Cluster diagrams - Revision history

Carl McBride at 14:28, 31 July 2007

2007-07-31T14:28:04Z

← Older revision		Revision as of 16:28, 31 July 2007
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	==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)]

Carl McBride: Reverted edits by 222.66.97.75 (Talk); changed back to last version by Carl McBride

2007-07-05T11:12:13Z

Reverted edits by 222.66.97.75 (Talk); changed back to last version by Carl McBride

← Older revision		Revision as of 13:12, 5 July 2007
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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==

222.66.97.75 at 22:50, 4 July 2007

2007-07-04T22:50:08Z

← Older revision		Revision as of 00:50, 5 July 2007
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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
Line 25:		Line 25:
	Similarly,		Similarly,

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

	==Lemmas==		==Lemmas==

Carl McBride at 10:32, 11 June 2007

2007-06-11T10:32:50Z

← Older revision		Revision as of 12:32, 11 June 2007
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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.
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	:<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]]

Carl McBride at 17:33, 5 June 2007

2007-06-05T17:33:33Z

← Older revision		Revision as of 19:33, 5 June 2007
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	==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)]
			[[category: integral equations]]
			[[category: statistical mechanics]]

Carl McBride at 11:02, 23 February 2007

2007-02-23T11:02:47Z

← Older revision		Revision as of 13:02, 23 February 2007
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	==References==		==References==
	#[http://dx.doi.org/10.1088/0034-4885/28/1/306		#[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)]

Carl McBride: New page: ==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...

2007-02-23T11:01:16Z

New page: ==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...

New page

==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.

:<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
''E(r)'' is the set of ''elementary'' clusters (Eq. 5.2 Ref. 1).
Similarly,

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

==References==
#[http://dx.doi.org/10.1088/0034-4885/28/1/306