Gyration tensor - Revision history

en>Sturgman: Corrected the acylindricity equation. For it to be positive we need to subtract lambda x from lambda y.

2014-05-13T19:16:49Z

Corrected the acylindricity equation. For it to be positive we need to subtract lambda x from lambda y.

← Older revision		Revision as of 21:16, 13 May 2014
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	'''Widom scaling''' is a hypothesis in [[statistical mechanics]] regarding the [[Thermodynamic free energy\|free energy]] of a [[magnetic system]] near its [[critical point (thermodynamics)\|critical point]] which leads to the [[critical exponent]]s becoming no longer independent so that they can be parameterized in terms of two values. The hypothesis can be seen to arise as a natural consequence of the block-spin renormalization procedure, when the ~~block size is chosen to be of the same size as the correlation length~~.~~<ref>Kerson Huang, Statistical Mechanics. John Wiley and Sons, 1987</ref>~~		Myrtle Benny is how I'm called and I feel comfy when people use the complete title. Years in the past we moved to North Dakota. Doing ceramics is what love doing. She is a librarian but she's always wanted her own company.<br><br>Here is my web blog - [http://musical.sehan.ac.kr/?document_srl=2110508 home std test kit]

	~~Widom scaling is an example of [[universality (dynamical systems)\|universality]].~~

	~~== Definitions ==~~

	~~The critical exponents <math> \alpha, \alpha', \beta, \gamma, \gamma' </math> and <math> \delta </math> are defined~~ in ~~terms of the behaviour of the order parameters and response functions near~~ the ~~critical point as follows~~

	~~:<math> M(t,0) \simeq (-t)^{\beta}</math>, for <math> t \uparrow 0 </math>~~
	~~:<math> M(0,H) \simeq \|H\|^{1/ \delta} \mathrm{sign}(H)</math>, for <math> H \rightarrow 0 </math>~~
	~~:<math> \chi_T(t,0) \simeq \begin{cases}~~
	~~(t)^{-\gamma}, & \textrm{for} \ t \downarrow 0 \\~~
	~~(-t)^{-\gamma'}, & \textrm{for} \ t \uparrow 0 \end{cases}~~
	~~</math>~~

	~~:<math> c_H(t,0) \simeq \begin{cases}~~
	~~(t)^{-\alpha} & \textrm{for} \ t \downarrow 0 \\~~
	~~(-t)^{-\alpha'} & \textrm{for} \ t \uparrow 0 \end{cases}~~
	~~</math>~~

	~~where~~

	~~:<math> t \equiv \frac{T-T_c}{T_c}</math> measures the temperature relative~~ to ~~the critical point~~.

	~~Near the critical point, Widom's scaling relation reads~~

	~~:<math> H(t) \simeq M\|M\|^{\delta-1} f(t/\|M\|^{1/\beta})</math>.~~

	~~where <math>f</math> has an expansion~~
	~~:<math> f(t/\|M\|^{1/\beta})\approx 1+{\rm const}\times( t/\|M\|^{1/\beta})^\omega +\dots~~
	~~</math>,~~
	~~with <math> \omega</math>~~
	~~being Wegner's exponent governing the [[approach to scaling]]~~.

	~~== Derivation ==~~

	~~The scaling hypothesis~~ is ~~that near the critical point, the free energy <math>f(t,H)</math>, in <math>d</math> dimensions, can be written as the sum of a slowly varying regular part <math>f_r</math> and~~ a ~~singular part <math>f_s</math>, with the singular part being a scaling function, i.e., a [[homogeneous function]], so that~~

	~~:<math> f_s(\lambda^p t, \lambda^q H) = \lambda^d f_s(t, H) \,</math>~~

	~~Then taking the [[partial derivative]] with respect to ''H'' and the form of ''M(t,H)'' gives~~

	~~:<math> \lambda^q M(\lambda^p t, \lambda^q H) = \lambda^d M(t, H) \,</math>~~

	~~Setting <math>H=0</math> and <math> \lambda = (-t)^{-1/p} </math> in the preceding equation yields~~

	~~:<math> M(t,0) = (-t)^{\frac{d-q}{p}} M(-1,0),</math> for <math> t \uparrow 0 </math>~~

	~~Comparing this with the definition of <math>\beta</math> yields its value,~~

	~~:<math> \beta = \frac{d-q}{p}\equiv \frac{\nu}2(d-2+\eta). </math>~~

	~~Similarly, putting <math>t=0</math> and <math> \lambda = H^{-1/q} </math> into the scaling relation for ''M'' yields~~

	~~:<math> \delta = \frac{q}{d-q} \equiv \frac{d+2-\eta}{d-2+\eta}.</math>~~

	~~Hence~~

	~~:<math> \frac{q}{p} = \frac{\nu}{2}~~
	~~(d+2-\eta),~\frac 1 p=\nu.</math>~~


	~~Applying the expression for the [[isothermal susceptibility]] <math> \chi_T </math> in terms of ''M'' to the scaling relation yields~~

	~~:<math> \lambda^{2q} \chi_T (\lambda^p t, \lambda^q H) = \lambda^d \chi_T (t, H) \,</math>~~

	~~Setting ''H=0'' and <math> \lambda = (t)^{-1/p}</math> for <math> t \downarrow 0</math> (resp. <math> \lambda = (-t)^{-1/p} </math> for <math> t \uparrow 0 </math>) yields~~

	~~:<math> \gamma = \gamma' = \frac{2q -d}{p} \,</math>~~

	~~Similarly for the expression for [[specific heat]] <math> c_H </math> in terms of ''M'' to the scaling relation yields~~

	~~:<math> \lambda^{2p} c_H ( \lambda^p t, \lambda^q H) = \lambda^d c_H(t, H) \, </math>~~

	~~Taking ''H=0'' and <math> \lambda = (t)^{-1/p} </math> for <math> t \downarrow 0 </math> (or <math> \lambda = (-t)^{-1/p} </math> for <math>t \uparrow 0)</math> yields~~

	~~:<math> \alpha = \alpha' = 2 -\frac{d}{p}=2-\nu d </math>~~

	~~As a consequence of Widom scaling, not all critical exponents are independent~~ but ~~they can be parameterized by two numbers <math> p, q \in \mathbb{R} </math> with the relations expressed as~~

	~~:<math> \alpha = \alpha~~' ~~= 2-\nu d,</math>~~
	~~:<math> \gamma = \gamma' = \beta(\delta -1)=\nu(2-\eta) .</math>~~

	~~The relations are experimentally well verified for magnetic systems and fluids~~.

	~~==References==~~
	*H. E. Stanley, ''Introduction to Phase Transitions and Critical Phenomena''
	*[[Hagen Kleinert\|H. Kleinert]] and V. Schulte-Frohlinde, ''Critical Properties of φ<~~sup~~>4<~~/sup~~>-~~Theories'',~~ [http://~~www~~.~~worldscibooks~~.~~com/physics/4733~~.~~html World Scientific (Singapore, 2001)]; Paperback ISBN 981-02-4658-7'' (also available [http://users.physik.fu-berlin.de/~kleinert/kleinert~~/?~~p=booklist&details~~=~~6 online])''~~
	~~{{Reflist\|2}}~~

	~~[[Category:Critical phenomena]]~~
	~~[[Category:Statistical mechanics]~~]

en>Mark viking: Added wl, context to first sentence

2013-09-26T03:57:34Z

Added wl, context to first sentence

← Older revision		Revision as of 05:57, 26 September 2013
Line 1:		Line 1:
	~~Myrtle Benny~~ is ~~how I~~'~~m called~~ and ~~I feel comfy when people use~~ the ~~complete title~~. ~~Puerto Rico~~ is ~~exactly where he~~ and ~~his spouse live~~. ~~Bookkeeping is her working day job now~~. ~~He is really fond~~ of ~~performing ceramics but he is having difficulties~~ to ~~discover time~~ for it.<br><br>~~Also visit my web site~~: ~~home std test kit~~ ([http://www.~~hooddirectory~~.com/~~how~~-~~you~~-~~can~~-~~cure~~-~~an-unpleasant-yeast-infection~~/ ~~related internet page~~])		'''Widom scaling''' is a hypothesis in [[statistical mechanics]] regarding the [[Thermodynamic free energy\|free energy]] of a [[magnetic system]] near its [[critical point (thermodynamics)\|critical point]] which leads to the [[critical exponent]]s becoming no longer independent so that they can be parameterized in terms of two values. The hypothesis can be seen to arise as a natural consequence of the block-spin renormalization procedure, when the block size is chosen to be of the same size as the correlation length.<ref>Kerson Huang, Statistical Mechanics. John Wiley and Sons, 1987</ref>

			Widom scaling is an example of [[universality (dynamical systems)\|universality]].

			== Definitions ==

			The critical exponents <math> \alpha, \alpha', \beta, \gamma, \gamma' </math> and <math> \delta </math> are defined in terms of the behaviour of the order parameters and response functions near the critical point as follows

			:<math> M(t,0) \simeq (-t)^{\beta}</math>, for <math> t \uparrow 0 </math>
			:<math> M(0,H) \simeq \|H\|^{1/ \delta} \mathrm{sign}(H)</math>, for <math> H \rightarrow 0 </math>
			:<math> \chi_T(t,0) \simeq \begin{cases}
			(t)^{-\gamma}, & \textrm{for} \ t \downarrow 0 \\
			(-t)^{-\gamma'}, & \textrm{for} \ t \uparrow 0 \end{cases}
			</math>

			:<math> c_H(t,0) \simeq \begin{cases}
			(t)^{-\alpha} & \textrm{for} \ t \downarrow 0 \\
			(-t)^{-\alpha'} & \textrm{for} \ t \uparrow 0 \end{cases}
			</math>

			where

			:<math> t \equiv \frac{T-T_c}{T_c}</math> measures the temperature relative to the critical point.

			Near the critical point, Widom's scaling relation reads

			:<math> H(t) \simeq M\|M\|^{\delta-1} f(t/\|M\|^{1/\beta})</math>.

			where <math>f</math> has an expansion
			:<math> f(t/\|M\|^{1/\beta})\approx 1+{\rm const}\times( t/\|M\|^{1/\beta})^\omega +\dots
			</math>,
			with <math> \omega</math>
			being Wegner's exponent governing the [[approach to scaling]].

			== Derivation ==

			The scaling hypothesis is that near the critical point, the free energy <math>f(t,H)</math>, in <math>d</math> dimensions, can be written as the sum of a slowly varying regular part <math>f_r</math> and a singular part <math>f_s</math>, with the singular part being a scaling function, i.e., a [[homogeneous function]], so that

			:<math> f_s(\lambda^p t, \lambda^q H) = \lambda^d f_s(t, H) \,</math>

			Then taking the [[partial derivative]] with respect to ''H'' and the form of ''M(t,H)'' gives

			:<math> \lambda^q M(\lambda^p t, \lambda^q H) = \lambda^d M(t, H) \,</math>

			Setting <math>H=0</math> and <math> \lambda = (-t)^{-1/p} </math> in the preceding equation yields

			:<math> M(t,0) = (-t)^{\frac{d-q}{p}} M(-1,0),</math> for <math> t \uparrow 0 </math>

			Comparing this with the definition of <math>\beta</math> yields its value,

			:<math> \beta = \frac{d-q}{p}\equiv \frac{\nu}2(d-2+\eta). </math>

			Similarly, putting <math>t=0</math> and <math> \lambda = H^{-1/q} </math> into the scaling relation for ''M'' yields

			:<math> \delta = \frac{q}{d-q} \equiv \frac{d+2-\eta}{d-2+\eta}.</math>

			Hence

			:<math> \frac{q}{p} = \frac{\nu}{2}
			(d+2-\eta),~\frac 1 p=\nu.</math>


			Applying the expression for the [[isothermal susceptibility]] <math> \chi_T </math> in terms of ''M'' to the scaling relation yields

			:<math> \lambda^{2q} \chi_T (\lambda^p t, \lambda^q H) = \lambda^d \chi_T (t, H) \,</math>

			Setting ''H=0'' and <math> \lambda = (t)^{-1/p}</math> for <math> t \downarrow 0</math> (resp. <math> \lambda = (-t)^{-1/p} </math> for <math> t \uparrow 0 </math>) yields

			:<math> \gamma = \gamma' = \frac{2q -d}{p} \,</math>

			Similarly for the expression for [[specific heat]] <math> c_H </math> in terms of ''M'' to the scaling relation yields

			:<math> \lambda^{2p} c_H ( \lambda^p t, \lambda^q H) = \lambda^d c_H(t, H) \, </math>

			Taking ''H=0'' and <math> \lambda = (t)^{-1/p} </math> for <math> t \downarrow 0 </math> (or <math> \lambda = (-t)^{-1/p} </math> for <math>t \uparrow 0)</math> yields

			:<math> \alpha = \alpha' = 2 -\frac{d}{p}=2-\nu d </math>

			As a consequence of Widom scaling, not all critical exponents are independent but they can be parameterized by two numbers <math> p, q \in \mathbb{R} </math> with the relations expressed as

			:<math> \alpha = \alpha' = 2-\nu d,</math>
			:<math> \gamma = \gamma' = \beta(\delta -1)=\nu(2-\eta) .</math>

			The relations are experimentally well verified for magnetic systems and fluids.

			==References==
			*H. E. Stanley, ''Introduction to Phase Transitions and Critical Phenomena''
			*[[Hagen Kleinert\|H. Kleinert]] and V. Schulte-Frohlinde, ''Critical Properties of φ<sup>4</sup>-Theories'', [http://www.worldscibooks.com/physics/4733.html World Scientific (Singapore, 2001)]; Paperback ISBN 981-02-4658-7'' (also available [http://users.physik.fu-berlin.de/~kleinert/kleinert/?p=booklist&details=6 online])''
			{{Reflist\|2}}

			[[Category:Critical phenomena]]
			[[Category:Statistical mechanics]]

en>Reedy Bot: /* References */Tagging for AFT v5

2012-01-04T20:30:13Z

References: Tagging for AFT v5

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Myrtle Benny is how I'm called and I feel comfy when people use the complete title. Puerto Rico is exactly where he and his spouse live. Bookkeeping is her working day job now. He is really fond of performing ceramics but he is having difficulties to discover time for it.<br><br>Also visit my web site: home std test kit ([http://www.hooddirectory.com/how-you-can-cure-an-unpleasant-yeast-infection/ related internet page])