APPROXIMATE METHODS in Java

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APPROXIMATE METHODS
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distinct pairs joined by lines are the pairs a, term
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/3, ... , X,
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then the graph represents the
(10.10)
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appearing in the expansion (10.9). If the set of distinct pairs {a, /3, ... , y} is joined by lines in a given graph, replacing this set by a set { a', /3', ... , y'} that is not identical with {a, /3, ... , y} gives rise to a graph that is counted as distinct from the original one (although the integrals represented by the respective graphs have the same numerical value). For example, for N = 3, the following graphs are distinct:
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but the following graphs are identical:
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CD---@
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We may regard a graph as a picturesque way of writing the integral (10.10). For example, we may write, for N = 10,
ep2 [
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(10.11)
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With such a convention, we can state that
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ZN = (sum of all distinct N-particle graphs)
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(10.12)
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The proof is obvious. Any graph can in general be decomposed into smaller units. For example, the graph (14.11) is a product of five factors, namely
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Each factor corresponds to a connected graph, in which every circle is attached to at least one line, and every circle is joined directly or indirectly to all other circles in the graph. It would facilitate the analysis of ZN if we first defined the basic units out of which an arbitrary graph can be composed. Accordingly we define an I-cluster to
STATISTICAL MECHANICS
be an I particle connected graph. For example, the following is a 6-cluster:
~ ep f
d3rl ... d 3r6 /12/23/14/46/56
(10.13)
We define a cluster integral bl(V, T) by
b l ( V, T) == l!X3/ - 3V (sum of all possible I-clusters)
(10.14)
The normalization factor is so chosen that
( a) bI (V, T) is dimensionless;
(b) DI(T) == lim bl(V, T) is a finite number. v --> 00 The property (b) follows from the fact that hj has a finite range, so that in an
I-cluster the only integration that gives rise to a factor V is the integration over the "center of gravity" of the I particles. Some of the cluster integrals are
b1 = b2 =
v[CD] = vf d
1 2!1\ V
r1 =
\ 3 f d3r12112
(10.15)
1 21\
~[CD---@] = ~fd3rld3r2/12 =
1\ 2V
(10.16)
b,- 3!~'V[~
(1017)
Any N-particle graph is a product of a number of clusters, of which m l are I-clusters, with
(10.18)
A given set of integers {m I} satisfying (10.18), however, does not uniquely specify a graph, because
(a) there are in general many ways to form an I-cluster, e.g.,
(b) there are in general many ways to assign which particle belongs to which
cluster, e.g.,
Thus a set of integers {m I} specifies a collection of graphs. Let the sum of all the
APPROXIMATE METHODS
graphs corresponding to {md be denoted by S{ m j }. Then
ZN =
{m,l
S{m j
(10.19)
where the summation extends over all sets {md satisfying (10.18). By definition, S{ md can be obtained as follows. First write down an arbitrary N-particle graph that contains m1 I-clusters, m 2 2-clusters, etc.; e.g.,
{[0] ... [O]} {[0-0] ... [0-0 ]}
factors
factors (10.20)
x {[
10 ][e&][10] [o1]}
factors
There are exactly N circles appearing in (10.20), and these N circles are to be filled in by the numbers 1,2, ... , N in an arbitrary but definite order. We can write down many more examples like (10.20); e.g., we may change the choice of some of the 3-clusters (there being four distinct topological shapes for a 3-cluster). Again we may permute the numbering of all the N circles in (10.20), and that would lead to a distinct graph. If we add up all these possibilities, we obtain S{m j }. Thus we may write
S {mj}
L [0]
ml [
0-0]
x[to cl d\ +~nr
(10.21)
The meaning of this formula is as follows. Each bracket contains the sum over all I-clusters. If all the brackets [ ... 1 are expanded in multinomial expansions, the m, summand of L will itself be a sum of a large number of terms in which every term contains exactly N circles. The sum I: extends over all distinct ways of numbering these circles from 1 to N. P Now each graph is an integral whose value is independent of the way its circles are numbered. Therefore S { m j} is equal to the number of terms in the sum I: times the value of any term in the sum. The number of terms in the sum I: