Fokker-Planck and Langevin equations for arbitrary slip velocities

PHYSICAL REVIEW AVOLUME 36, NUMBER 10 NOVEMBER 15, 1987 Fokker-Planck and Langevin equations for arbitrary slip velocities R. Fernandez-Feria and P. Riesco-Chueca Department ofMechanical Engineering, Yale Uniuersity, Post Ofhce Box 2159 Yale Station, Xew Hauen, Connecticut 06520-2159 (Received 26 June 1987) An expression for the Fokker-Planck equation governing the velocity distribution function of particles or heavy molecules immersed in ahost light gas valid for arbitrary mean velocities of the heavy component is given. This expression generalizes previous results which were limited to small differences between the mean velocities of the heavy and light components compared with the thermal velocity of the light gas. The derivation assumes aMaxwellian velocity distribution function for the light gas, elastic heavy-light collisions, and makes use of integrals computed by Riesco-Chueca, Fernandez-Feria, and Fernandez de la Mora in Ref. 1. The stochastic Langevin equation associated with this Fokker-Planck collision operator is also obtained. More in general, we derive the Langevin equation corresponding to the general form of the Fokker-Planck collision operator, and particularize it to the present case. I. INTRODUCTION As is well known, the Fokker-Planck kinetic equation governing the velocity distribution function of particles or heavy molecules diluted in ahost light gas can be derived from two different points of view. ' ' The more traditional one is based on the theory of stochastic processes and the Langevin equation, while the other approach makes use of the Boltzmann equation for the heavy component and expands the cross-collision integrals in powers of the small mass ratio. ''Based on this second procedure, we give in Sec. II an expression for the Fokker-Planck equation not restricted to small differences between the mean velocities of the heavy and light components compared to the thermal velocity of the light gas, but assuming aMaxwellian velocity distribution function for the light gas. In order to connect this kinetically derived Fokker-Planck equation with the stochastic approach, the corresponding stochastic Langevin equation is derived in Sec. III. where Up and up are the molecular velocities of the heavy component before and after the collision with a light molecule and 4g=g' — g, g=Up UgUp To first order in the mass ratio M, the cross-collision integral may be written as 1=V„IB(f;u )f +—,V„[II(f;u)f ]]+ (2) where fand fare the velocity distribution functions of the light and heavy components. — mBand mII represent, respectively, the rate of momentum and energy tensor transfer from the carrier gas to aparticle moving with agiven velocity U; for plastic collisions Band II can be expressed as [see Eq. (1)] B(f;u )— =fdufdQo(g, g)gbgf(u) m+m dggg '"g up — g p 2 II. THE FOKKER-PLANCK EQUATION FOR ARBITRARY SLIP VELOCITIES 11(f;u,)—= m+m 2 dudQu g,L9 gAgA u The cross-collision integral appearing in the Boltzmann equation corresponding to the heavy component (denoted by the subscript p) of abinary mixture whose constituents have very different molecular masses can be simplified to aFokker-Planck form, 'after an expansion in the ratio of molecular weights M — :mjmp «1. This expansion is based on the small recoil velocity of the heavy molecule or particle upon collision with amuch lighter molecule: From the momentum conservation it follows that, for elastic collisions, m+m fd'g g[-, '(g'I — 3gg)Q"'(g) +2ggQ"'(g)] Xf(u~ — g) . In the above expressions, Q"(g)— =2~ fdg(1 — cos'8)o(g, g) sing, 0 (4) Up — Up =— m+mp cr is the differential scattering cross section for heavylight collisions, dQ= singdgdg, and (g, g, p) are the 36 4940 FOKKER-PLANCK AND LANGEVIN EQUATIONS FOR. . . 4941 iU~ — U[ (2kT/m)'" '(6) where Uand U~ are the mean velocities of the light and heavy components, Tis the temperature of the light gas, and kis Boltzmann's constant. The assumption v«1 [more precisely, U=O(M' )] has been used, to our knowledge, in all previous works on the subject (Refs. 6— 10). Notice that the assumption u~/u =O(M' ) used in Refs. 6— 8is equivalent to U=O(M' ), since a frame in which U=0 was used, and since c~/c=O(M' ), where c— :u— Uand c~—=u~ — U~ are the thermal velocities of the light and heavy components [more accurately, c~ /c=O{(mT~/m~T)'i ), but we assume that T~/T=O(1)]. To compute the integrals (3) and (4) one needs to specify the light-gas distribution function f. In Refs. 6— 8fwas taken to be aMaxwellian distribution, f(u) =f()(u) — =n2mkT 3/2 miu — U[2 2kT spherical coordinates of g' in areference frame in which gis along the polar axis. In this section we shall give ageneral expression for the cross-collision operator (2) not restricted to small values of the slip velocity parameter U, defined as tern). However, this assumption is appropriate in most practical situations and, moreover, it does not constrain the heavy-gas distribution function since, as shown in Ref. 10, the Boltzmann equation of the light gas is kinetically uncoupled from the kinetic equation of the heavy gas. The distribution function of the light gas used in Refs. 9and 10 contains additional terms proportional to Kn, but the assumption U«1is used. When fis given by Eq. (7), the integrals (3) and (4) have been computed in Ref. 1for arbitrary values of U. In that reference the transfer of momentum and energy between species, involving integrals of fBand fII in cspace, were evaluated by expansion of B(f0;u)and II(fo;u )in powers of c/c around B(fo,U)and II(fo;U ). However, since no integration of Band II are required here, these expansions need not be made, so that the parameter Uappearing in the expression for B and II given in Ref. 1must be substituted here by (2kT/m )' (8) [Notice that u~ enters into the integrals (3) and (4) as a parameter, so that the substitution of uz by U~, and therefore of 6by U, does not change at all the form of these integrals. ]Using aLennard-Jones potential to describe the interaction between light and heavy components, 12 6 where nis the number density of the light gas. More general expressions for fwere used in Refs. 9and 10. In particular, Ref. 9used the first order of the Chapman- Enskog expansion for f(considering the light gas as a pure gas), while in Ref. 10 the effect of the heavy species on the light-gas distribution function fwas taken into account to first order in the Knudsen number of the light gas. Here we shall assume that fis the Maxwellian distribution (7) but arbitrary values of vwill be allowed in the evaluation of (3) and (4). The assumption (7) will restrict our results (as in previous works) to situations in which the light gas is in near-equilibrium conditions (i.e., Kn «1, where Kn is the Knudsen number of the light gas, defined as the ratio between the frequency of lightlight collisions and acharacteristic frequency of the sysp(r) =4e (9) one obtains' " u— U P B=v~ (10) rr=n, +rr, , 2kT [v2) (I— eses )+vn) eses ], 7m 2kT II2— —5Gvn2(I — 3e&e&), 'Dmin (1la) (1lb) (1lc) where Iis the unit tensor, e~ is the unit vector along the direction of u~ — U, 3 Vg =dxx exp[ — (x +5 )]Q'" (xT*' ) 2Sn" "(T*) asinha CK (12a) 5 &rr2 =dx xexp[ — (x +6 )]Q' '*(xT"' ) 450'' '(T )cz 3cosho. 3sinha + Qf 2Q3 (12b) 8lnvg v~1 — — v~ 1+25 +6 kTmp a=2xs, e(m+m~) '3m' 2 16n(r II""(T*) 1/2 (12c) 2II('2) (T+) G=,II" '(Z')= fdx x+exp( —)xQ2(i) (xTel/2) 5II(1,1) (T» )(j+1)t (12d) 4942 R. FERNANDEZ-FERIA AND P. RIESCO-CHUECA 36 The dimensionless quantities g" are related to the integrals defined by Eq. (5) through Eq. (8.2-7) of Ref. 12. On the other hand, the characteristic collision time ~ between heavy and light components is related to the first approximation of the binary diffusion coefficient Dvia (e.g.,Ref. 9) m~D (n +np) 7=kT n where n~ is the number density of the heavy gas. Then, for arbitrary values of U, the Fokker-Planck equation which results from neglecting the self-collision term in the kinetic equation for the heavy component may be written as +up 'Vfp at 'V„(u~ U)vs f~— P kT +V„f~ [vii (I— eses )+vti, eses +5 Gv„z(I— 3eses)] of far-from-equilibrium disparate-mass mixtures under conditions of considerable practical interest. III. LANGEVIN EQUATION In the stochastic treatment of the Brownian motion of particles or heavy molecules, the Fokker-Planck equation governing the velocity distribution function of these particles is obtained from the stochastic Langevin equation, applying the theory of Markov processes. 'In this section we shall proceed inversely: Given the Fokker- Planck equation (13), which has been derived from the kinetic Boltzrnann equation, we shall obtain the Langevin equation governing the motion of the individual particles or heavy molecules whose distribution function satisfies that equation. More in general, we will derive the Langevin equation associated with arbitrary momentum and energy transfer functions B(f;u)and II(f;u~ )appearing in the Fokker-Planck collision operator (2), of which the right-hand side of Eq. (13) is aparticular case [when the distribution function of the light gas is the Maxwellian (7)]. Let us write the equation of motion of the individual particles (Langevin equation) as (13) de =F(u~;r, t)+ A(r, t), dt (15) In the particular limit where 6«1,the vcoefficients approach unity and the above equation reduces to the standard form of the Fokker-Planck equation. 'In effect, the integral expressions (12) can be expanded to show that for 6«1 and T*))1 vp 1+ &g6 gl, 6+ +—, ', 6— —, '„6...,while for 6«1 and T* «1, must be noticed that, in the limit U«1, 5=u +O(M, vM' ), so that the above expansions show the equivalence between Eq. (13) and the standard form of the Fokker-Planck equation for small slip velocity (u «1). Fitting expressions for the vcoefficients covering the full range of values of 6in the high- and lowtemperature limits are'" Au~ =FAt+I (16) where the total acceleration of the particles due to collisions with the light molecules has been divided in two terms: amean acceleration F, and afluctuating or stochastic acceleration A, which by definition has zero mean. As in Eq. (13), the heavy component is assumed so diluted that heavy-heavy collisions can be neglected. In addition, we have assumed that there are no external forces [the inclusion of external forces in both Eq. (13) and Eq. (15) is astraightforward matter]. For an interval of time At long compared to the period of fluctuation of the acceleration Abut short compared to the intervals during which the mean acceleration Fchanges appreciably, we can write Eq. (15) as v~ — — (1+0.45965 )', 1&& T* or 1&&5T*'~ vz — — (1+0.47605 )'i, 1»T*or 1»5T*'i (14a) where (17) vn2 — (1+0.31252)'~3, 1&& T* or 1&&5T*' vugg — — (1+0.3225 )'~, 1&&T* or 1&&5T*'~ (14b) More general interpolated formulas are given in Ref. 1 for arbitrary values of T* and v. Although Eqs. (14a) and (14b) for the vcoefficients are computed for a Lennard- Jones potential, expressions (12a)— (12c) are general (for elastic collisions) and can be particularized to any potential of interaction. The limit T' »1is relevant in most far-from-equilibrium physical situations (shock waves, impingement of aflow in aplate, etc.). The opposite limit is more uncommon; it can, however, be used to describe the final stages of the expansion of a jet into avacuum. The Fokker-Planck equation given in Eq. (13) thus provides abroad description of the kinetics (bu, 4u~) (18) where (b,u~ )lb tand (bu~ b.u~ )lb.tare, respectively, the average time rate of change of Au& and hu~ Au„. Since Ihas zero mean, from Eq. (16) Using the theory of Markov processes, it can be shown (e.g.,Ref. 3) that, neglecting terms O(b,t)and 0(b,u/b, t), the Fokker-Planck equation governing the velocity distribution function of the heavy component is FOKKER-PLANCK AND LANGEVIN EQUATIONS FOR. ..4943 (b,u~ )=Fb,t, (b,ubu )=FF(ht) +(rr&, (19a) (19b) so that identifying the right-hand side of Eq. (18) with the Fokker-Planck collision operator (2) one obtains F=— B, (rr&=at 11, (20a) (20b) de vg dt (u — U)+ A, (21) where (I I)=b,tII, with II given by Eqs. (11). Further, it can be shown that, under the assumption that the interval of time At is long compared to the periods I where higher-order terms in At have been neglected in Eq. (20b). [Clearly, the relations between (hu )and (b,ubu )and the collision integrals Band II could have been obtained directly from the definitions (3) and (4); see, e.g.,Ref. 13.]Therefore, in the particular case in which the distribution function of the light gas is the Maxwellian (7), the Langevin Eq. (15) becomes of fluctuations of the stochastic acceleration A(in other words, if the number Xof light molecules that collide with the particle or heavy molecule during At is alarge number), the probability distribution function of 1is the Gaussian p(r)= r,r, 2xt II,„ (2~) (At detII )' (22) where repeated subscripts are summed. In effect, writing N mpbu =gp;, (23) where p; is the momentum interchanged by asingle heavy-light collision, and assuming that the p; are independent random variables, for Nlarge one can apply the central limit theorem (see, e.g.,Ref. 14; we assume that the probability distribution functions of the variables p; are well behaved so that this theorem applies) to obtain the probability distribution function of mz Au~ as 11 P(m b,u)= exp (277) '— dety (p p)'2 m, b,u„— g(p,„)'m, au,,—g(p,,)' (p ~pI ) t +O(1/N) .(24) In this expression (p; )and (p;p; )are the first two moments of the probability function of p;. Thus g; &(p; ) and g+ &(p;p; )are, respectively, the average total momentum and the average total tensor pp delivered by the light molecules to aparticle or heavy molecule with velocity uz during b,t. From Eqs. (3) and (4) we have N g(p, &= — m, Bat, (25) and de =— — (u — U)+ A(r, t) dt (27a) p(r)= p2 4kT At /mp~ (47rkT b,t/m~r) exp (27b) N g(p,p, )=m'list . i=1 Therefore, substituting Eqs. (16), (25), and (26) into Eq. (24), and making use of the fact that the probability distribution function of Ihas zero mean, one readily gets F=— B[Eq. (20a)] and the Gaussian distribution (22) for P( I)[with errors 0( 1/N )]. The Langevin equation derived above [Eq. (15) with Eqs. (20a) and (22)] contains previous results on the subject. For instance, when the distribution function of the light gas is the Maxwellian (7) and, in addition, the slip velocity parameter vis small, in first approximation in v we have B=(u — U)/r and II=(2kT/rm )I, so that (Ref. 2, pp. 22— 24; in this reference the host gas is at rest, U=0). In the case in which the first approximation of the Chapman-Enskog expansion of the velocity distribution function for the light gas is used and v«1, B=(u„U DaTV l— nT)— /r and II=(2kT/rm )I (see Ref. 9; aT is the thermal diffusion factor and it has been assumed that the heavy component is very dilute). Then de 1 =— — (u — UarDV'lnT)+— A, dt (28) with the probability distribution function of Igiven by Eq. (27b). The Langevin equation and the distribution of Igiven in Ref. 5for this same case of a"Chapman- Enskog host gas" and v«1 contain additional terms proportional to the stress tensor of the light gas. However, these terms are O(vKn), where Kn is the Knudsen number of the light gas, so that they ought not to be taken into account in afirst-order theory in both Kn and v. Finally, in the case considered in this note, where the distribution function of the light gas is Maxwellian but arbitrary slip velocities are allowed, the Langevin equation is given by Eq. (21), and the distribution of Iis the Gaussian (22) with the tensor II given by Eqs. (11). It is interesting to estimate the range of intervals of time b,tfor which the stochastic difference Eq. (16) [with Eqs. (20a) and (22)] is valid in terms of the parameters of the problem. To this end we know that, in order for the Gaussian distribution (22) to hold, (b,t) 'must be small compared to the frequency coi~ of heavy-light collisions [cote — n(kT/m)'r cr~, where cr, is the diameter of the 4944 R. FERNANDEZ-FERIA AND P. RIESCO-CHUECA 36 particle or heavy molecule] and b,tmust be small compared to the time in which Bchanges appreciably. Thus ht must be small compared to (i) the time win which the properties of the heavy component change by collisions with the light molecules [r-(Mco&~) ', since on the order of M'light-heavy collisions are required to change u~ by an amount of the same order as itself] and to (ii) a characteristic macroscopic time t, [in terms of the lightgas Knudsen number, t, '— nKn(kTlm )' o, where o is the diameter of alight molecule]. Therefore, 1«htcoI «M 1«AtmIp «Kn o In the case of amonatomic heavy molecule, the last condition becomes ((~tcolp ((Kn 'M since, based on the fact that the experimental viscosities of the noble gases are roughly mass independent (see, e.g.,Ref. 10), (cr/op) -M' .All the above conditions for At can, in principle, be fulfilled because, by hypothesis, M«1, Kn «1, and o.lo p«1. ACKNOWLEDGMENTS We are indebted to Professor J. Fernandez de la Mora for suggesting the problem and for many useful discussions. This work has been supported by acooperative research grant from Schmitt Technologies Associates and the State of Connecticut (No. 885-176), by the U.S.— Spanish Joint Committee for Cultural and Educational Cooperation, and by Grant No. CBT-86-12143 from the U.S. National Science Foundation (NSF). P. Riesco-Chueca, R. Fernandez-Feria, and J. Fernandez de la Mora, Phys. Fluids 30, 45 (1987). Selected Papers on Noise and Stochastic Processes, edited by N. Wax (Dover, New York, 1954). 3P. Resibois and M. DeLeener, Classical Kinetic Theory of Fluids (Wiley, New York, 1977), Chap. 2. 4R. M. Mazo, J. Stat. Phys. 1, 101 (1969). 5W. G. N. Slinn and S. F. Shen, J. Stat. Phys. 3, 291 (1971). T. Kihara, Rev. Mod. Phys. 25, 844 (1953). 7C. S. Wang Chang and G. E. Uhlenbeck, in Studies in Statistical Mechanics, edited by J. de Boer and G. E. Uhlenbeck (North-Holland, Amsterdam, 1970), Vol. 5, Chap. V. L. Ferrari, Physica 101A, 491 (1980); 115A, 232 (1982). J. Fernandez de la Mora and J. M. 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