Critical exponents - Revision history

93.207.220.254: /* Magnetic order parameter exponent: \beta */

2013-02-17T16:49:32Z

Magnetic order parameter exponent: \beta

← Older revision		Revision as of 18:49, 17 February 2013
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	Theoretically one has <math>\beta =0.32653(10)</math><ref name="Campostrini2002"> </ref> for the [[Universality classes#Ising \|three dimensional Ising model]], and <math>\beta = 0.3485(2)</math><ref name="Campostrini2001"> </ref> for the three-dimensional XY universality class.		Theoretically one has <math>\beta =0.32653(10)</math><ref name="Campostrini2002"> </ref> for the [[Universality classes#Ising \|three dimensional Ising model]], and <math>\beta = 0.3485(2)</math><ref name="Campostrini2001"> </ref> for the three-dimensional XY universality class.

	==Susceptibility exponent: <math>\gamma</math>==		==Susceptibility exponent: <math>\gamma</math>==
	[[Susceptibility]]		[[Susceptibility]]

Carl McBride: Reverted edits by 70.135.118.126 (talk) to last revision by Carl McBride

2012-01-04T14:32:37Z

Reverted edits by 70.135.118.126 (talk) to last revision by Carl McBride

← Older revision		Revision as of 16:32, 4 January 2012
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	<references/>		<references/>
	[[category: statistical mechanics]]		[[category: statistical mechanics]]
	~~[http://www.prlog.org/11289974-phone-number-lookup-verizon-phone-number-reverse-lookup-to-get-information-you-need-quickly.html phone address lookup]~~

	~~[http://thetvtopc.com/Reverse_Cell_Phone_Lookup_Number reverse phone number lookup]~~

70.135.118.126 at 05:59, 4 January 2012

2012-01-04T05:59:27Z

← Older revision		Revision as of 07:59, 4 January 2012
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	<references/>		<references/>
	[[category: statistical mechanics]]		[[category: statistical mechanics]]
			[http://www.prlog.org/11289974-phone-number-lookup-verizon-phone-number-reverse-lookup-to-get-information-you-need-quickly.html phone address lookup]

			[http://thetvtopc.com/Reverse_Cell_Phone_Lookup_Number reverse phone number lookup]

Carl McBride: Started introduction

2011-07-26T12:32:02Z

Started introduction

← Older revision		Revision as of 14:32, 26 July 2011
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			'''Critical exponents'''. Groups of critical exponents form [[universality classes]].
	==Reduced distance: <math>\epsilon</math>==		==Reduced distance: <math>\epsilon</math>==
	<math>\epsilon</math> is the reduced distance from the critical [[temperature]], i.e.		<math>\epsilon</math> is the reduced distance from the critical [[temperature]], i.e.
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	where <math>\epsilon</math> is 2/3 for the [[Van der Waals equation of state]], and is usually 0.75 to 0.8.		where <math>\epsilon</math> is 2/3 for the [[Van der Waals equation of state]], and is usually 0.75 to 0.8.
	~~==See also==~~
	*[[Universality classes]]
	==References==		==References==
	<references/>		<references/>
	[[category: statistical mechanics]]		[[category: statistical mechanics]]

Carl McBride: added an internal link

2011-07-26T10:35:06Z

added an internal link

← Older revision		Revision as of 12:35, 26 July 2011
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	:<math>\left. m\right. = m_0 \epsilon^\beta</math>		:<math>\left. m\right. = m_0 \epsilon^\beta</math>

	Theoretically one has <math>\beta =0.32653(10)</math><ref name="Campostrini2002"> </ref> for the three dimensional Ising model, and <math>\beta = 0.3485(2)</math><ref name="Campostrini2001"> </ref> for the three-dimensional XY universality class.		Theoretically one has <math>\beta =0.32653(10)</math><ref name="Campostrini2002"> </ref> for the [[Universality classes#Ising \|three dimensional Ising model]], and <math>\beta = 0.3485(2)</math><ref name="Campostrini2001"> </ref> for the three-dimensional XY universality class.
	==Susceptibility exponent: <math>\gamma</math>==		==Susceptibility exponent: <math>\gamma</math>==
	[[Susceptibility]]		[[Susceptibility]]
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	:<math>\left. \chi \right. = \chi_0 \epsilon^{-\gamma}</math>		:<math>\left. \chi \right. = \chi_0 \epsilon^{-\gamma}</math>

	Theoretically one has <math>\gamma = 1.2373(2)</math><ref name="Campostrini2002"> </ref> for the three dimensional Ising model, and <math>\gamma = 1.3177(5)</math><ref name="Campostrini2001"> </ref> for the three-dimensional XY universality class.		Theoretically one has <math>\gamma = 1.2373(2)</math><ref name="Campostrini2002"> </ref> for the [[Universality classes#Ising \|three dimensional Ising model]], and <math>\gamma = 1.3177(5)</math><ref name="Campostrini2001"> </ref> for the three-dimensional XY universality class.
	==Correlation length==		==Correlation length==

	:<math>\left. \xi \right.= \xi_0 \epsilon^{-\nu}</math>		:<math>\left. \xi \right.= \xi_0 \epsilon^{-\nu}</math>

	Theoretically one has <math>\nu = 0.63012(16)</math><ref name="Campostrini2002"> </ref> for the three dimensional Ising model, and <math>\nu = 0.67155(27)</math><ref name="Campostrini2001"> </ref> for the three-dimensional XY universality class.		Theoretically one has <math>\nu = 0.63012(16)</math><ref name="Campostrini2002"> </ref> for the [[Universality classes#Ising \|three dimensional Ising model]], and <math>\nu = 0.67155(27)</math><ref name="Campostrini2001"> </ref> for the three-dimensional XY universality class.
	==Inequalities==		==Inequalities==
	====Fisher inequality====		====Fisher inequality====

Carl McBride: /* Gamma divergence */ Updated internal link

2011-07-15T15:47:39Z

Gamma divergence: Updated internal link

← Older revision		Revision as of 17:47, 15 July 2011
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	:<math>\left. \right. \kappa_T \sim (T-T_c)^{-\gamma} \sim (p-p_c)^{-\gamma}</math>		:<math>\left. \right. \kappa_T \sim (T-T_c)^{-\gamma} \sim (p-p_c)^{-\gamma}</math>

	where <math>\kappa_T</math> is the [[Compressibility#Isothermal compressibility \| isothermal compressibility]]. <math>\gamma</math> is 1.0 for the [[Van der Waals equation of state]], and is usually 1.2 to 1.3.		where <math>\kappa_T</math> is the [[Compressibility#Isothermal compressibility \| isothermal compressibility]]. <math>\gamma</math> is 1.0 for the [[Van der Waals equation of state#Critical exponents \| Van der Waals equation of state]], and is usually 1.2 to 1.3.

	==Epsilon divergence==		==Epsilon divergence==

Carl McBride: /* Gamma divergence */

2011-07-15T15:32:59Z

Gamma divergence

← Older revision		Revision as of 17:32, 15 July 2011
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	When approaching the critical point along the critical isochore (<math>T > T_c</math>) the divergence is of the form		When approaching the critical point along the critical isochore (<math>T > T_c</math>) the divergence is of the form

	:<math>\left. \right. ~~C_p \sim~~ \kappa_T \sim (T-T_c)^{-\gamma} \sim (p-p_c)^{-\gamma}</math>		:<math>\left. \right. \kappa_T \sim (T-T_c)^{-\gamma} \sim (p-p_c)^{-\gamma}</math>

	where <math>\gamma</math> is 1.0 for the [[Van der Waals equation of state]], and is usually 1.2 to 1.3.		where <math>\kappa_T</math> is the [[Compressibility#Isothermal compressibility \| isothermal compressibility]]. <math>\gamma</math> is 1.0 for the [[Van der Waals equation of state]], and is usually 1.2 to 1.3.

	==Epsilon divergence==		==Epsilon divergence==

Carl McBride at 13:40, 26 November 2009

2009-11-26T13:40:08Z

← Older revision		Revision as of 15:40, 26 November 2009
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	Theoretically one has <math>\nu = 0.63012(16)</math><ref name="Campostrini2002"> </ref> for the three dimensional Ising model, and <math>\nu = 0.67155(27)</math><ref name="Campostrini2001"> </ref> for the three-dimensional XY universality class.		Theoretically one has <math>\nu = 0.63012(16)</math><ref name="Campostrini2002"> </ref> for the three dimensional Ising model, and <math>\nu = 0.67155(27)</math><ref name="Campostrini2001"> </ref> for the three-dimensional XY universality class.
	==Inequalities==		==Inequalities==
	====Fisher ====		====Fisher inequality====
			The Fisher inequality (Eq. 5 <ref>[http://dx.doi.org/10.1103/PhysRev.180.594 Michael E. Fisher "Rigorous Inequalities for Critical-Point Correlation Exponents", Physical Review '''180''' pp. 594-600 (1969)]</ref>)

			:<math>\gamma \le (2-\eta) \nu</math>
			====Griffiths inequality====
			The Griffiths inequality (Eq. 3 <ref>[http://dx.doi.org/10.1103/PhysRevLett.14.623 Robert B. Griffiths "Thermodynamic Inequality Near the Critical Point for Ferromagnets and Fluids", Physical Review Letters '''14''' 623-624 (1965)]</ref>):

			:<math>(1+\delta)\beta \ge 2-\alpha'</math>
	====Josephson inequality====		====Josephson inequality====
	The Josephson inequality <ref>[http://dx.doi.org/10.1088/0370-1328/92/2/301 B. D. Josephson "Inequality for the specific heat: I. Derivation", Proceedings of the Physical Society '''92''' pp. 269-275 (1967)]</ref><ref>[http://dx.doi.org/10.1088/0370-1328/92/2/302 B. D. Josephson "Inequality for the specific heat: II. Application to critical phenomena", Proceedings of the Physical Society '''92''' pp. 276-284 (1967)]</ref><ref>[http://dx.doi.org/10.1007/BF01008478 Alan D. Sokal "Rigorous proof of the high-temperature Josephson inequality for critical exponents", Journal of Statistical Physics '''25''' pp. 51-56 (1981)]</ref>		The Josephson inequality <ref>[http://dx.doi.org/10.1088/0370-1328/92/2/301 B. D. Josephson "Inequality for the specific heat: I. Derivation", Proceedings of the Physical Society '''92''' pp. 269-275 (1967)]</ref><ref>[http://dx.doi.org/10.1088/0370-1328/92/2/302 B. D. Josephson "Inequality for the specific heat: II. Application to critical phenomena", Proceedings of the Physical Society '''92''' pp. 276-284 (1967)]</ref><ref>[http://dx.doi.org/10.1007/BF01008478 Alan D. Sokal "Rigorous proof of the high-temperature Josephson inequality for critical exponents", Journal of Statistical Physics '''25''' pp. 51-56 (1981)]</ref>

	:<math>d\nu \ge 2-\alpha</math>		:<math>d\nu \ge 2-\alpha</math>
			====Liberman inequality====
			<ref>[http://dx.doi.org/10.1063/1.1726488 David A. Liberman "Another Relation Between Thermodynamic Functions Near the Critical Point of a Simple Fluid", Journal of Chemical Physics '''44''' 419-420 (1966)]</ref>
	====Rushbrooke inequality====		====Rushbrooke inequality====
	The Rushbrooke inequality (Eq. 2 <ref>[http://dx.doi.org/10.1063/1.1734338 G. S. Rushbrooke "On the Thermodynamics of the Critical Region for the Ising Problem", Journal of Chemical Physics 39, 842-843 (1963)]</ref>), based on the work of Essam and Fisher (Eq. 38 <ref>[http://dx.doi.org/10.1063/1.1733766 John W. Essam and Michael E. Fisher "Padé Approximant Studies of the Lattice Gas and Ising Ferromagnet below the Critical Point", Journal of Chemical Physics 38, 802-812 (1963)]</ref>) is given by		The Rushbrooke inequality (Eq. 2 <ref>[http://dx.doi.org/10.1063/1.1734338 G. S. Rushbrooke "On the Thermodynamics of the Critical Region for the Ising Problem", Journal of Chemical Physics 39, 842-843 (1963)]</ref>), based on the work of Essam and Fisher (Eq. 38 <ref>[http://dx.doi.org/10.1063/1.1733766 John W. Essam and Michael E. Fisher "Padé Approximant Studies of the Lattice Gas and Ising Ferromagnet below the Critical Point", Journal of Chemical Physics 38, 802-812 (1963)]</ref>) is given by

	:<math>\alpha + 2\beta + \gamma \ge 2</math>.		:<math>\alpha' + 2\beta + \gamma' \ge 2</math>.

	Using the above-mentioned values<ref name="Campostrini2002"> </ref> one has:		Using the above-mentioned values<ref name="Campostrini2002"> </ref> one has:

	:<math>0.1096 + (2\times0.32653) + 1.2373 = 1.99996</math>		:<math>0.1096 + (2\times0.32653) + 1.2373 = 1.99996</math>
	====Widom ~~relation~~====		====Widom inequality====
			The Widom inequality <ref>[http://dx.doi.org/10.1063/1.1726135 B. Widom "Degree of the Critical Isotherm", Journal of Chemical Physics '''41''' pp. 1633-1634 (1964)]</ref>

			:<math>\gamma' \ge \beta(\delta -1)</math>
	==Hyperscaling==		==Hyperscaling==
	==Gamma divergence==		==Gamma divergence==

Carl McBride: partial save

2009-11-26T13:07:10Z

partial save

← Older revision		Revision as of 15:07, 26 November 2009
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	Theoretically one has <math>\nu = 0.63012(16)</math><ref name="Campostrini2002"> </ref> for the three dimensional Ising model, and <math>\nu = 0.67155(27)</math><ref name="Campostrini2001"> </ref> for the three-dimensional XY universality class.		Theoretically one has <math>\nu = 0.63012(16)</math><ref name="Campostrini2002"> </ref> for the three dimensional Ising model, and <math>\nu = 0.67155(27)</math><ref name="Campostrini2001"> </ref> for the three-dimensional XY universality class.
			==Inequalities==
			====Fisher ====
			====Josephson inequality====
			The Josephson inequality <ref>[http://dx.doi.org/10.1088/0370-1328/92/2/301 B. D. Josephson "Inequality for the specific heat: I. Derivation", Proceedings of the Physical Society '''92''' pp. 269-275 (1967)]</ref><ref>[http://dx.doi.org/10.1088/0370-1328/92/2/302 B. D. Josephson "Inequality for the specific heat: II. Application to critical phenomena", Proceedings of the Physical Society '''92''' pp. 276-284 (1967)]</ref><ref>[http://dx.doi.org/10.1007/BF01008478 Alan D. Sokal "Rigorous proof of the high-temperature Josephson inequality for critical exponents", Journal of Statistical Physics '''25''' pp. 51-56 (1981)]</ref>

			:<math>d\nu \ge 2-\alpha</math>
			====Rushbrooke inequality====
			The Rushbrooke inequality (Eq. 2 <ref>[http://dx.doi.org/10.1063/1.1734338 G. S. Rushbrooke "On the Thermodynamics of the Critical Region for the Ising Problem", Journal of Chemical Physics 39, 842-843 (1963)]</ref>), based on the work of Essam and Fisher (Eq. 38 <ref>[http://dx.doi.org/10.1063/1.1733766 John W. Essam and Michael E. Fisher "Padé Approximant Studies of the Lattice Gas and Ising Ferromagnet below the Critical Point", Journal of Chemical Physics 38, 802-812 (1963)]</ref>) is given by

	~~==Rushbrooke equality==~~		:<math>\alpha + 2\beta + \gamma \ge 2</math>.
	~~The Rushbrooke equality~~ <~~ref~~>~~[http://dx.doi.org/10.1063/1.1734338 G. S. Rushbrooke "On the Thermodynamics of the Critical Region for the Ising Problem", Journal of Chemical Physics~~ ~~39, 842-843 (1963)]~~</~~ref~~> ~~, proposed by Essam and Fisher (Eq~~. 38 <ref>[http://dx.doi.org/10.1063/1.1733766 John W. Essam and Michael E. Fisher "Padé Approximant Studies of the Lattice Gas and Ising Ferromagnet below the Critical Point", Journal of Chemical Physics 38, 802-812 (1963)]</ref>) is given by

	~~:<math>\alpha + 2\beta + \gamma =2</math>.~~		Using the above-mentioned values<ref name="Campostrini2002"> </ref> one has:

	Using the above-mentioned values one has:

	:<math>0.1096 + (2\times0.32653) + 1.2373 = 1.99996</math>		:<math>0.1096 + (2\times0.32653) + 1.2373 = 1.99996</math>
			====Widom relation====
			==Hyperscaling==
	==Gamma divergence==		==Gamma divergence==
	When approaching the critical point along the critical isochore (<math>T > T_c</math>) the divergence is of the form		When approaching the critical point along the critical isochore (<math>T > T_c</math>) the divergence is of the form

Carl McBride: /* Correlation length */ Inseted leading zero

2009-11-26T12:04:39Z

Correlation length: Inseted leading zero

← Older revision		Revision as of 14:04, 26 November 2009
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	:<math>\left. \xi \right.= \xi_0 \epsilon^{-\nu}</math>		:<math>\left. \xi \right.= \xi_0 \epsilon^{-\nu}</math>

	Theoretically one has <math>\nu = 0.63012(16)</math><ref name="Campostrini2002"> </ref> for the three dimensional Ising model, and <math>\nu = .~~671 55~~(27)</math><ref name="Campostrini2001"> </ref> for the three-dimensional XY universality class.		Theoretically one has <math>\nu = 0.63012(16)</math><ref name="Campostrini2002"> </ref> for the three dimensional Ising model, and <math>\nu = 0.67155(27)</math><ref name="Campostrini2001"> </ref> for the three-dimensional XY universality class.

	==Rushbrooke equality==		==Rushbrooke equality==
	The Rushbrooke equality <ref>[http://dx.doi.org/10.1063/1.1734338 G. S. Rushbrooke "On the Thermodynamics of the Critical Region for the Ising Problem", Journal of Chemical Physics 39, 842-843 (1963)]</ref> , proposed by Essam and Fisher (Eq. 38 <ref>[http://dx.doi.org/10.1063/1.1733766 John W. Essam and Michael E. Fisher "Padé Approximant Studies of the Lattice Gas and Ising Ferromagnet below the Critical Point", Journal of Chemical Physics 38, 802-812 (1963)]</ref>) is given by		The Rushbrooke equality <ref>[http://dx.doi.org/10.1063/1.1734338 G. S. Rushbrooke "On the Thermodynamics of the Critical Region for the Ising Problem", Journal of Chemical Physics 39, 842-843 (1963)]</ref> , proposed by Essam and Fisher (Eq. 38 <ref>[http://dx.doi.org/10.1063/1.1733766 John W. Essam and Michael E. Fisher "Padé Approximant Studies of the Lattice Gas and Ising Ferromagnet below the Critical Point", Journal of Chemical Physics 38, 802-812 (1963)]</ref>) is given by