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Van Kampen, N. G. 1961
Physica 27 783-
A SIMPLIFIED CLUSTER EXPANSION FOR THE
CLASSICAL REAL GAS
by N. G. VAN KAMPEN Instituut voor Theoretische Fysica der Rijksuniversiteit te Utrecht, Nederland
synopsis Mayer’s expansion of the partition function of a classical real gas in terms of irreducible cluster integrals is derived by a simpler and more direct method. The two principal features of this method are the following. (i) The partition function is expanded in an infinite Product rather than a series. As a result the exponential form is obtained immediately: there is no need to sum up infinite sets of graphs. Disconnected graphs never enter. (ii) The calculation leads directly to the canonical N-particle partition function. Neither the fugacity nor the reducible cluster integrals are introduced. The same method is also used to find the expansion of the pair-correlation function. Finally it is applied to the partition function of a real gas in an external potential field.
- The first a@roximatiolz (second virial coefficient). In classical theory the problem of finding the equation of state of a real monatomic gas amounts to evaluating the configurational partition function QN = f e-@(v10+~13+. ..Wf-1.~) drl dr2... d,.,.
The domain of this 3N-dimensional integral is determined by the fact that each particle may move throughout the volume I’, irrespective of the positions of the others. Hence the integral may also be regarded as an average over N-particle configurations,
QN = VN W.Y?Q~... FN--I,N, (^) (1) where ~1s = e-@‘ls (^) etc., and the bar denotes the average over all positions of the particles inside V. The v’s are functions of the random variables ri,.. ., rN and v12 -+ 1
as Irr - r2l + co. It is clear that 9~s and q134are statistically independent,
and also ~12 and ~13. Thus
**ql2yl3 = ‘?‘12ql3 = 9)12”.**
However, the three functions ~12, 913, ~23 are not mutually independent. Yet one may write (^) --- 4)12q13’$‘23 = ‘?‘12(p13’$‘23 (^) (2)
784 N. G. VAN KAMPEN
as a first approximation for low density. The rationale is, that this equality would be correct if one of the ~J’S were replaced with 1; but the configura- tions for which all three q’s differ from 1 are rare if the density is low. For products of 9’s involving more than three particles a similar argument applies. One is thus led to the first approximation -- ~ Qg)/VN = ~123713... TN-~,N = (z2)tN(N-1)
This yields the usual result for the second virial coefficient in the following way.
In the limit of a large system (i.e., N + co, V --f 00, N/V = n = constant)
lirn CQ~') 1’N V
= exp [+z s
{e- ‘v(r) - 1) dr] = exp[+/?r],
where fix is Mayer’s first irreducible cluster integral 1)s). Thus
Q$) = VN exp [$ ~11.
This is the familiar first term in the expansion in powers of the density, which leads to the second virial coefficient. It should be noted that the correct exponential form is obtained without summing over an infinite set of graphs with cumbersome combinatorial factors.
- The second approximation (order ns). The above method of arriving at the second virial coefficient is not new 3) ; however, our real problem is to find the higher orders in the expansion. This will be done by supplying successive correction factors, rather than additive terms. These correction factors describe successively the statistical correlations between the q’s, which have been neglected in the first approximation. It is reasonable to expect (and it will be verified by the result) that the next approximation involves three-particle correlations. The triplet of particles 1, 2, 3 gave rise in the first approximation to the factor (2). TO make up for the error committed there one has to multiply by the correction factor 912(?‘139)23 (^) **= q124)139)
q12p113q’23 G”**
As there are (f) triplets of particles, this factor has to be raised to the power (F). Writing, as usually, ~$3 = 1 + fti one finds for the total second order
786 N. G. VAN KAMPEN
All terms of the classes (i) and (ii) belong to lower approximations and are therefore also present in D. The terms of class (iii) are necessarily of order V-(k-1). There are also corresponding terms in D, that is, terms made up with the same factors f but erroneously treated as reducible. It is clear that such terms are of order V-k or smaller. In general, each term in D is identical with a term of class (i) or (ii) in the numerator, or else O(V-k). Hence (4) becomes
1 + c f12f13 a.. + o(v-k)~
where the summation extends over all irreducible k-particle terms. This is just the usual irreducible cluster integral:
2 fl2fl3 a.. = Ik)
@l/n_:)! /9,&l.
The total correction factor to the configurational partition function per particle, (QN)~/N, is obtained by raising (4) to the power
The result is
Combining these correction factors with the first approximation (3) one obtains the familiar result
QN = VN exp [Nk!I (G)” $&I.
- The pair correlation function. The^ pair^ correlation^ function^ gis^ is defined by
-=^ g12 _^1 V2 QN s
e-~(V~a+Vls+. ..~~N-I,N) dr3 dr4... drN.
This integral can be evaluated by the same method. First we denote the two fixed particles by a and b rather than by 1 and 2, to exhibit clearly that they are not involved in the averaging. Second we note that all factors that do not involve both a and b may be discarded, because the normalization of gab can be found a posteriori from the condition *)
Hence we write gab = conk Tab vu1... TaN vbl... FbN v12v13. * * p7~-1, N.
*) All terms of order l/V have already been omitted.
A SIMPLIFIED CLUSTER EXPANSION 787
Again the first afiproximation is obtained by regarding all factors as statistically independent. In the result the only factor that depends on both a and b is q&, so that
&,’ = const. I$7ab= e-8vaa. (^) (8) The next a#woximation takes into account correlations that arise from one additional particle. These can be accounted for by supplying the cor- rection factor
val vbl N [ 1
--. Val Vbl Expansion in f’s yields
1 + 2% + falfbl N
- 1 + 2fal + G2 I.
As both fai and falfbl are of order V-l, this is equal to
(1 + falfbl + o(J’-2))N = exp [q/falfbl dri].
Combining this correction factor with (8), one finds the familiar result 4)5)s)
(e-Bv.l (^) - 1) (e--Babl - 1) dri 1
.
It is readily seen how this process continues step by step. For the k-th correction factor one selects k particles in addition to a and b, and writes the product of all v’s that connect two of these k + 2 particles, excepting qab.
p= gg’l’ qal^ .-a^ ~alc~bl^...^ ‘$‘bk’?‘l2... vk-l,k^ (f) g’k,’ a D 1 * (9)
On expanding in f’s the numerator consists of four classes of terms: (i) terms involving less than k + 2 particles; (ii) terms that involve all k + 2 particles but are reducible, in the sense that a factor can be split off which does not contain a or b; (iii) irreducible terms involving all k + 2 particles, which however decompose in two or more disconnected parts when a and b are erased; (iv) irreducible terms involving all k + 2 particles which remain con- nected on erasing a and b. It is clear that the terms of classes (i) and (ii) belong to a lower approxi- mation and therefore also occur in D. Each term of class (iii) decomposes in two or more products which have no other particles in common than a
A SIMPLIFIED CLUSTER EXPANSION^789
In the next approximation correlations between two particles have to be taken into account. The new correction factor is
- &N(N-1)
A calculation along the same lines as before shows that this is equal to
(12)
There is, however, an alternative method for computing QG, which is very convenient for obtaining a general expansion. We define a new averaging process (distinguished from the previous one by,an asterisk) as follows. For any function xi of the coordinates of particle 1 we write
-* j-x1$1dtl x1+ xl = / t,bl drl = 7 ’
(13)
Moreover it is convenient to put
The definition of this average for more particles is obvious. Equation (11) may now be written
Q; = (v)Np112ff'13 ... TN-l,N>**
in complete analogy with (1). The calculation of QL can now be copied from the calculation of QN in section 3. The result is
Q& = V*Nexp[Nkgi(g)X$&-],
where the & are defined, in analogy (6), by
(
(15)
the summation extending again over all irreducible K-particle terms. The first term in (14) is identical with the approximation (12). The result (14) can be employed for an alternative calculation of the pair- distribution function. Indeed, according to the definition (7) one has
i?ub Vz=e
wBvob (^) -_Q&-- QN ’ where Q&, is the partition function of the remaining N - 2 particles in
790 N. G. VAN KAMPEN
the field caused by a and b. From this identity follows
Because of the special form of the external field,
v* -
- (^) V = .& = ~‘arv%r = 1 + + /!Q+ +- hz,l(a, b).
Hence
(T)“= exp[F/h+ Gh,l(a.b)].
Substitute this in (16) and rearrange terms:
gab = e-BuaoeXp (^) ;kz,l (a, b) x 1
It is shown in appendix II that this result is identical with the usual formula (10).
Appendix I. Explicit verification of the third approximation. Consider the particles 1, 2, 3, 4; the factor to be computed is
~12($‘13qJl4($‘23q’24~34. (^) (19)
This factor can be represented by the graph in fig. 1. The first approximation is (^) ------ ql2 vl3 ‘7’14v23 F24 9)34 = G6.
w
Fig. 1. The graph for the q’s.
In second approximation this was corrected by the factor
[g11W$3]4;
792 A SIMPLIFIED CLUSTER EXPANSION
If one writes vai = 1 + fal, etc., this takes the form
h-1 =
Vk- Iz fl2..., (k - l)! & (k>;a,b
the summation extending over all graphs which connect the particles 1, 2,.... k in an irreducible fashion and are connected by any number (from zero to 2k) of lines with a and b. The contribution to &_i of those graphs that are not connected with a or
b is (a-lbk-i. All other contributions are proportional to V-k. Hence the right-hand side of (23) is, apart from terms O(V-I),
v 1 -( )
V’” -
_- 1 bk-1 + k &
(
The first term is equal to (compare (17))
-^ V k - $ pl - $h2,1) pk-1 + o(v-l) (^) (25)
The second term contains the contributions of the following four types of graphs. (i) Irreducible k-graphs linked once with either a or b. Each of them contributes to this term
____^ Vk 81 (k - I)! k!
Vk pk_l = !!$s.
As there are 2k of such graphs, their contributions exactly cancel the first term of (25). (ii) Irreducible k- graphs of which one vertex is linked with a and b. It is easily seen that they cancel the second term of (25). (iii) I rre duci ‘bl e k-graphs of which two or more vertices are linked with a and none with b (or vice versa). These are the irreducible (k + 1 )-graphs ; their contributions cancel the last term in (24). (iv) The remaining graphs are just the ones that make up hs,k(a, b) - which proves the identity (23).
Received: 6-5-6 1
REFERENCES Mayer, J. E. and Mayer, M. G., Statistical Mechanics (New York 1940). De Boer, J., Rep. Prog. Phys. 12 (1948/1949) 305; Miinster, A., Statistische Thermodynamik (Berlin 1956); Mayer, J. E., Handb. Physik 12, 73 (Berlin 1958). Landau, L. D. and Lifshitz, E. M., Statistical Physics (English translation, London 1958) p. 220; Brout, R., Proc. Intern. Cong. Many-Particle Problems, Utrecht 1960 (Physica 26, 1960, Supp.) page S 28. Brout also finds the higher terms by expanding in powers of j3 and summing ladder graphs. De Boer, J. and Michels, A., Physica 6 (1939) 97. Mayer, J. E. and Montroll, E., J. them. Phys. 9 (1941) 2. KBnig, D., The&e der endlichen. und mendlichen Graphen (Leipzig 1936; New York 1950) page 244.
