ENGINEERING RESEARCH IN:':TITUTE THE UNIVERSITY OF MICHIGAN ANN ARBOR Technical Report THE KIINFETIC THEORY OF A GAS IN ALTERNATING OUTSIDE FORCE FIELDS: A GENERALIZATION OF THE RAYLEIGH PROBLEM Co S. Wang Chang G. E:. eU lenbeck Project 2457 OFFICE OF NAVAL RES:EARCH, Uo SO NAVY DEPARTMENT CONTRACT NO o Nonr-1224- (15) October 1956

The University of Michigan ~ Engineering Research Institute TABLE OF CONTENTS Page ABSTRACT ii i OBJECTIVE 111 I. STATEMENT OF PROBLEM 1 II. THE BROWNIAN MOTION LIMIT 3 III. THE STRONG COUPLING APPROXIMATION 4 IV. DISCUSSION OF TEE APPROXIMATE RESULTS 5 V. THE EXACT SOLUTION FOR MAXWELL MOLECULES 6 VI. THE SOLUTION FOR ELASTIC SPHERES IN THE LORENTZ LIMIT IF o = 0 8 VII. CONCLUDING REMARK 12 APPENDICES 13 Appendix I. Proof of the BrowTiian Motion Form of the Collision Operator 14 Appendix II. Eigenvalues and Eigenfunctions of the Maxwell Collision Operator 18 Appendix III. Eigenvalues and Eigenfunctions of the Brownian Motion Form of the C(ollision Operator 21 Appendix ITV. The Bracket Expression H[r~,lmu,\ trRm] in the Lorentz Limit 22...._ __ ~ ii....

The University of Michigan ~ Engineering Research Institute ABSTRACT The problem of the absorption of power by a set of harmonic oscillators surrounded by a dilute gas is investigated on the basis of the linear Boltzmann equation as a function of the frequencies of the oscillator and of the outside electric field, and as a function of the type of interaction between the oscillator and the gas molecule. For the case where this interaction is an inverse fifth-power repulsion (Maxwell molecules), an exact solution of the problem is presented (Section V). Various limiting cases and approximate results are discussed. OBJECTIVE The main purpose was to elucidate the validity of the so-called strong collision approximation, which is often used in the kinetic theory of gases. iii

The University of Michigan ~ Engineering Research Institute I. STATEMENT OF THE PROBLEM In this report we will be concerned with the following problem. Suppose a particle of mass m is bound harmonically to a fixed point with proper frequency uo; it is surrounded by a gas of particles of mass M against which it collides according to some given force law; the gas is supposed to be in equilibrium at temperature T and the equilibrium is not affected by the motion of the particle m.l Finally, an outside alternating force mEo cos wt acts, say in the x-direction, on the particle m (not on the molecules of the surrounding gas). One wants to know the average power absorbed by the particle as a function of wo, Lithe ratio of the masses m/M, and the type of force law between the particle and the molecules of the surrounding gas. Clearly the problem is a generalization of the well-known problem of Rayleigh,2 in which it will go over if wo = O, no outside force is present, and the motion is one dimensional. The relation to the theory of the shape of absorption lines and to the theory of metals will be more or less evident and will not be further elaborated. The problem was in fact suggested in a discussion with Dr. J. M. Luttinger, because of a paradox which he encountered in the theory of metals. In the classical form in which we stated the problem, the mathematical formulation is given by the so-called linear Boltzmann equation. Let f(x, v, t) dx dv be the probability at time t.that the particle m is in the space' and velocity range dx dv, then f will"rfulfill the equation af af 6f +t v -+ + a = J(f) (1) where a is the acceleration produced by the forces acting on the particle m, so that 1. One may think that the gas is sufficiently dilute so that the velocity distribution of the molecules around the particle m remains the Maxwell distribution 20 Rayleigh, Scientific Papers, Volo 3, p. 473. See also: Ming Chen Wang, A Study of Various Solutions of the Boltzmann Equation, Dissertation, Univ. of Micho, Ann Arbor, 1942o

The University of Michigan ~ Engineering Research Institute ax = - o2X + Eo cos ot (2) ay = - o y; az = -o2z J(f) is the collision term: J(f) = ~dV dS gI (g,) [fF' - fF] (3) where F(V) =N (M)3/2 _-MV2/2kT is the distribution function of the surrounding gas and the primes refer to the velocity variables; the collision (v, V) + (v', V') turns the relative velocity g = Iv - vI over the angle G and I(g,G) is the differential collision cross section. The outside force Eo cos wt must be considered as the perturbation which prevents the distribution function f from going to the equilibrium distribution: o 3 e -(mv2 + m o2 r2)/2kT (4) ~ v gkT/ In the steady state we now want to calculate the time average of P = Eo cos t, (5) where Vx(t) = ( dx dv vxf (X, vt) Clearly f = fo + fi, where the perturbation fl of the distribution function will in the steady state be proportional to Eo and vary in time like the outside force, although, of course, it will not be in phase because of the friction with the surrounding gas. An exact solution of the problem we have found only for the case of the so-called Maxwell molecules, where one assumes that the interaction between particle m and a gas molecule is a repulsion ^r l/r5. Before turning to this special case we first will discuss some approximate solutions, assuming some approximate expressions for the collision operator which are current in the liturature.

The University of Michigan * Engineering Research Institute IIo THE BROWNIAN MOTION LIMIT If the particle m is very heavy compared to the gas molecule (m/M >> 1), and if, in addition, we assume that the velocity v is never very different from the equipartition value, so that v/V is always of order (M/m)1/2, then one finds, by an expansion in powers of M/m, that the collision term J(f) can be approximated by the well-known Rayleigh or Brownian motion form K ra kT af 1 J(f): (vYf) + P v (6) [6va m. (6) t where the friction coefficient 13 is given by 16=.NM. dG sin Q (1-cos G) dWVV e I(V,@) Since for general I(g,G) the proof of (6) is not easily available, we give the details in Appendix Io With the collision term (6) it is simple to solve the problem. Multiplying the Boltzmann equation ff a aft a6 kT 6f at-: - ~ - va +, ~~f a+ T - a + m a (7)| with xi or vi, respectively, and integrating over the coordinate and velocity space, assuming that for large xi and vi f vanishes sufficiently fast, one obtains for the average values xi and Vi the equations dii dt dTi 2 - _ =i,o xi + Eo cos wt il - Vi (8) dt which have an obvious physical interpretation. In the steady state 3. The assumption that v/V is always of order (M/m)1/2 implies that on the average the particle m will feel a frictional force proportional to its velocity; the proportionality constant is no For very large v, this will not be true anymore; the friction will then become "Newtonian," proportional to V2o.., — 3

The University of Michigan ~ Engineering Research Institute EoT w2' Eo wo(w2 - W02) Vi = (2 2) 2 2 cos t + (2 2)2 +2W2 sin ut so that the average power absorbed is given by B.M 2 (P2 _ o2)2 + ~2(W2 For a discussion of this result, see Section IV. III o THE STRONG COUPLING APPROXIMATION Especially in the theory of metals, it is customary to approximate the collision term by assuming J(f) o (10) T where fo is the equilibrium distribution (4) and T is the relaxation time, which is a measure of the time required for the collisions to establish equilibrium. One of the main questions we will have to discuss is the question under which circumstances (10) may be used as an approximation of the collision term. With Eq. (10), one obtains for the equation of motion of the average values xi, vi instead of (8), the equations dxi Xi a-t = vi - dv i 2i dt 1 O di _ - Cio2 xi + Eo cos at 11 - Vi (11) at T Note especially the first of these two equations. It says that the average position of the probability distribution does not change with time according to the average velocity io. The origin of this paradoxical result is the fact that with (10) dv J(f) = S d (f- f) which is not necessarily zero, while from the exact expression (3) follows

The University of Michigan ~ Engineering Research Institute Sdv J(f) = o (12) Egq. (12) is an expression of the fact that in a collision the number of particles does not change. One must say therefore that the strong collision approximation (10) violates this conservation law. A consequence of this is, as Luttinger has pointed out, that it makes a difference whether one calculates the average power absorbed with the help of the average velocity or with the help of dxi/dt. Using the average velocity, one obtains from Eqo (11) the resuit first derived by Van Vleck and Weisskopf:4 VW = it 10)2 + ()1 ++1o)2 2 (5) 4 + (W~o)2~ l+ (T +~o) +2 while, using dx/dt, one obtains, t4 cO 1+ ( O_)2T2 1 + (. + o ] ) T (14) first given by Luttinger.5 IV.o DISCUSSION OF THE APPROXIMATE RESULTS If one puts in the Brownian motion resltt (9) q = 1/7, then the three results, (9), (13), and (14), can be directly compared with each other. Irn Figs. 1 to 5 we have plotted P as a f'unction of XT for various values of eoS One easily verifies the following facts~ a. For all three forrms, the area f'nder the curve is the same and equal to:ioj / 4. b. For small %'oT PVW will be a monotonic decreasing function of )To. O'itly for WuoT > l./3, P7 will have a maximun. Since both PL and PB.M are zero for (DT = 0, if 'oT ~ 0, they always will show a maximum. c. For Nor = 0, PVW and PBoM are identical. do For wOT >> 1, PVW and PL become nearly the same, especially near the resonance peak. The PBoM gives an essentially sharper resonance peak~ 4- J- HO Vanr Vleck and V. F. Weisskopf, Rev, Modo Phys~, 17, 227 (1945), 5o JO Ma Luttinger, Private Communication~ "~~~~~

The University of Michigan ~ Engineering Research Institute V. THEE EXACT SOLUTION FOR MAXWELL MOLECULES Although not quite necessary for the solution of our problem, it seems worth-while first to point out that for Maxwell molecules and for arbitrary ratio of the masses, the eigenfunctions and eigenvalues of the collision operator can be determined. For Maxwell molecules gI(g,G) is independent of g. Putting gI(g G) = AI2K (M + m) gI(g,~G) 8tl F(G) then F(G) is the dimensionless function discussed in a previous report. Introducing in the exact collision term (3) mv2 f = e 2kT h (16) then from the energy conservation in the collision (v,V) + (v', V') it follows that nn2 J(f) = N e 2kT I(h) (17) where I(h) is the dimensionless collision operator I(h) = 72 dC e dl F(G) (h' h) (18) and C = V (M/2kT)/2. It is also conveniEnt to consider h as a function of the dimensi:orless relociaty z _ (m/2'kT)1/' and then he means h(-C), where C: =c + M - * ( Mt - c] (19) is the dimensionless velocity of the particle after collision. Since the operator I does not depend on the velocity c, and since the connection between c and. c is linear, it is clear that the eigenfunctions of I must be polynomials in c. Since the operator I is spherically symmetric in the velocity space c, the dependence of the eigenfunctions on the direction 6. Co S. Wang Chang and G. Eo Uhlenbeck, "On the Propagation of Sound in Monoatomic Gases,? Eng. Rese Inst. Proj. M999, Univ. of Mich., Ann Arbor, October, 1952, Appendix I. 7. The symbol s is the unit vector in the direction of closest approach; s makes an angle (ic + )/2 with = - c, 6S

The University of Michigan * Engineering Research Institute of c must be like a spherical harmonic. In fact, one can prove that the eigenfunctions are of the form trem(c) = Nr~m c y2m (c,) S +1/2 () C (20) where (r) S +)1/2 (C2) is the Sonine polynomial of degree r and order 2 + 1/2 and Nrm is a normalization factor. The ram form a complete orthogonal set of functions with the weight factor exp (-c2). The corresponding eigenvalues are given by ri = 2Ji dG sin G F(G) [ sin- (s + 2 (1 (21) P2 is the Legendre polynomial, and F mM 2 2M ( 2M cos V = - - sin — s. (21a) (m+M)2 im+M 2 For a proof of these statements see Appendix II. Note that the eigenfunctions (20) are independent of the mass ratio m/M, which enters only in the eigenvalues (21). The first eigenvalue koo = O, corresponding to oo000(c) = constant; this expresses the conservation of the number of particles in a collision. All other eigenvalues are negative, and this expresses the tendency of f to go to the Maxwell distribution. The reason why our problem of the power loss can be solved exactly for Maxwll molecules lies in the fact that the velocity ~ is ag eigenfunction of the collision operator I, corresponding to r = 0, and 2 = i. By developing h in the eigenfunctions trl.m, one sees that because of the orthogonality property of the trim d ci e I(h) = ko, dc ci e h. (22) As a consequence, one obtains from the Boltzmann equation for the average valN ues xi and Vi the equations dt di =_ xi + Eo cos V t in vi, 8. Note that Sk(O)(X) = 1. 7

The University of Michigan ~ Engineering Research Institute where T is given by = - N /2K (M + m) 01.=...N (M+. dG sin $ (1 - cos M) F() - n'- m+M o We see therefore that for all ratios of the masses one obtains the same functional dependence as in the Brownian motion limit. In the limit m/M >> 1, the value of 71 goes over into the value found in the Brownian motion limit (see Appendix I). In addition, one can show as a check that the eigenfunctions of the Brownian motion form (6) of the collision operator are again the ArRm given by (20), while the eigenvalues are equidistant and equal to Ar2 = - + (2r +,) For the proof, see Appendix III. It is therefore clear that the strong collision approximation discussed in Section III cannot have a general validity independent of the intermolecular forces. Especially, it cannot be true that in the limit m/M << 1, which is opposite to the Brownian motion limit,9 the collision term can be approximated by the strong coupling form (10) for all types of intermolecular force laws. Of course, it may be that the inverse fifth power law gives too "soft" collisions. It is therefore of interest to investigate other force laws and especially the case of elastic spheres. VI. THE SOLUTION FOR ELASTIC SPHERES IN THE LORENTZ LIMIT IF o = 0 If m/M << 1 and wo = O, then our problem can be solved by an adaptation of the perturbation method used in the Lorentz theory of electronic conduction in metals. The distribution function f now depends only on the velocity v and the time and fulfills the equation 6f iit af + Ee e x = J(f) (24) Considering the second term as the perturbation which prevents f from reaching the equilibrium distribution 9. It may be called the Lorentz limit, since it corresponds to the situation considered in the Lorentz theory of electronic conduction in metals. 8..

The University of Michigan ~ Engineering Research Institute fo ( 3/2 exp (-mv2/2kT) one obtains, by putting f = f (1 + h), (25) for h the inhomogeneous equation -ah mvX EO et = SfV:F(V) ddQ gI(gG) (h' h) (26) t kT Because of the linearity, h will be " exp (iwt) in the steady state, and in the limit m/M << 1, h will have the form iRot h ho(v2) vx e, (27) since in the lowest order of m/M the velocity v does not change in magnitude in a collision, and g may be replaced by v. Hence, substituting (27) in the right-hand side of (26), one gets jdV F(V)/ da vI(v,G) ho(v2) (v - vx) 2tN ho(v2) vx dG sin G(1-coS Q) vI (,G ) Therefore, for elastic spheres [I (v,G) = 1/4 o2, where a is the average of the diameters of the spheres with masses m and M] one obtains m Eo 1 kT' i1) + It g2 Nv Introducing the relaxation time T 1 4L IT (29) -o c2 N 2kT one gets for the average velocity in the x-direction 8E = el~t dc e 1 (30) ad3 IC4e o c + ica r and. hence for the average pover absorbed

The University of Michigan. Engineering Research Institute 4 Eoa 2~ P d= c c5 eC.2 (31) o2 + W2T2 2 Eo 2T -K2T2 [1 - a2T2+ W4T4 e Ei(-a2T2)] 3 qT where Ei(y) is the exponential integral (On as Ei(-y) = dx e P as function of a is shown in Fig. 1. One easily verifies that the area under the curve is again DEo02/4, just as for all the other curves. One sees that the elastic-sphere result lies between the Van Vleck-Weisskopf and the Luttinger results. In fact, for large oTr - 2 4 PVW rv ----- PL - ------- W2T2 L 09T2 while from (31) one obtains 16 3.01 P 5 j2v3 = W2m -D2T2 One can solve the problem also in another way, which is of interest since it may be generalizable to the case where wo is not zero. Expand the perturbation h in the eigehfunctions (20) of the collision operator for Maxwell molecules. Then we can write f = fo + A Crer,,(C) ej,a (32) where c = v (m/2kT)1/2 and ark are the development coefficients.l0 Substituting in the Boltzmann equation (24), multiplying by *trlv exp (-c2), and integrating over c, one obtains a set of linear equations in orolll 10. Because of the axial symmetry of the problem we can take m = 0. 11. The Nrl are normalization constants, determined by /di exp (-c2) r2 = 1, which gives Nr - = (r' ( 2 + 1 r).... Jl_ 1_, _10

The University of Michigan ~ Engineering Research Institute fTh 2 E2 I 2i a NoW Cro i^ V*er [r r'ef] ar'P ' (33) rv,,i where the bracket symbols are defined by: [trg, 'r, eS =,dc e C2:r~e(c) dV F(V) d2 gI(g,Q) [tr'P'(c) *- ir'(C)] For arbitrary ratio of m/M the calculation of the bracket symbols is complicated, but in the Lorentz limit m/M << 1 the result is again simple As shown in Appendix 4 one obtains in this limit for elastic spheres: 21 N' N (s++) s (rl - -s)' (r- ) ________ 7& & (s+~+l)'(r -3/2-s).(r-3 -s) (34) 2r+1 s! (r - () r (rs)3 Since the equations (33) are not coupled through the index,. and since the inhomogeneous part contains only ~ = 1, we can restrict ourselves throughout to ~ = 1. The equations (33) can. then be written in the form i UOr - X brr rt =. ro, (35) r = 0 N=O with r = O, I, 2 o.. and where brrs is the bracket symbol T[rli, *r l]~ The average velocity in the x-direction depends on a%; in fact, v _ N01 a~o 4, it From (35) follows C o = 2 Eo T D0(icT ~ a No01 Do(i~) ' where D is the determinant I| icT 6rr brr II and Doo is the minor of the (o,o) element~ Thus Doo (ior) =x EoT ](iAT) eST (356) which must be compared with Eq. (30)o The identity of (30) and (56) for = 0O... 1....

The University of Michigan ~ Engineering Research Institute has been shown by Chapman.12 We also verified the identity for large w. A complete formal proof of the identity is lacking. VII. CONCLUDING REMARKS If wo is not zero, then the perturbation h will depend on the coordinates as well as on v and t. It seems feasible to generalize the second method of the previous section by developing h into products rm(x) v using the same type of functions in both x and v, and again considering the Lorentz limit. However, the details have not yet been worked out. Presumably for elastic spheres and in the Lorentz limit, the result for wO f O will always lie between the Van Vleck-Weisskopf and the Luttinger result. There is one feature which the exact result will have in common with the Luttinger result, namely, that for wo J 0 and X = 0, will be zero. This is clear, because in this case the constant outside force will only polarize the oscillator and in the steady state f will be the Maxwell-Boltzmann distribution so that vx is zero. 12. S. Chapman, J. London Math, Soc., 8, 266 (1933). _.,.. v.. 12

APPENDICES

The University of Michigan ~ Engineering Research Institute APPENDIX I PROOF OF THE BROWNIAN MOTION FORM OF THE COLLISION OPERATOR As in Section V we put in the exact collision term (3) my2 f = e 2kT h (l-A) then again it follows that mv2 /M 3/2 2kT J(f) = N( - e I(h), (2-A) \2:r kT/ where MV2 I(h) = dV e dekT dG sin G gI(g,G) (h' - h). (3-A) Since for m/M >> 1, vv differs very little from v, one may make a Taylor expansion: w1~ 1 __ _ h' - h = (vI v) + ' (v" - v) (v - ) v ac +.. (4-A) From the momentum conservation follows v' -V = - m+M -g (5-A) wherV, g= - O v= are the relative velocities before and after collision1i3 Introducing (4-A) and (5-A) one can integrate over the azimuthal angle c, since only g' - g depends on e. Next, introduce in the velocity space V polar coordinates with the direction of v as polar axis. One then can again integrate over the azimuthal angle of V. In these two integrations over azimuthal angles, the following general formulas are used, which are easily verifiede 135 Of course, g' I = Igl, and Q is the angle between g' and g. 14

The University of Michigan ~ Engineering Research Institute dO (Ai - Bi) = 2 cos 8 - Bi jo2n dO (Ai - Bi) (Aj - Bj) = 2i BiB cos 0 - 1 12 A2 1 Asin2 '8 2 2 J 2 i2 In here B is a fixed vector and 0, 0 are the polar angles of the vector A with respect to B. Now it is convenient to use dimensionless variables: m1+ / _M -+ c = m2T v C = V. (6-A) 2kT 2kT Note that we use different units for v and V. Assuming, as we will from now on, that c and C are of the same order of magnitude means that we have introduced our second assumption, namely, that v never differs very much from the equipartition value. With these units 4.~ + M 21/2 12T / 1i c M c g A= '- C - 2 - cos + m C where- is the angle between C and c. Up to order (M/m)1/2, one has therefore 12kT,I r 1 C 0 C gI(g,G) = 'M CI( M- C, - 7 cos+ Developing also the rest of the integrand in powers of M/m, one obtains M /2kT\2 00 2 I(h) 4 m 2 M JO d9 sin (1 - cos ) dCC e o c do sin 0 I ( 1 Cos [ ( [ Acoas C ca- Lca + 8 (3 cos 9 - 1) (3 cos2 - l) o C2 aa + I cos2 0 - 3 - cos (3 cos - ) aaah +c c } o MI~~~! 15 o2; aa2-k

The University of Michigan * Engineering Research Institute Carrying out the 0 integration, keeping under the integral sign only the terms of order one,4 and making the partial integration p00 _C2 aq i2C J dCdC e dC e I (4C3 - 2C5), one obtains lt2 M 'A 00 2 I(h) = 8 M (k' A dG sin (1 - cos c ) dC e 3 mn C5 I (M / f 3) caa' - 2ca.Introducing this expression in (2-A), and going back to the original velocity variables v and V, and to the original distribution function f, one gets J(f) =, '- f+ — with MV2 i6 J M 2M i0 2kT 6- - dG sin G (1 - cos G) dW5 e I(V,@) 3 m kT o 0 For a repulsive force Kr S, we write for the differential cross section15 4 I(g,G) = g -1 F(,Ks) and then 8 A; NM IT 2 ( - 1r s* 8 - J M- (M) — 2 s0- dG sin G (1 - cos @) F(G,K,s) Especially for Maxwell molecules (s = 5), 16 14. It needs some further argument to show that it would be inconsistent to include higher-order terms in M/m, since then also the further terms in the Taylor development (4-A) would have to be included. 150 C. So Wang Chang and Go E. Uhlenbeck, "Transport Phenomena in Very Dilute Gases," CM 579, UMH-3F, Univ. of Mich. 16. F(G) = K"+ F(G,K,5) 7 XF(G,K,5) is the dimensionless collision cross section used in Section Vo 16

The University of Michigan * Engineering Research Institute = 2N m dG sin G (1 - cos G) F(G) and for elastic spheres (s = oo; a = diameter), 7 8 J- NMv2 /2kT = 3 m VM l7~ For elastic spheres the Brownian motion form of the collision operator was first derived by M, So Green, J. Chemo Phys., 19, 1036 (1951)o 17

The University of Michigan ~ Engineering Research Institute APPENDIX II EIGENVALUES AND EIGENFUNCTIONS OF THE MAXWELL COLLISION OPERATOR The dimensionless collision operatorl8 1 de I(h) 3/2 de esc 2 d F ()[h(') - h J can be written in the standard form1l8 1 M)3/2 c2 -I(h) = A h + 3 (/2 / e de' K(cce) h(c') where AO 20 t dG sin G F(e) dn cos -csc2 2 F(6) exp L ( A,) c - c ) cot2 27j 2 2 L m+M(- I xnt-M 3-B- otG d Jo si - cc' sin (c,) cot 2 2 We now will verify that rimr = c Sr I(C2) Ym (0X) 2 is an eigenfunction of I. Let B, X' be the polar angles of C' with respect to the same set of axes as used for c. For the integration over ' we take the direction of c as polar axis and let 0, X1 then be the polar angles. 18, D. Enskog, "Kinetische Theorie der Vorggnge in mnssig verdunnten Gasen," Dissertation, Uppsala, 1917, p. 154. Note that Ao is divergent and really should be kept together with the second term. 18

The University of Michigan * Engineering Research Institute The integration over x1 can be performed, using 2T dX1 YA m(, ',x') = 2i Pi (cos 1) Y~ m (~,X), (7-A) o and one obtains M2 (m-M)ec2 I/ M+.m 3'~ 4.mM I(Arlm) = - Aoir1m + e( ) (M+m)2 C2 C et2 Yrm(,-) ~X) dG cos csc2 - F(G) e cmM 2 2 o 2+2 (r) (+M) C'2 CSC dc' c' Sr l(c2) e 24mM 2 0 dol sin 01 P2(cos 01) Jo -i m+Mcc sin cot (+M c sin ~l cot. o exp 2cct [ (m+M)2 t 2 cos }.ep L + 21ot C0 mm2 Putting = (mM)2 csc2 1 4mM sir cc ' - c 2mM 2 (m+M)2 2 1 2M sn2 m+M 2 cos = 4mM (m-+M)2I 2 then the last integral can be carried out, using do sin 01 Pi (cos 1) e c(z csin l sin i - 0 t ii Pi (cos *) JQ+ 1 (z) - 2 Writing z = c~ c', the integral over c' can be carried out next with the help of the formula 19

The University of Michigan ~ Engineering Research Institute + (r) q2c (c') dc' c' 2 SR+ 1 (c2) e- 2c2 + 1 ( 2 2 o 2 _ () 2... -a 42 a +31 ' q) e Sq 4 r 4q2(q_2-j (2q2) 2 where q2 is an abbreviation for 2 (m+M) S2 _ 4mrM 2 Putting everything together, one obtains I(Arlm) = - Ao* rim + 21 trim itd sin F(G) P2 ((COS ) (rm) 2 S 1fL2 2 which is krQ *rfm, where the eigenvalue Xkr is given by the equations (21), (21a). 20

The University of Michigan ~ Engineering Research Institute APPENDIX III EIGENVALUES AND EIGENFUNCTIONS OF THE BROWNIAN MOTION FORM OF THE COLLISION OPERATOR The question is to find the eigenfunctions and eigenvalues of the differential equation Oyef + kT fU = f Ova (a m Av C l Introducing the dimensionless velocity c = v (m/2kT)1/, we find Af + Ca + 6f f The angular dependence is clearly like a spherical harmonic, and putting f = c R(c) e-c Yem(e,), one gets for the radial function R. Ld2R +2(+l) 2-2 cdR 1 + 2J R = 0 dc2 c c dc L 1 or using x = c2 as independent variable, x d + 2.+ - x-= 0.:2 + 2 dx 2 dx'2 St~t r(r) dx2 + (t + 1- x) + r St O (r) I I one sees that R = S +(c2) and that r = - + ), so that the eigenvalue A2r = - rj(2r + ) It is also easy to verify that the exact formula (21) in Section V leads in the BrowTnian motion limit to the same result~

The University of Michigan ~ Engineering Research Institute APPENDIX IV THE BRACKET EXPRESSION [r t'm't, trim] IN THE LORENTZ LIMIT In the limit m/M << 1 we will calculate the bracket expression for a repulsive intermolecular force equal to Kr-S, since this contains the Maxwell model and the elastic-sphere model as limiting cases. Writing as in Appendix I, s-5 gI(g,G) = gs-1 F(G,K,s), the bracket expression becomes 3/2 cMis/2 d le [~~r Q RN m r dm ek (- ~ 2k_ //kT / fdV e 2T de dG sin G gS-l F(G,K,s) 0o -2kTW Rm( W m ( 2kT 2 l Since mv2 is of the same order of magnitude as MV2, in the limit m/M << 1 and to the lowest order in m/M, one can replace g' by -v' and g by -v. Using Eqo (7A) of Appendix II, the integral over e can be carried out, and also the integral over IT is immediate. With dimensionless velocity variables, one then obtains 2o [\1frtsmvVrrml 2,ftN(,4In dG sin 9 F(G,,K,s) s-5 -c2 s-1 [P (cos 9) - 1] d e c s r,,m, (c) *rim() Since the angular dependence of the *rim is a spherical harmonic, the integral over the directions of c is immediate. Putting in the explicit expression for the Sonine polynomials, the integral over c can be carried out in each term and the result is a double sum, of which one sum can be evaluated with the help of the formula 22

The University of Michigan ~ Engineering Research Institute s One obtains [F ='m r 4~~2 ~(t2,2kT 2(s-1) L[~sR?2tiVr2 2I+1 =Nr1 Nr Nr''6t 6mlm' dG9 sin G F(G,K,s)[P(cos G) - 1] s+ 2 2 2(s-l r - 2(s-) r - p 2( sp)(r' - p)r - p) p Since 2 - 2> s-5 )> O for oo > s 2> 5 only the 2 - 2(s-l) - case of Maxwell molecules (s = 5) needs special consideration since then factorials of negative integers appear. In this case one finds either directly or by a limit consideration, ['r'~'m''Vrem] = 2Q+1 NrQ Nr'tI brrt I625 6mm' -- dG sin G F(G,K,5)[PS(cos G) - 1] o Finally, for elastic spheres (s = ), one gets [r' I m' Irm] = 8itc2N t2kT ~r8m' trem] = 2~+1 Nr Nr Nr't ' 622, 6mm m (p + 2 + 1)'(r -2 2 )(' - p) ' ~L p'(r' - p)'(r -p) p which reduces to the expression (34) used in Section VI. 23

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