Widom scaling

 ( Critical Phenomena ) 

Widom scaling (after Benjamin Widom) is a hypothesis in statistical mechanics regarding the free energy of a magnetic system near its critical point which leads to the critical exponents 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.Kerson Huang, Statistical Mechanics. John Wiley and Sons, 1987

Widom scaling is an example of universality.

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Definitions The critical exponents $\backslash alpha,\; \backslash alpha\text{'},\; \backslash beta,\; \backslash gamma,\; \backslash gamma\text{'}$ and $\backslash delta$ are defined in terms of the behaviour of the order parameters and response functions near the critical point as follows

$M(t,0)\; \backslash simeq\; (-t)^\{\backslash beta\}$, for $t\; \backslash uparrow\; 0$

$M(0,H)\; \backslash simeq\; |H|^\{1/\; \backslash delta\}\; \backslash mathrm\{sign\}(H)$, for $H\; \backslash rightarrow\; 0$

$\backslash chi\_T(t,0)\; \backslash simeq\; \backslash begin\{cases\}$

(t)^{-\gamma}, & \textrm{for} \ t \downarrow 0 \\
(-t)^{-\gamma'}, & \textrm{for} \ t \uparrow 0 \end{cases}
 
     

$c\_H(t,0)\; \backslash simeq\; \backslash begin\{cases\}$

(t)^{-\alpha} & \textrm{for} \ t \downarrow 0 \\
(-t)^{-\alpha'} & \textrm{for} \ t \uparrow 0 \end{cases}
     
     

where

$t\; \backslash equiv\; \backslash frac\{T-T\_c\}\{T\_c\}$ measures the temperature relative to the critical point.

Near the critical point, Widom's scaling relation reads

$H(t)\; \backslash simeq\; M|M|^\{\backslash delta-1\}\; f(t/|M|^\{1/\backslash beta\})$.

where $f$ has an expansion

$f(t/|M|^\{1/\backslash beta\})\backslash approx\; 1+\{\backslash rm\; const\}\backslash times(\; t/|M|^\{1/\backslash beta\})^\backslash omega\; +\backslash dots$

, with $\backslash omega$ being Wegner's exponent governing the approach to scaling.

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Derivation The scaling hypothesis is that near the critical point, the free energy $f(t,H)$, in $d$ dimensions, can be written as the sum of a slowly varying regular part $f\_r$ and a singular part $f\_s$, with the singular part being a scaling function, i.e., a homogeneous function, so that

$f\_s(\backslash lambda^p\; t,\; \backslash lambda^q\; H)\; =\; \backslash lambda^d\; f\_s(t,\; H)\; \backslash ,$

Then taking the partial derivative with respect to H and the form of M(t,H) gives

$\backslash lambda^q\; M(\backslash lambda^p\; t,\; \backslash lambda^q\; H)\; =\; \backslash lambda^d\; M(t,\; H)\; \backslash ,$

Setting $H=0$ and $\backslash lambda\; =\; (-t)^\{-1/p\}$ in the preceding equation yields

$M(t,0)\; =\; (-t)^\{\backslash frac\{d-q\}\{p\}\}\; M(-1,0),$ for $t\; \backslash uparrow\; 0$

Comparing this with the definition of $\backslash beta$ yields its value,

$\backslash beta\; =\; \backslash frac\{d-q\}\{p\}\backslash equiv\; \backslash frac\{\backslash nu\}2(d-2+\backslash eta).$

Similarly, putting $t=0$ and $\backslash lambda\; =\; H^\{-1/q\}$ into the scaling relation for M yields

$\backslash delta\; =\; \backslash frac\{q\}\{d-q\}\; \backslash equiv\; \backslash frac\{d+2-\backslash eta\}\{d-2+\backslash eta\}.$

Hence

$\backslash frac\{q\}\{p\}\; =\; \backslash frac\{\backslash nu\}\{2\}$

(d+2-\eta),~\frac 1 p=\nu.

Applying the expression for the isothermal susceptibility $\backslash chi\_T$ in terms of M to the scaling relation yields

$\backslash lambda^\{2q\}\; \backslash chi\_T\; (\backslash lambda^p\; t,\; \backslash lambda^q\; H)\; =\; \backslash lambda^d\; \backslash chi\_T\; (t,\; H)\; \backslash ,$

Setting H=0 and $\backslash lambda\; =\; (t)^\{-1/p\}$ for $t\; \backslash downarrow\; 0$ (resp. $\backslash lambda\; =\; (-t)^\{-1/p\}$ for $t\; \backslash uparrow\; 0$) yields

$\backslash gamma\; =\; \backslash gamma\text{'}\; =\; \backslash frac\{2q\; -d\}\{p\}\; \backslash ,$

Similarly for the expression for specific heat $c\_H$ in terms of M to the scaling relation yields

$\backslash lambda^\{2p\}\; c\_H\; (\; \backslash lambda^p\; t,\; \backslash lambda^q\; H)\; =\; \backslash lambda^d\; c\_H(t,\; H)\; \backslash ,$

Taking H=0 and $\backslash lambda\; =\; (t)^\{-1/p\}$ for $t\; \backslash downarrow\; 0$ (or $\backslash lambda\; =\; (-t)^\{-1/p\}$ for $t\; \backslash uparrow\; 0)$ yields

$\backslash alpha\; =\; \backslash alpha\text{'}\; =\; 2\; -\backslash frac\{d\}\{p\}=2-\backslash nu\; d$

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

$\backslash alpha\; =\; \backslash alpha\text{'}\; =\; 2-\backslash nu\; d,$

$\backslash gamma\; =\; \backslash gamma\text{'}\; =\; \backslash beta(\backslash delta\; -1)=\backslash nu(2-\backslash eta)\; .$

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

• H. E. Stanley, Introduction to Phase Transitions and Critical Phenomena
• Hagen Kleinert and V. Schulte-Frohlinde, Critical Properties of φ⁴-Theories, World Scientific (Singapore, 2001); Paperback (also available online )

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