ion unc ion o he ligh gas is used and «1,
B=(u„U DaTV l—
nT)—
/ and II=(2kT/ m )I (see
Re . 9; aT is he he mal di usion ac o and i has been
assumed ha he hea y componen is e y dilu e). Then
de 1
=—
—
(u —
Ua DV'lnT)+—
A,
d (28)
wi h he p obabili y dis ibu ion unc ion o Igi en by
Eq. (27b). The Lange in equa ion and he dis ibu ion
o Igi en in Re . 5 o his same case o a"Chapman-
Enskog hos gas" and «1 con ain addi ional e ms
p opo ional o he s ess enso o he ligh gas. How-
e e , hese e ms a e O( Kn), whe e Kn is he Knudsen
numbe o he ligh gas, so ha hey ough no o be
aken in o accoun in a i s -o de heo y in bo h Kn and
. Finally, in he case conside ed in his no e, whe e he
dis ibu ion unc ion o he ligh gas is Maxwellian bu
a bi a y slip eloci ies a e allowed, he Lange in equa-
ion is gi en by Eq. (21), and he dis ibu ion o Iis he
Gaussian (22) wi h he enso II gi en by Eqs. (11).
I is in e es ing o es ima e he ange o in e als o
ime b, o which he s ochas ic di e ence Eq. (16) [wi h
Eqs. (20a) and (22)] is alid in e ms o he pa ame e s o
he p oblem. To his end we know ha , in o de o he
Gaussian dis ibu ion (22) o hold, (b, ) 'mus be small
compa ed o he equency coi~ o hea y-ligh collisions
[co e —
n(kT/m)' c ~, whe e c , is he diame e o he
4944 R. FERNANDEZ-FERIA AND P. RIESCO-CHUECA 36
pa icle o hea y molecule] and b, mus be small com-
pa ed o he ime in which Bchanges app eciably. Thus
h mus be small compa ed o (i) he ime win which he
p ope ies o he hea y componen change by collisions
wi h he ligh molecules [ -(Mco&~) ', since on he o -
de o M'ligh -hea y collisions a e equi ed o change
u~ by an amoun o he same o de as i sel ] and o (ii) a
cha ac e is ic mac oscopic ime , [in e ms o he ligh -
gas Knudsen numbe , , '—
nKn(kTlm )' o, whe e o
is he diame e o aligh molecule]. The e o e,
1«h coI «M
1«A mIp «Kn o
In he case o amona omic hea y molecule, he las con-
di ion becomes
((~ colp ((Kn 'M
since, based on he ac ha he expe imen al iscosi ies
o he noble gases a e oughly mass independen (see,
e.g.,Re . 10), (c /op) -M' .All he abo e condi ions
o A can, in p inciple, be ul illed because, by hy-
po hesis, M«1, Kn «1, and o.lo p«1.
ACKNOWLEDGMENTS
We a e indeb ed o P o esso J. Fe nandez de la Mo a
o sugges ing he p oblem and o many use ul discus-
sions. This wo k has been suppo ed by acoope a i e
esea ch g an om Schmi Technologies Associa es
and he S a e o Connec icu (No. 885-176), by he
U.S.—
Spanish Join Commi ee o Cul u al and Educa-
ional Coope a ion, and by G an No. CBT-86-12143
om he U.S. Na ional Science Founda ion (NSF).
P. Riesco-Chueca, R. Fe nandez-Fe ia, and J. Fe nandez de la
Mo a, Phys. Fluids 30, 45 (1987).
Selec ed Pape s on Noise and S ochas ic P ocesses, edi ed by N.
Wax (Do e , New Yo k, 1954).
3P. Resibois and M. DeLeene , Classical Kine ic Theo y o
Fluids (Wiley, New Yo k, 1977), Chap. 2.
4R. M. Mazo, J. S a . Phys. 1, 101 (1969).
5W. G. N. Slinn and S. F. Shen, J. S a . Phys. 3, 291 (1971).
T. Kiha a, Re . Mod. Phys. 25, 844 (1953).
7C. S. Wang Chang and G. E. Uhlenbeck, in S udies in S a is i-
cal Mechanics, edi ed by J. de Boe and G. E. Uhlenbeck
(No h-Holland, Ams e dam, 1970), Vol. 5, Chap. V.
L. Fe a i, Physica 101A, 491 (1980); 115A, 232 (1982).
J. Fe nandez de la Mo a and J. M. Me ce , Phys. Re . A26,
2178 (1982).
'J. Fe nandez de la Mo a and R. Fe nandez-Fe ia, Phys.
Fluids 30, 740 (1987).
"P.Riesco-Chueca, R. Fe nandez-Fe ia, and J. Fe nandez de
la Mo a, in Ra e ied Gas Dynamics, edi ed by V. Bo i and C.
Ce cignani (Teubne , S u ga , 1986), Vol. 1, p. 283.
J. O. Hi sch elde , C. F. Cu iss, and R. B. Bi d, Molecula
Theo y o Gases and Liquids {Wiley, New Yo k, 1954), pp.
526-527.
'I. B. Be ns ein, Lec u e No es, P ince on Plasma Physics
Summe Ins i u e (1964), Chap. II.
'4A. I. Khinchin, Ma hema ical Founda ions o S a is ical
Mechanics (Do e , New Yo k, 1949), Appendix.