# Difference between revisions of "Cluster diagrams"

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+ | '''Diagrams''' are sometimes known as ''graphs''. | ||

==Chain clusters== | ==Chain clusters== | ||

A ''chain'' is a cluster with at least one node. | A ''chain'' is a cluster with at least one node. | ||

Line 26: | Line 27: | ||

:<math>\left.c(r)\right. = B(r) + E(r)</math> | :<math>\left.c(r)\right. = B(r) + E(r)</math> | ||

− | + | ==Lemmas== | |

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

==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)] | ||

+ | #[http://dx.doi.org/10.1143/PTP.25.537 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)] | ||

+ | #[http://dx.doi.org/10.1063/1.1724313 Cyrano De Dominicis "Variational Formulations of Equilibrium Statistical Mechanics", Journal of Mathematical Physics '''3''' pp. 983-1002 (1962)] | ||

+ | #[http://dx.doi.org/10.1063/1.1703949 Cyrano De Dominicis "Variational Statistical Mechanics in Terms of "Observables" for Normal and Superfluid Systems", Journal of Mathematical Physics '''4''' pp. 255-265 (1963)] | ||

[[category: integral equations]] | [[category: integral equations]] | ||

[[category: statistical mechanics]] | [[category: statistical mechanics]] |

## Revision as of 11:32, 11 June 2007

**Diagrams** are sometimes known as *graphs*.

## Chain clusters

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

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

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

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,

## Lemmas

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

## References

- J. S. Rowlinson "The equation of state of dense systems", Reports on Progress in Physics
*28'*pp. 169-199 (1965) - 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) - Cyrano De Dominicis "Variational Formulations of Equilibrium Statistical Mechanics", Journal of Mathematical Physics
**3**pp. 983-1002 (1962) - Cyrano De Dominicis "Variational Statistical Mechanics in Terms of "Observables" for Normal and Superfluid Systems", Journal of Mathematical Physics
**4**pp. 255-265 (1963)