Fokker-Planck and Langevin equations for arbitrary slip velocities

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 . 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