1-dimensional Ising model - Revision history

147.96.6.43: minor index change

2009-02-19T17:05:06Z

minor index change

← Older revision		Revision as of 19:05, 19 February 2009
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	Performing the sum of the possible values of <math> S_{N} </math> we get:		Performing the sum of the possible values of <math> S_{N} </math> we get:

	:<math> Q_{N} = \sum_{S_1} \sum_{S_2} e^{K S_1S_2} \sum_{S_3} e^{K S_2 S_3} \cdots \sum_{S_{N-2}} e^{K S_{N-2} S_{N-1}} \left[ 2 \cosh ( K S_{N-1} ) \right]		:<math> Q_{N} = \sum_{S_1} \sum_{S_2} e^{K S_1S_2} \sum_{S_3} e^{K S_2 S_3} \cdots \sum_{S_{N-1}} e^{K S_{N-2} S_{N-1}} \left[ 2 \cosh ( K S_{N-1} ) \right]
	</math>		</math>

Nice and Tidy: Added internal link to Ising models page.

2009-01-14T18:57:09Z

Added internal link to Ising models page.

← Older revision		Revision as of 20:57, 14 January 2009
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	~~Consider~~ a system with <math> N </math> spins in a row. The energy of the system ~~will be~~ given by		The '''1-dimensional Ising model''' is an [[Ising Models\| Ising model]] that consists of a system with <math> N </math> spins in a row. The energy of the system is given by

	:<math> U = -J \sum_{i=1}^{N-1} S_{i} S_{i+1} </math>,		:<math> U = -J \sum_{i=1}^{N-1} S_{i} S_{i+1} </math>,

Carl McBride: /* References */ Added a reference

2008-07-07T10:43:12Z

References: Added a reference

← Older revision		Revision as of 12:43, 7 July 2008
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	:<math> A = - N k_B T \log \left( 2 \cosh K \right) </math>		:<math> A = - N k_B T \log \left( 2 \cosh K \right) </math>
	==References==		==References==
			# Rodney J. Baxter "Exactly Solved Models in Statistical Mechanics", Academic Press (1982) ISBN 0120831821 Chapter 2 (freely available [http://tpsrv.anu.edu.au/Members/baxter/book/Exactly.pdf pdf])
	[[Category: Models]]		[[Category: Models]]

Carl McBride at 11:08, 28 May 2007

2007-05-28T11:08:20Z

← Older revision		Revision as of 13:08, 28 May 2007
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	~~Model:~~		Consider a system with <math> N </math> spins in a row. The energy of the system will be given by
	Consider a system with <math> N </math> spins in a row.

	The energy of the system will be given by

	:<math> U = -J \sum_{i=1}^{N-1} S_{i} S_{i+1} </math>,		:<math> U = -J \sum_{i=1}^{N-1} S_{i} S_{i+1} </math>,
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	where each variable <math> S_j </math> can be either -1 or +1.		where each variable <math> S_j </math> can be either -1 or +1.

	The partition function of the system will be:		The [[partition function]] of the system will be:

	:<math> Q_N = \sum_{\Omega^N } \exp \left[ K \sum_{i=1}^{N-1} S_i S_{i+1} \right]</math>,		:<math> Q_N = \sum_{\Omega^N } \exp \left[ K \sum_{i=1}^{N-1} S_i S_{i+1} \right]</math>,
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	:<math> Q_N = 2^{N} \left( \cosh K \right)^{N-1} \approx ( 2 \cosh K )^N </math>		:<math> Q_N = 2^{N} \left( \cosh K \right)^{N-1} \approx ( 2 \cosh K )^N </math>

	The [[Helmholtz energy function]] in the thermodynamic limit will be		The [[Helmholtz energy function]] in the [[thermodynamic limit]] will be

	:<math> A = - N k_B T \log \left( 2 \cosh K \right) </math>		:<math> A = - N k_B T \log \left( 2 \cosh K \right) </math>
			==References==
	[[Category: Models]]		[[Category: Models]]

Nice and Tidy at 16:23, 27 February 2007

2007-02-27T16:23:23Z

← Older revision		Revision as of 18:23, 27 February 2007
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	The energy of the system will be given by		The energy of the system will be given by

	<math> U = -J \sum_{i=1}^{N-1} S_{i} S_{i+1} </math>,		:<math> U = -J \sum_{i=1}^{N-1} S_{i} S_{i+1} </math>,

	where each variable <math> S_j </math> can be either -1 or +1.		where each variable <math> S_j </math> can be either -1 or +1.
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	The partition function of the system will be:		The partition function of the system will be:

	<math> Q_N = \sum_{\Omega^N } \exp \left[ K \sum_{i=1}^{N-1} S_i S_{i+1} \right]</math>,		:<math> Q_N = \sum_{\Omega^N } \exp \left[ K \sum_{i=1}^{N-1} S_i S_{i+1} \right]</math>,


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	and <math> K = J/k_B T </math>		and <math> K = J/k_B T </math>

	<math> Q_{N} = \sum_{S_1} \sum_{S_2} e^{K S_1S_2} \sum_{S_3} e^{K S_2 S_3} \cdots \sum_{S_{N-1}} e^{K S_{N-2} S_{N-1}}\sum_{S_{N}} e^{K S_{N-1} S_{N} }		:<math> Q_{N} = \sum_{S_1} \sum_{S_2} e^{K S_1S_2} \sum_{S_3} e^{K S_2 S_3} \cdots \sum_{S_{N-1}} e^{K S_{N-2} S_{N-1}}\sum_{S_{N}} e^{K S_{N-1} S_{N} }
	</math>		</math>

	Performing the sum of the possible values of <math> S_{N} </math> we get:		Performing the sum of the possible values of <math> S_{N} </math> we get:

	<math> Q_{N} = \sum_{S_1} \sum_{S_2} e^{K S_1S_2} \sum_{S_3} e^{K S_2 S_3} \cdots \sum_{S_{N-2}} e^{K S_{N-2} S_{N-1}} \left[ 2 \cosh ( K S_{N-1} ) \right]		:<math> Q_{N} = \sum_{S_1} \sum_{S_2} e^{K S_1S_2} \sum_{S_3} e^{K S_2 S_3} \cdots \sum_{S_{N-2}} e^{K S_{N-2} S_{N-1}} \left[ 2 \cosh ( K S_{N-1} ) \right]
	</math>		</math>

	Taking into account that <math> \cosh(K) = \cosh(-K) </math>		Taking into account that <math> \cosh(K) = \cosh(-K) </math>

	<math> Q_{N} = \sum_{S_1} \sum_{S_2} e^{K S_1S_2} \sum_{S_3} e^{K S_2 S_3} \cdots \sum_{S_{N-1}} e^{K S_{N-2} S_{N-1}} \left[ 2 \cosh ( K ) \right]		:<math> Q_{N} = \sum_{S_1} \sum_{S_2} e^{K S_1S_2} \sum_{S_3} e^{K S_2 S_3} \cdots \sum_{S_{N-1}} e^{K S_{N-2} S_{N-1}} \left[ 2 \cosh ( K ) \right]
	</math>		</math>

	Therefore:		Therefore:

	<math> Q_{N} = \left( 2 \cosh K \right) Q_{N-1} </math>		:<math> Q_{N} = \left( 2 \cosh K \right) Q_{N-1} </math>

	<math> Q_N = 2^{N} \left( \cosh K \right)^{N-1} \approx ( 2 \cosh K )^N </math>		:<math> Q_N = 2^{N} \left( \cosh K \right)^{N-1} \approx ( 2 \cosh K )^N </math>

	The Helmholtz ~~free~~ energy in the thermodynamic limit will be		The [[Helmholtz energy function]] in the thermodynamic limit will be

	<math> A = - N k_B T \log \left( 2 \cosh K \right) </math>		:<math> A = - N k_B T \log \left( 2 \cosh K \right) </math>

			[[Category: Models]]

Carl McBride: 1dimensional Ising model moved to 1-dimensional Ising model

2007-02-23T14:34:47Z

1dimensional Ising model moved to 1-dimensional Ising model

← Older revision	Revision as of 16:34, 23 February 2007
(No difference)

Carl McBride: Ising Model 1dimensional moved to 1dimensional Ising model

2007-02-23T14:34:15Z

Ising Model 1dimensional moved to 1dimensional Ising model

← Older revision	Revision as of 16:34, 23 February 2007
(No difference)

Carl McBride: Ising Model 1d moved to Ising Model 1dimensional

2007-02-23T14:31:55Z

Ising Model 1d moved to Ising Model 1dimensional

← Older revision	Revision as of 16:31, 23 February 2007
(No difference)

Noe at 12:03, 23 February 2007

2007-02-23T12:03:08Z

← Older revision		Revision as of 14:03, 23 February 2007
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	and <math> K = J/k_B T </math>		and <math> K = J/k_B T </math>

	<math> Q_{N+1} = \sum_{S_1} \sum_{S_2} e^{K S_1S_2} \sum_{S_3} e^{K S_2 S_3} \cdots \sum_{~~S_N~~} e^{K S_{N-1} ~~S_N~~}\sum_{S_{N+1}} e^{K ~~S_N~~ S_{N+1} }		<math> Q_{N} = \sum_{S_1} \sum_{S_2} e^{K S_1S_2} \sum_{S_3} e^{K S_2 S_3} \cdots \sum_{S_{N-1}} e^{K S_{N-2} S_{N-1}}\sum_{S_{N}} e^{K S_{N-1} S_{N} }
	</math>		</math>

	Performing the sum of the possible values of <math> S_{N+1} </math> we get:		Performing the sum of the possible values of <math> S_{N} </math> we get:

	<math> Q_{N+1} = \sum_{S_1} \sum_{S_2} e^{K S_1S_2} \sum_{S_3} e^{K S_2 S_3} \cdots \sum_{~~S_N~~} e^{K S_{N-1} ~~S_N~~} \left[ 2 \cosh ( K ~~S_N~~ ) \right]		<math> Q_{N} = \sum_{S_1} \sum_{S_2} e^{K S_1S_2} \sum_{S_3} e^{K S_2 S_3} \cdots \sum_{S_{N-2}} e^{K S_{N-2} S_{N-1}} \left[ 2 \cosh ( K S_{N-1} ) \right]
	</math>		</math>

	Taking into account that <math> \cosh(K) = \cosh(-K) </math>		Taking into account that <math> \cosh(K) = \cosh(-K) </math>

	<math> Q_{N+1} = \sum_{S_1} \sum_{S_2} e^{K S_1S_2} \sum_{S_3} e^{K S_2 S_3} \cdots \sum_{~~S_N~~} e^{K S_{N-1} ~~S_N~~} \left[ 2 \cosh ( K ) \right]		<math> Q_{N} = \sum_{S_1} \sum_{S_2} e^{K S_1S_2} \sum_{S_3} e^{K S_2 S_3} \cdots \sum_{S_{N-1}} e^{K S_{N-2} S_{N-1}} \left[ 2 \cosh ( K ) \right]
	</math>		</math>

	Therefore:		Therefore:

	<math> Q_{N+1} = \left( 2 \cosh K \right) Q_{N} </math>		<math> Q_{N} = \left( 2 \cosh K \right) Q_{N-1} </math>

	<math> Q_N = 2^{N} \left( \cosh K \right)^{N-1} \approx ( 2 \cosh K )^N </math>		<math> Q_N = 2^{N} \left( \cosh K \right)^{N-1} \approx ( 2 \cosh K )^N </math>

Noe at 11:57, 23 February 2007

2007-02-23T11:57:40Z

← Older revision		Revision as of 13:57, 23 February 2007
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	Therefore:		Therefore:

	<math> Q_{N+1} = \left( 2 \cosh K \right) Q_{N+1} </math>		<math> Q_{N+1} = \left( 2 \cosh K \right) Q_{N} </math>

	<math> Q_N = 2^{N} \left( \cosh K \right)^{N-1} \approx ( 2 \cosh K )^N </math>		<math> Q_N = 2^{N} \left( \cosh K \right)^{N-1} \approx ( 2 \cosh K )^N </math>