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Widom scaling
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<h2 id="definitions">Definitions</h2> <p>The critical exponents α , α ′ , β , γ , γ ′ {\displaystyle \alpha ,\alpha ',\beta ,\gamma ,\gamma '} and δ {\displaystyle \delta } are defined in terms of the behaviour of the order parameters and response functions near the critical point as follows </p> M ( t , 0 ) ≃ ( − t ) β {\displaystyle M(t,0)\simeq (-t)^{\beta }} , for t ↑ 0 {\displaystyle t\uparrow 0} M ( 0 , H ) ≃ | H | 1 / δ s i g n ( H ) {\displaystyle M(0,H)\simeq |H|^{1/\delta }\mathrm {sign} (H)} , for H → 0 {\displaystyle H\rightarrow 0} χ T ( t , 0 ) ≃ { ( t ) − γ , for t ↓ 0 ( − t ) − γ ′ , for t ↑ 0 {\displaystyle \chi _{T}(t,0)\simeq {\begin{cases}(t)^{-\gamma },&{\textrm {for}}\ t\downarrow 0\\(-t)^{-\gamma '},&{\textrm {for}}\ t\uparrow 0\end{cases}}} c H ( t , 0 ) ≃ { ( t ) − α for t ↓ 0 ( − t ) − α ′ for t ↑ 0 {\displaystyle c_{H}(t,0)\simeq {\begin{cases}(t)^{-\alpha }&{\textrm {for}}\ t\downarrow 0\\(-t)^{-\alpha '}&{\textrm {for}}\ t\uparrow 0\end{cases}}} <p>where </p> t ≡ T − T c T c {\displaystyle t\equiv {\frac {T-T_{c}}{T_{c}}}} measures the temperature relative to the critical point. <p>Near the critical point, Widom's scaling relation reads </p> H ( t ) ≃ M | M | δ − 1 f ( t / | M | 1 / β ) {\displaystyle H(t)\simeq M|M|^{\delta -1}f(t/|M|^{1/\beta })} . <p>where f {\displaystyle f} has an expansion </p> f ( t / | M | 1 / β ) ≈ 1 + c o n s t × ( t / | M | 1 / β ) ω + … {\displaystyle f(t/|M|^{1/\beta })\approx 1+{\rm {const}}\times (t/|M|^{1/\beta })^{\omega }+\dots } , <p>with ω {\displaystyle \omega } being Wegner's exponent governing the approach to scaling. </p> <h2 id="derivation">Derivation</h2> <p>The scaling hypothesis is that near the critical point, the free energy f ( t , H ) {\displaystyle f(t,H)} , in d {\displaystyle d} dimensions, can be written as the sum of a slowly varying regular part f r {\displaystyle f_{r}} and a singular part f s {\displaystyle f_{s}} , with the singular part being a scaling function, i.e., a <a href="/facts/Homogeneous_function/wF9XX7s5">homogeneous function</a>, so that </p> f s ( λ p t , λ q H ) = λ d f s ( t , H ) {\displaystyle f_{s}(\lambda ^{p}t,\lambda ^{q}H)=\lambda ^{d}f_{s}(t,H)\,} <p>Then taking the <a href="/facts/Partial_derivative/JWHsBkAu">partial derivative</a> with respect to <i>H</i> and the form of <i>M(t,H)</i> gives </p> λ q M ( λ p t , λ q H ) = λ d M ( t , H ) {\displaystyle \lambda ^{q}M(\lambda ^{p}t,\lambda ^{q}H)=\lambda ^{d}M(t,H)\,} <p>Setting H = 0 {\displaystyle H=0} and λ = ( − t ) − 1 / p {\displaystyle \lambda =(-t)^{-1/p}} in the preceding equation yields </p> M ( t , 0 ) = ( − t ) d − q p M ( − 1 , 0 ) , {\displaystyle M(t,0)=(-t)^{\frac {d-q}{p}}M(-1,0),} for t ↑ 0 {\displaystyle t\uparrow 0} <p>Comparing this with the definition of β {\displaystyle \beta } yields its value, </p> β = d − q p ≡ ν 2 ( d − 2 + η ) . {\displaystyle \beta ={\frac {d-q}{p}}\equiv {\frac {\nu }{2}}(d-2+\eta ).} <p>Similarly, putting t = 0 {\displaystyle t=0} and λ = H − 1 / q {\displaystyle \lambda =H^{-1/q}} into the scaling relation for <i>M</i> yields </p> δ = q d − q ≡ d + 2 − η d − 2 + η . {\displaystyle \delta ={\frac {q}{d-q}}\equiv {\frac {d+2-\eta }{d-2+\eta }}.} <p>Hence </p> q p = ν 2 ( d + 2 − η ) , 1 p = ν . {\displaystyle {\frac {q}{p}}={\frac {\nu }{2}}(d+2-\eta ),~{\frac {1}{p}}=\nu .} <p> Applying the expression for the isothermal susceptibility χ T {\displaystyle \chi _{T}} in terms of <i>M</i> to the scaling relation yields </p> λ 2 q χ T ( λ p t , λ q H ) = λ d χ T ( t , H ) {\displaystyle \lambda ^{2q}\chi _{T}(\lambda ^{p}t,\lambda ^{q}H)=\lambda ^{d}\chi _{T}(t,H)\,} <p>Setting <i>H=0</i> and λ = ( t ) − 1 / p {\displaystyle \lambda =(t)^{-1/p}} for t ↓ 0 {\displaystyle t\downarrow 0} (resp. λ = ( − t ) − 1 / p {\displaystyle \lambda =(-t)^{-1/p}} for t ↑ 0 {\displaystyle t\uparrow 0} ) yields </p> γ = γ ′ = 2 q − d p {\displaystyle \gamma =\gamma '={\frac {2q-d}{p}}\,} <p>Similarly for the expression for <a href="/facts/Specific_heat/9P0U6YgI">specific heat</a> c H {\displaystyle c_{H}} in terms of <i>M</i> to the scaling relation yields </p> λ 2 p c H ( λ p t , λ q H ) = λ d c H ( t , H ) {\displaystyle \lambda ^{2p}c_{H}(\lambda ^{p}t,\lambda ^{q}H)=\lambda ^{d}c_{H}(t,H)\,} <p>Taking <i>H=0</i> and λ = ( t ) − 1 / p {\displaystyle \lambda =(t)^{-1/p}} for t ↓ 0 {\displaystyle t\downarrow 0} (or λ = ( − t ) − 1 / p {\displaystyle \lambda =(-t)^{-1/p}} for t ↑ 0 ) {\displaystyle t\uparrow 0)} yields </p> α = α ′ = 2 − d p = 2 − ν d {\displaystyle \alpha =\alpha '=2-{\frac {d}{p}}=2-\nu d} <p>As a consequence of Widom scaling, not all critical exponents are independent but they can be parameterized by two numbers p , q ∈ R {\displaystyle p,q\in \mathbb {R} } with the relations expressed as </p> α = α ′ = 2 − ν d , {\displaystyle \alpha =\alpha '=2-\nu d,} γ = γ ′ = β ( δ − 1 ) = ν ( 2 − η ) . {\displaystyle \gamma =\gamma '=\beta (\delta -1)=\nu (2-\eta ).} <p>The relations are experimentally well verified for magnetic systems and fluids. </p> <ul><li>H. E. Stanley, <i>Introduction to Phase Transitions and Critical Phenomena</i></li> <li><a href="/facts/Hagen_Kleinert/1vF2CVlD">H. Kleinert</a> and V. Schulte-Frohlinde, <i>Critical Properties of φ4-Theories</i>, <a href="http://www.worldscibooks.com/physics/4733.html">World Scientific (Singapore, 2001)</a>; Paperback <a href="/facts/ISBN_(identifier)/15AdSPa9">ISBN</a> 981-02-4658-7<i> (also available <a href="http://users.physik.fu-berlin.de/~kleinert/kleinert/?p=booklist&details=6">online</a>)</i></li></ul> <h2 id="references">References</h2> <ol> <li id="fn:1"><p>Kerson Huang, Statistical Mechanics. John Wiley and Sons, 1987 <a href="#fnref:1" class="footnote-back-ref">↩</a></p></li> </ol>