Difference between revisions of "Cluster diagrams"

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An ''elementary'' cluster is that which is neither a chain or a bundle.
 
An ''elementary'' cluster is that which is neither a chain or a bundle.
  
:<math>\left.h(r)\right. = C(r)   B(r)   E(r)</math>
+
:<math>\left.h(r)\right. = C(r) + B(r) + E(r)</math>
  
 
where ''C(r)'' is the set of ''chain'' clusters,  ''B(r)'' is the set of ''bundles'', and
 
where ''C(r)'' is the set of ''chain'' clusters,  ''B(r)'' is the set of ''bundles'', and
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Similarly,
 
Similarly,
  
:<math>\left.c(r)\right. = B(r)   E(r)</math>
+
:<math>\left.c(r)\right. = B(r) + E(r)</math>
  
 
==Lemmas==
 
==Lemmas==
 
There are five lemmas (see Ref.s 2,3 and 4).
 
There are five lemmas (see Ref.s 2,3 and 4).
 +
==See also==
 +
*[[Cluster integrals]]
 
==References==
 
==References==
 
#[http://dx.doi.org/10.1088/0034-4885/28/1/306  J. S. Rowlinson "The equation of state of dense systems", Reports on Progress in Physics ''28''' pp. 169-199 (1965)]
 
#[http://dx.doi.org/10.1088/0034-4885/28/1/306  J. S. Rowlinson "The equation of state of dense systems", Reports on Progress in Physics ''28''' pp. 169-199 (1965)]

Latest revision as of 15:28, 31 July 2007

Diagrams are sometimes known as graphs.

Chain clusters[edit]

A chain is a cluster with at least one node. All routes from one base point to the other pass through at least one nodal field point. Cutting at this point separates the diagram into two parts, each of which contains a base point. A simple chain is one in which every field point is a node. A netted chain is one that can be formed from a simple chain by adding not more than one field point across each link of the simple chain.

Bundles[edit]

A bundle is a parallel collection of links. It has no nodes since there is always more than one independent path from one base point to the other.

Elementary Clusters[edit]

An elementary cluster is that which is neither a chain or a bundle.

\left.h(r)\right. = C(r) + B(r) + E(r)

where C(r) is the set of chain clusters, B(r) is the set of bundles, and E(r) is the set of elementary clusters (Eq. 5.2 Ref. 1). Similarly,

\left.c(r)\right. = B(r) + E(r)

Lemmas[edit]

There are five lemmas (see Ref.s 2,3 and 4).

See also[edit]

References[edit]

  1. J. S. Rowlinson "The equation of state of dense systems", Reports on Progress in Physics 28' pp. 169-199 (1965)
  2. Tohru Morita and Kazuo Hiroike "A New Approach to the Theory of Classical Fluids. III General Treatment of Classical Systems", Progress of Theoretical Physics 25 pp. 537-578 (1961)
  3. Cyrano De Dominicis "Variational Formulations of Equilibrium Statistical Mechanics", Journal of Mathematical Physics 3 pp. 983-1002 (1962)
  4. Cyrano De Dominicis "Variational Statistical Mechanics in Terms of "Observables" for Normal and Superfluid Systems", Journal of Mathematical Physics 4 pp. 255-265 (1963)