APPROXIMATE METHODS

Get qrcode for javause java qr code 2d barcode encoder tocreate qr codes for java

distinct pairs joined by lines are the pairs a, term

Bar Code barcode library on javause java bar code implement todraw bar code with java

/3, ... , X,

Barcode barcode library for javaUsing Barcode decoder for Java Control to read, scan read, scan image in Java applications.

then the graph represents the

(10.10)

Control qr bidimensional barcode image in .netuse an asp.net form qrcode generation toproduce qr code jis x 0510 for .net

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:

QR-Code encoder with .netusing .net framework crystal toaccess qr codes in asp.net web,windows application

but the following graphs are identical:

Include qr-codes with vb.netusing barcode integrated for .net vs 2010 control to generate, create qr code image in .net vs 2010 applications.

CD---@

Control code 128b image on javause java barcode code 128 drawer togenerate code 128 code set a with java

We may regard a graph as a picturesque way of writing the integral (10.10). For example, we may write, for N = 10,

ep2 [

c1@~G(@

ANSI/AIM Code 128 printer for javausing barcode creator for java control to generate, create ansi/aim code 128 image in java applications.

---0)

Control data matrix 2d barcode data with javato render datamatrix and data matrix data, size, image with java barcode sdk

f d 3r1 ... d3rlOf12f39f67f6sf8,Jof6,lOf78

Java standard 2 of 5 maker on javausing barcode generating for java control to generate, create 2/5 industrial image in java applications.

(10.11)

Control ansi/aim code 39 image for .netuse asp.net web pages code 39 generating toconnect code 3/9 for .net

With such a convention, we can state that

Receive pdf 417 for excelusing barcode integrating for excel control to generate, create barcode pdf417 image in excel applications.

ZN = (sum of all distinct N-particle graphs)

Asp.net Web Pages Crystal data matrix barcodes implementation with c#.netuse asp.net web forms crystal ecc200 encoding touse datamatrix on visual c#

(10.12)

Font barcode integration with fontusing font todevelop barcode in asp.net web,windows application

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

Control ean 128 barcode image on visual basicusing barcode printer for visual studio .net control to generate, create gs1 barcode image in visual studio .net applications.

ep ~ ~j reb @ ~ "

Code128 recognizer in noneUsing Barcode Control SDK for None Control to generate, create, read, scan barcode image in None applications.

~ [@] o[al) o[(!)-@] o[QJ---0)] o[

Bar Code barcode library in .netusing barcode maker for sql server 2005 reporting services control to generate, create bar code image in sql server 2005 reporting services applications.

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: