connected-diagram technique.5 This expansion reduces exactly to the density expansion obtained by Kawasaki and Oppenheim6 in the classical limit. By com-.
ON THE NONPOWER DENSITY EXPANSION OF TRANSPORT COEFFICIENTS* BY SHIGEJI FUJITA DEPARTMENT OF PHYSICS, UNIVERSITY OF OREGON, EUGENE
Communicated by Terrell L. Hill, June 13, 1966 1. Introduction.-Recently, several investigators, including Dorfman, Cohen, Sengers, Kawasaki and Oppenheim,1-3 demonstrated that the formal density expansion of transport coefficients for a classical imperfect gas is term-by-term divergent. Those authors,2' 3 cited above explicitly, went one step further and argued the existence of terms involving In n, where n is the density. In an earlier paper,4 the present author derived the formal density expansion of a transport coefficient for a quantum gas obeying the Boltzmann statistics by the connected-diagram technique.5 This expansion reduces exactly to the density expansion obtained by Kawasaki and Oppenheim6 in the classical limit. By comparing these two treatments, one can see that whether the system is classical or quantum-mechanical does not change the formal mathematical structure in an essential way. This is not surprising and could be expected in advance. However, in the actual calculation by means of a perturbation theory the differences of classical and quantum systems become more explicit. For example, in classical mechanics the calculation of the scattering crosssection by a potential (perturbation) must be made by an infinite-order perturbation theory, since finite-order perturbations can produce only an infinitesimal amount of change in the momentum of the particle. On the other hand, a quanltum-mechanical potential can induce a finite amount of change in momentum, and this fact makes the finite-order perturbation calculation in quantum mechanics physically interesting. It is in fact by the use of this simplifying nature of the quantum theory that we attempt to investigate the question of the nonpower density expansion in further detail than hitherto attempted. We shall show that in contrast to the earlier investigator's arguments, terms in the combination of n and In n for a transport coefficient in a quantum gas are not likely to arise. This is based on the calculation of the momentum transfers between particles explicitly in the lowest order in potential. The differences in system, quantum or classical, and in potential, weak or strong, do not seem to change the character of the nonpower density expansion of a transport coefficient. In the text the same notations as in A4 are used whenever possible. 2. Calculation of a Viscosity Coefficient. In B,5 the discussion of the generalized Boltzmann equation was given in the position representation. With the assumption of the absence of magnetic fields, the calculation is simplest in the momentum representation. The results obtained in the position representation can be simply transformed, of course. We shall summarize the main points reinterpreted in the momentum representation. The prescription of drawing diagrams is unchanged since the diagrams do not depend on the representation. A diagram is composed of (solid) particle, (dotted) interaction, and (broken) correlation lines. A particle line is called free or nonfree according to whether or not the diagram is broken in two by cutting it. 794
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Before giving the prescription of obtaining the corresponding expressions, we shall introduce the P-k representation which will facilitate the practical calculation. Let us specify the p' - p' element,
I=-- (P1'(1),P2'1(2) ... YPNI(N) of an arbitrary N-body operator A by the set of numbers (P,k) such that (p'IA IP') Ak(P)-AA,'.p"(1/2(p' + p')) p
1/2(P' + p'), k = pl _
P=
(2.1)
-
or P (o
=
1/2(PI(°) + p (J)), k U°
Let us define a matrix 3C with elements
(k3jCjk')
p 'U) - p fi). such that
(2.2)
Ia;C(P)Ik')
(k _ qk'Hk-k,(P)n-k - OlkHk.k,(P)lk where +k is a displacement operator acting on a function of P such that n±k(J)f(p(J)) = f(p(i) :1:/2k(J))nj>k(j)
(2.3) (2.4)
More explicitly,
jfio(') + X E 1(Jk)
3C
j
(k(j)J N(J (P(J) 'k'(J))
(2.5)
j>k
M-lP(J) * k(0'b(3)k(),k(i) (k(JOk(k)lb(jk)(p(J) p(k))~k/(JOk (k))
-
=
v(k(1k(k)k/(Jk'(k)) (7k(j) -k( )17kk(k)-k(k)
-
lj-k(J)+k(°k -k/(k)+k(k))
(2.6)
v(k 1, m n) = v(k-m)6(3)(k + I-m-n ). (2.7) The time derivative of the momentum distribution function F*(P,t) -= no*(P,t) can be written as anO*(P~t) = G(Pt;n*) + D(P,t;n*,X*), (2.8) which is obtained by transforming (B.3.11) and assuming zero magnetic fields. The part G( ;n*) can be formally written as [see (B.3.9) ], (3) l) ))etiot {-iX(OjVu(t)|k) G(P,t;n*) =[ (Pf-P o
+
(01 [¶(t)V(T) .V(rTk) ](c)lk)
}
rt (rk-1
3(-iX)k+l dT... 1o
I[d3k(J)d3P(J) exp [i o(J) (°]ink(*(P(i),r()) ],
j=l
o
dTk
(2.9)
(t) = eiX0e''-iaco, (2.10) where T(J denotes the smallest of those T. which characterize the interaction 3U involving the particle j.
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It is found that all density operators n* become diagonal in momentum. Then, the integrations with respect to k-variables can be simply carried out and, as a result, G becomes a functional of the momentum distribution function no*(P,t) F*(P,t) only, as mentioned in A. The other part D is given by [see (B.3.10) ]
D(Pt;x*)
=
Tr
(npe-'I3Ct{
iX [eU(t)p* ](C)
+ (-iX)k+lfdrTi dr2...
drk [1(t)(r) .. .. V(rk)p*]()}). (2.11)
Since these connected products involve particle correlations existing all the time from t = 0, the contributions contained in D cannot be expressed in terms of n* only. The Laplace transform of G(P,t;F*) can be simply calculated by using the inversion formula in (A.2.5). G(PzF*) =
-if
f
(3)(p(1)
-
P)(0lg(zz(1). .,z(m);F*)Ik)[z(l)
- z
mi
X II 1=1
27r
d3P()d3k(1)dz('),
(2.12)
where the operator g can be read from diagrams as follows: The particle lines may be numbered 1,2,...m from below. Corresponding to the interaction line at the far left, connecting the particle lines 1 and 2, write a factor b(12). Corresponding to the parallel particle lines between the first and second interaction lines from the left, write a factor [3Co - z]-1. Corresponding to a further interaction line connecting the particle lines j and k, write b(jk). Corresponding to a further interval between interaction lines, write a factor H F,* [3Co + Ez(z) 1 1 - z] ', where 1 runs over those indices of the set (1,2,. . m) whose particle lines are free. Continue to write factors until the final interaction line is reached. Sum over the particle indices. Take the 0 - k element in the k-representation. This gives a contribution to the (Og(z,z(1),. .z(m); F*) |k). The Laplace transform of D in (2.11) may be written as
(P) P)(Oid(zz('),. Z(k);F*,X*)Ik)[z(z)
D(Pz;F*,x*) = f f f
-
- z]
k
X HI -d3P()d3k(')dz(')Ild3P(j)d'k(j), 1
2vr
(2.13)
i
where the index 1 runs over the number k of those free particle lines found on the right of the last interaction line, and j runs over the rest of the particle lines; the operator d should be read from diagrams in the same way as for the operator g, except that additional factors of cluster operators x* are given as explained below. According to the analysis by Ursell and others,7 a density operator PN corresponding to the N-particles has the product and cluster properties such that PN =
2IlPNo, 1 _ Ni < N;
(2.14)
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and each PNi can be decomposed into the sum of cluster operators x such that Wl'pill) _ (1'lX111) (2.15) (1'2'Ip2I1 2) (1'1XIl1) (2'Ixij2) + (1'2'1X211 2)
where ji> are a complete set of kets corresponding to the particle i. Substituting (2.15) into (2.14), we see that PN can be decomposed into the sum of products of x. These x may be represented by broken lines connecting clusters of particles. Let us now discuss the calculation of the part X2. The quantity J2*(t) has a structure similar to bnk*(P,t)/bt from the viewpoint of the connected-diagram analysis. It can be written as a sum of two contributions:
J2*(t)
=
Gl(t;F*) + DI(tF*,x*).
(2.16)
For G' and D' we may draw diagrams identical with those for G and D, respectively. The prescription of obtaining the corresponding expressions is the same, except for the rule that the first interaction line should contribute J2(12) instead of 12p(lI)
The Laplace transformation and the u-differentiation can be carried out in an analogous manner. We can easily verify that the contribution of X02 begins with a term in the order n'. 3. Divergence and Its Elimination.-According to the analysis of the earlier investigators,1-3 the formal density expansion for a transport coefficient has divergence difficulty. In order to save space and to make direct comparison with the
r
0() b
a L I
L
l
I
. T~~~~~~~~~~~~~~~
c
d FIG. I.-Diagrams representing components of G. Diagrams b, c, and d contain self-energy diagrams and lead to term-by-term divergence.
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PROC. N. A. S.
work of Kawasaki and Oppenheim,3 we shall follow their notations as closely as possible. The diagrams of divergent contributions are found (see below) to be those like b, c, d in Figure 1, which contain indefinite numbers of self-energy diagrams decorating the particle lines 1 and 2. These were referred to as ring diagrams with loops. After summing these ring diagrams, they arrived at
12 )2 fd3Afd3p2VT(12)(Ok-k)ga(k,-k)A(k -k) (2ir)2 X [1 + pA(k,-k)]-1T(12)(k,-kO)sP(p2). (K-O-III, 3.12) In our notations, = Q,p n, P- P, P2 = p(2),
