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

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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....

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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

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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

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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.

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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

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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

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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~
"~~~~~

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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

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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

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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..

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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

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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

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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....

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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

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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

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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

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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

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= 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

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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

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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

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+ (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

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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~

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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

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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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