Potts model - Revision history

Nice and Tidy: Added DOI to a paper.

2009-11-11T11:22:37Z

Added DOI to a paper.

← Older revision		Revision as of 13:22, 11 November 2009
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	The '''Potts model''', proposed by [[Renfrey B. Potts]] in 1952 <ref>Renfrey B. Potts "Some generalized order-disorder transformations", Proceedings of the Cambridge Philosophical Society '''48''' pp. ~~106−109~~ (1952)</ref><ref>Rodney J. Baxter "Exactly Solved Models in Statistical Mechanics", Academic Press (1982) ISBN 0120831821 Chapter 12 (freely available [http://tpsrv.anu.edu.au/Members/baxter/book/Exactly.pdf pdf])</ref>, is a generalisation of the [[Ising Models \| Ising model]] to more than two components. For a general discussion on Potts models see Refs <ref>[http://dx.doi.org/10.1103/RevModPhys.54.235 F. Y. Wu "The Potts model", Reviews of Modern Physics '''54''' pp. 235-268 (1982)]</ref><ref>[http://dx.doi.org/10.1103/RevModPhys.55.315 F. Y. Wu "Erratum: The Potts model", Reviews of Modern Physics '''55''' p. 315 (1983)]</ref>.		The '''Potts model''', proposed by Renfrey B. Potts in 1952 <ref>[http://dx.doi.org/10.1017/S0305004100027419 Renfrey B. Potts "Some generalized order-disorder transformations", Proceedings of the Cambridge Philosophical Society '''48''' pp. 106-109 (1952)]</ref><ref>Rodney J. Baxter "Exactly Solved Models in Statistical Mechanics", Academic Press (1982) ISBN 0120831821 Chapter 12 (freely available [http://tpsrv.anu.edu.au/Members/baxter/book/Exactly.pdf pdf])</ref>, is a generalisation of the [[Ising Models \| Ising model]] to more than two components. For a general discussion on Potts models see Refs <ref>[http://dx.doi.org/10.1103/RevModPhys.54.235 F. Y. Wu "The Potts model", Reviews of Modern Physics '''54''' pp. 235-268 (1982)]</ref><ref>[http://dx.doi.org/10.1103/RevModPhys.55.315 F. Y. Wu "Erratum: The Potts model", Reviews of Modern Physics '''55''' p. 315 (1983)]</ref>.
	In practice one has a lattice system. The sites of the lattice can be occupied by		In practice one has a lattice system. The sites of the lattice can be occupied by
	particles of different ''species'', <math> S=1,2, \cdots, q </math>.		particles of different ''species'', <math> S=1,2, \cdots, q </math>.

Nice and Tidy: Slight tidy

2009-11-10T14:46:56Z

Slight tidy

← Older revision		Revision as of 16:46, 10 November 2009
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	The '''Potts model''' ~~was~~ proposed by Renfrey B. Potts in 1952 <ref>Renfrey B. Potts "Some generalized order-disorder transformations", Proceedings of the Cambridge Philosophical Society '''48''' pp. 106−109 (1952)</ref><ref>Rodney J. Baxter "Exactly Solved Models in Statistical Mechanics", Academic Press (1982) ISBN 0120831821 Chapter 12 (freely available [http://tpsrv.anu.edu.au/Members/baxter/book/Exactly.pdf pdf])</ref>~~. The Potts model~~ is a generalisation of the [[Ising Models \| Ising model]] to more than two components. For a general discussion on Potts models see Refs <ref>[http://dx.doi.org/10.1103/RevModPhys.54.235 F. Y. Wu "The Potts model", Reviews of Modern Physics '''54''' pp. 235-268 (1982)]</ref><ref>[http://dx.doi.org/10.1103/RevModPhys.55.315 F. Y. Wu "Erratum: The Potts model", Reviews of Modern Physics '''55''' p. 315 (1983)]</ref>.		The '''Potts model''', proposed by [[Renfrey B. Potts]] in 1952 <ref>Renfrey B. Potts "Some generalized order-disorder transformations", Proceedings of the Cambridge Philosophical Society '''48''' pp. 106−109 (1952)</ref><ref>Rodney J. Baxter "Exactly Solved Models in Statistical Mechanics", Academic Press (1982) ISBN 0120831821 Chapter 12 (freely available [http://tpsrv.anu.edu.au/Members/baxter/book/Exactly.pdf pdf])</ref>, is a generalisation of the [[Ising Models \| Ising model]] to more than two components. For a general discussion on Potts models see Refs <ref>[http://dx.doi.org/10.1103/RevModPhys.54.235 F. Y. Wu "The Potts model", Reviews of Modern Physics '''54''' pp. 235-268 (1982)]</ref><ref>[http://dx.doi.org/10.1103/RevModPhys.55.315 F. Y. Wu "Erratum: The Potts model", Reviews of Modern Physics '''55''' p. 315 (1983)]</ref>.
	In practice one has a lattice system. The sites of the lattice can be occupied by		In practice one has a lattice system. The sites of the lattice can be occupied by
	particles of different ''species'', <math> S=1,2, \cdots, q </math>.		particles of different ''species'', <math> S=1,2, \cdots, q </math>.

Carl McBride: Added a new reference

2009-11-10T11:41:27Z

Added a new reference

← Older revision		Revision as of 13:41, 10 November 2009
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	The '''Potts model''' was proposed by Renfrey B. Potts in 1952 (~~Ref~~. 1). The Potts model is a generalisation of the [[Ising Models \| Ising model]] to more than two components. For a general discussion on Potts models see Refs. ~~2 and 3~~.		The '''Potts model''' was proposed by Renfrey B. Potts in 1952 <ref>Renfrey B. Potts "Some generalized order-disorder transformations", Proceedings of the Cambridge Philosophical Society '''48''' pp. 106−109 (1952)</ref><ref>Rodney J. Baxter "Exactly Solved Models in Statistical Mechanics", Academic Press (1982) ISBN 0120831821 Chapter 12 (freely available [http://tpsrv.anu.edu.au/Members/baxter/book/Exactly.pdf pdf])</ref>. The Potts model is a generalisation of the [[Ising Models \| Ising model]] to more than two components. For a general discussion on Potts models see Refs <ref>[http://dx.doi.org/10.1103/RevModPhys.54.235 F. Y. Wu "The Potts model", Reviews of Modern Physics '''54''' pp. 235-268 (1982)]</ref><ref>[http://dx.doi.org/10.1103/RevModPhys.55.315 F. Y. Wu "Erratum: The Potts model", Reviews of Modern Physics '''55''' p. 315 (1983)]</ref>.
	In practice one has a lattice system. The sites of the lattice can be occupied by		In practice one has a lattice system. The sites of the lattice can be occupied by
	particles of different ''species'', <math> S=1,2, \cdots, q </math>.		particles of different ''species'', <math> S=1,2, \cdots, q </math>.
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	the Potts model exhibits order-disorder [[phase transitions]]. For space dimensionality <math> d=2 </math>, and low values of <math> q </math> the transitions are continuous (<math> E(T) </math> is a continuous function), but the [[heat capacity]], <math> C(T) = (\partial E/\partial T) </math>, diverges at the transition [[temperature]]. The critical behaviour of		the Potts model exhibits order-disorder [[phase transitions]]. For space dimensionality <math> d=2 </math>, and low values of <math> q </math> the transitions are continuous (<math> E(T) </math> is a continuous function), but the [[heat capacity]], <math> C(T) = (\partial E/\partial T) </math>, diverges at the transition [[temperature]]. The critical behaviour of
	different values of <math> q </math> belong to (or define) different [[universality classes]] of criticality		different values of <math> q </math> belong to (or define) different [[universality classes]] of criticality

	For space dimensionality <math> d=3 </math>, the transitions for <math> q \ge 3 </math> are [[First-order transitions \|first order]] (<math> E </math> shows a discontinuity at the transition temperature).		For space dimensionality <math> d=3 </math>, the transitions for <math> q \ge 3 </math> are [[First-order transitions \|first order]] (<math> E </math> shows a discontinuity at the transition temperature).

	==See also==		==See also==
	*[[Ashkin-Teller model]]		*[[Ashkin-Teller model]]
	*[[Kac model]]		*[[Kac model]]
	==References==		==References==
	~~#Renfrey B. Potts "Some generalized order-disorder transformations", Proceedings of the Cambridge Philosophical Society~~ '''48''' ~~pp. 106−109 (1952)~~		<references/>
	#[http://dx.doi.org/10.~~1103~~/~~RevModPhys~~.~~54.235~~ ~~F. Y. Wu~~ "~~The~~ Potts model~~", Reviews~~ of ~~Modern Physics '''54''' pp. 235-268 (1982)]~~		'''Related reading'''
	~~#[http://dx.doi.org/10.1103/RevModPhys.55.315 F. Y. Wu "Erratum: The Potts model~~", ~~Reviews~~ of ~~Modern~~ Physics '''55''' ~~p. 315~~ (~~1983~~)]		*[http://dx.doi.org/10.1063/1.3250934 Nathan Duff and Baron Peters "Nucleation in a Potts lattice gas model of crystallization from solution", Journal of Chemical Physics '''131''' 184101 (2009)]
	# Rodney J. Baxter "Exactly Solved Models in Statistical Mechanics", Academic Press (1982) ISBN 0120831821 Chapter 12 (freely available [http://tpsrv.anu.edu.au/Members/baxter/book/Exactly.pdf pdf])
	[[category:models]]		[[category:models]]

Carl McBride: Added internal links and a reference.

2008-07-07T12:18:42Z

Added internal links and a reference.

← Older revision		Revision as of 14:18, 7 July 2008
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	Note that the particular case <math> q=2 </math> is equivalent to the [[Ising Models \| Ising model]].		Note that the particular case <math> q=2 </math> is equivalent to the [[Ising Models \| Ising model]].

	===Phase ~~transtions=~~==		==Phase transitions==
	Considering a symmetric situation (i.e. equal chemical potential for all the species):		Considering a symmetric situation (i.e. equal [[chemical potential]] for all the species):

	<math> \mu_1 = \mu_2 = \cdots = \mu_q </math>;		:<math> \mu_1 = \mu_2 = \cdots = \mu_q </math>;

	the Potts ~~models exhibit~~ order-disorder phase transitions. For space dimensionality <math> d=2 </math>, and low		the Potts model exhibits order-disorder [[phase transitions]]. For space dimensionality <math> d=2 </math>, and low values of <math> q </math> the transitions are continuous (<math> E(T) </math> is a continuous function), but the [[heat capacity]], <math> C(T) = (\partial E/\partial T) </math>, diverges at the transition [[temperature]]. The critical behaviour of
	values of <math> q </math> the transitions are continuous (<math> E(T) </math> is a continuous function), but the heat capacity		different values of <math> q </math> belong to (or define) different [[universality classes]] of criticality
	<math> C(T) = (\partial E/\partial T) </math> diverges at the transition temperature. The critical ~~behavior~~ of
	different values of <math> q </math> belong to (or define) different universality classes of criticality

	For space dimensionality <math> d=3 </math>, the transitions for <math> q \ge 3 </math> are first order (<math> E </math>		For space dimensionality <math> d=3 </math>, the transitions for <math> q \ge 3 </math> are [[First-order transitions \|first order]] (<math> E </math> shows a discontinuity at the transition temperature).
	shows a discontinuity at the transition temperature).

	==See also==		==See also==
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	#[http://dx.doi.org/10.1103/RevModPhys.54.235 F. Y. Wu "The Potts model", Reviews of Modern Physics '''54''' pp. 235-268 (1982)]		#[http://dx.doi.org/10.1103/RevModPhys.54.235 F. Y. Wu "The Potts model", Reviews of Modern Physics '''54''' pp. 235-268 (1982)]
	#[http://dx.doi.org/10.1103/RevModPhys.55.315 F. Y. Wu "Erratum: The Potts model", Reviews of Modern Physics '''55''' p. 315 (1983)]		#[http://dx.doi.org/10.1103/RevModPhys.55.315 F. Y. Wu "Erratum: The Potts model", Reviews of Modern Physics '''55''' p. 315 (1983)]
			# Rodney J. Baxter "Exactly Solved Models in Statistical Mechanics", Academic Press (1982) ISBN 0120831821 Chapter 12 (freely available [http://tpsrv.anu.edu.au/Members/baxter/book/Exactly.pdf pdf])
	[[category:models]]		[[category:models]]

Noe at 10:04, 7 July 2008

2008-07-07T10:04:44Z

← Older revision		Revision as of 12:04, 7 July 2008
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	Note that the particular case <math> q=2 </math> is equivalent to the [[Ising Models \| Ising model]].		Note that the particular case <math> q=2 </math> is equivalent to the [[Ising Models \| Ising model]].

			===Phase transtions===
	Considering a symmetric situation (i.e. equal chemical potential for all the species):		Considering a symmetric situation (i.e. equal chemical potential for all the species):

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	values of <math> q </math> the transitions are continuous (<math> E(T) </math> is a continuous function), but the heat capacity		values of <math> q </math> the transitions are continuous (<math> E(T) </math> is a continuous function), but the heat capacity
	<math> C(T) = (\partial E/\partial T) </math> diverges at the transition temperature. The critical behavior of		<math> C(T) = (\partial E/\partial T) </math> diverges at the transition temperature. The critical behavior of
	different values of <math> q </math> ~~corresponds~~ to different ~~critical~~ universality classes.		different values of <math> q </math> belong to (or define) different universality classes of criticality

	For space dimensionality <math> d=3 </math>, the transitions for <math> q \ge 3 </math> are first order (<math> E </math>		For space dimensionality <math> d=3 </math>, the transitions for <math> q \ge 3 </math> are first order (<math> E </math>

Noe: some details about phase behavior

2008-07-07T09:59:11Z

some details about phase behavior

@@ Line 8: / Line 8: @@
 where <math> K </math> is the coupling constant, <math> \langle ij \rangle </math> indicates
 that the sum is performed exclusively over pairs of nearest neighbour sites,  and <math> \delta(S_i,S_j) </math> is the [[Kronecker delta|Kronecker delta]].
-Note that the particular case <math> q=2 </math> is equivalent to the [[Ising Models | Ising model]]
+Note that the particular case <math> q=2 </math> is equivalent to the [[Ising Models | Ising model]].
 ==See also==
 *[[Ashkin-Teller model]]

Carl McBride: Trivial tidy up.

2008-07-04T15:38:32Z

Trivial tidy up.

← Older revision		Revision as of 17:38, 4 July 2008
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			The '''Potts model''' was proposed by Renfrey B. Potts in 1952 (Ref. 1). The Potts model is a generalisation of the [[Ising Models \| Ising model]] to more than two components. For a general discussion on Potts models see Refs. 2 and 3.
	The '''Potts model''' was proposed by Renfrey B. Potts in 1952 (Ref. 1). The Potts model is a generalisation of the [[Ising Models \| Ising model]] to more than two components.

	For a general discussion on Potts models see ~~Ref~~.2.

	In practice one has a lattice system. The sites of the lattice can be occupied by		In practice one has a lattice system. The sites of the lattice can be occupied by
	particles of different '''species''', <math> S=1,2, \cdots, q </math>.		particles of different ''species'', <math> S=1,2, \cdots, q </math>.

	The energy of the system, <math> E </math>, is defined as:		The energy of the system, <math> E </math>, is defined as:

	:<math> E = - K \sum_{<ij>} \delta (S_i,S_j) </math>		:<math> E = - K \sum_{ \langle ij \rangle } \delta (S_i,S_j) </math>
	where <math> K </math> is the coupling constant, <math> <ij> </math> indicates		where <math> K </math> is the coupling constant, <math> \langle ij \rangle </math> indicates
	that the sum is ~~done~~ exclusively over pairs of nearest ~~neighbor~~ sites, and <math> \delta(S_i,S_j) </math> is the [[Kronecker delta\|Kronecker delta]].		that the sum is performed exclusively over pairs of nearest neighbour sites, and <math> \delta(S_i,S_j) </math> is the [[Kronecker delta\|Kronecker delta]].
			Note that the particular case <math> q=2 </math> is equivalent to the [[Ising Models \| Ising model]]
	~~The~~ particular case <math> q=2 </math> is equivalent to the [[Ising Models \| Ising model]]
	==See also==		==See also==
	*[[Ashkin-Teller model]]		*[[Ashkin-Teller model]]

Noe: stubb removed

2008-07-04T13:35:54Z

stubb removed

← Older revision		Revision as of 15:35, 4 July 2008
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	~~{{Stub-general}}~~
	The '''Potts model''' was proposed by Renfrey B. Potts in 1952 (Ref. 1). The Potts model is a generalisation of the [[Ising Models \| Ising model]] to more than two components.		The '''Potts model''' was proposed by Renfrey B. Potts in 1952 (Ref. 1). The Potts model is a generalisation of the [[Ising Models \| Ising model]] to more than two components.

Noe at 13:20, 4 July 2008

2008-07-04T13:20:47Z

@@ Line 1: / Line 1: @@
 {{Stub-general}}
 The '''Potts model''' was proposed by Renfrey B. Potts in 1952 (Ref. 1). The Potts model is a generalisation of the [[Ising Models | Ising model]] to more than two components.
 ==See also==
 *[[Ashkin-Teller model]]

Carl McBride at 16:31, 22 January 2008

2008-01-22T16:31:49Z

← Older revision		Revision as of 18:31, 22 January 2008
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	{{Stub-general}}		{{Stub-general}}
			The '''Potts model''' was proposed by Renfrey B. Potts in 1952 (Ref. 1). The Potts model is a generalisation of the [[Ising Models \| Ising model]] to more than two components.
			==See also==
			*[[Ashkin-Teller model]]
			*[[Kac model]]
	==References==		==References==
	#[http://dx.doi.org/10.1103/RevModPhys.54.235 F. Y. Wu "The Potts model", Reviews of Modern Physics '''54''' pp. 235 - 268 (1982)]		#Renfrey B. Potts "Some generalized order-disorder transformations", Proceedings of the Cambridge Philosophical Society '''48''' pp. 106−109 (1952)
			#[http://dx.doi.org/10.1103/RevModPhys.54.235 F. Y. Wu "The Potts model", Reviews of Modern Physics '''54''' pp. 235-268 (1982)]
	#[http://dx.doi.org/10.1103/RevModPhys.55.315 F. Y. Wu "Erratum: The Potts model", Reviews of Modern Physics '''55''' p. 315 (1983)]		#[http://dx.doi.org/10.1103/RevModPhys.55.315 F. Y. Wu "Erratum: The Potts model", Reviews of Modern Physics '''55''' p. 315 (1983)]
	[[category:models]]		[[category:models]]