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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] |
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| Widom scaling is an example of [[universality (dynamical systems)|universality]].
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| == Definitions ==
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| 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
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| :<math> M(t,0) \simeq (-t)^{\beta}</math>, for <math> t \uparrow 0 </math>
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| :<math> M(0,H) \simeq |H|^{1/ \delta} \mathrm{sign}(H)</math>, for <math> H \rightarrow 0 </math>
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| :<math> \chi_T(t,0) \simeq \begin{cases}
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| (t)^{-\gamma}, & \textrm{for} \ t \downarrow 0 \\
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| (-t)^{-\gamma'}, & \textrm{for} \ t \uparrow 0 \end{cases}
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| </math>
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| :<math> c_H(t,0) \simeq \begin{cases}
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| (t)^{-\alpha} & \textrm{for} \ t \downarrow 0 \\
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| (-t)^{-\alpha'} & \textrm{for} \ t \uparrow 0 \end{cases}
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| </math>
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| where
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| :<math> t \equiv \frac{T-T_c}{T_c}</math> measures the temperature relative to the critical point.
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| Near the critical point, Widom's scaling relation reads
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| :<math> H(t) \simeq M|M|^{\delta-1} f(t/|M|^{1/\beta})</math>.
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| where <math>f</math> has an expansion
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| :<math> f(t/|M|^{1/\beta})\approx 1+{\rm const}\times( t/|M|^{1/\beta})^\omega +\dots
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| </math>,
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| with <math> \omega</math>
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| being Wegner's exponent governing the [[approach to scaling]].
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| == Derivation ==
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| 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
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| :<math> f_s(\lambda^p t, \lambda^q H) = \lambda^d f_s(t, H) \,</math>
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| Then taking the [[partial derivative]] with respect to ''H'' and the form of ''M(t,H)'' gives
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| :<math> \lambda^q M(\lambda^p t, \lambda^q H) = \lambda^d M(t, H) \,</math>
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| Setting <math>H=0</math> and <math> \lambda = (-t)^{-1/p} </math> in the preceding equation yields
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| :<math> M(t,0) = (-t)^{\frac{d-q}{p}} M(-1,0),</math> for <math> t \uparrow 0 </math>
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| Comparing this with the definition of <math>\beta</math> yields its value,
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| :<math> \beta = \frac{d-q}{p}\equiv \frac{\nu}2(d-2+\eta). </math>
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| Similarly, putting <math>t=0</math> and <math> \lambda = H^{-1/q} </math> into the scaling relation for ''M'' yields
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| :<math> \delta = \frac{q}{d-q} \equiv \frac{d+2-\eta}{d-2+\eta}.</math>
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| Hence
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| :<math> \frac{q}{p} = \frac{\nu}{2}
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| (d+2-\eta),~\frac 1 p=\nu.</math>
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| Applying the expression for the [[isothermal susceptibility]] <math> \chi_T </math> in terms of ''M'' to the scaling relation yields
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| :<math> \lambda^{2q} \chi_T (\lambda^p t, \lambda^q H) = \lambda^d \chi_T (t, H) \,</math>
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| 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
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| :<math> \gamma = \gamma' = \frac{2q -d}{p} \,</math>
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| Similarly for the expression for [[specific heat]] <math> c_H </math> in terms of ''M'' to the scaling relation yields
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| :<math> \lambda^{2p} c_H ( \lambda^p t, \lambda^q H) = \lambda^d c_H(t, H) \, </math>
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| 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
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| :<math> \alpha = \alpha' = 2 -\frac{d}{p}=2-\nu d </math>
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| 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
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| :<math> \alpha = \alpha' = 2-\nu d,</math>
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| :<math> \gamma = \gamma' = \beta(\delta -1)=\nu(2-\eta) .</math>
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| The relations are experimentally well verified for magnetic systems and fluids.
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| ==References==
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| *H. E. Stanley, ''Introduction to Phase Transitions and Critical Phenomena''
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| *[[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])''
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| {{Reflist|2}}
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| [[Category:Critical phenomena]]
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| [[Category:Statistical mechanics]]
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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.
Here is my web blog - home std test kit