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Widom scaling



Widom scaling 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 paramaterized in terms of two values.

Additional recommended knowledge

Definitions

The critical exponents α,α',β,γ,γ' and δ are defined in terms of the behaviour of the order parameters and response functions near the critical point as follows

M(t,0) \simeq (-t)^{\beta}, for t \uparrow 0
M(0,H) \simeq |H|^{1/ \delta} sign(H), for H \rightarrow 0
\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) \simeq \begin{cases} 	(t)^{-\alpha} & \textrm{for} \ t \downarrow 0 \\ 	(-t)^{-\alpha'} & \textrm{for} \ t \uparrow 0 \end{cases}

where

t \equiv \frac{T-T_c}{T_c} measures the temperature relative to the critical point.

Derivation

The scaling hypothesis is that near the critical point, the free energy f(t,H) can be written as the sum of a slowly varying regular part fr and a singular part fs, with the singular part being a scaling function, ie, a homogeneous function, so that

fsptqH) = λfs(t,H)

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

λqMptqH) = λM(t,H)

Setting H = 0 and λ = ( − t) − 1 / p in the preceding equation yields

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

Comparing this with the definition of β yields its value,

\beta = \frac{1-q}{p}

Similarly, putting t = 0 and λ = H − 1 / q into the scaling relation for M yields

\delta = \frac{q}{1-q}

Applying the expression for the isothermal susceptibility χT in terms of M to the scaling relation yields

λ2qχTptqH) = λχT(t,H)

Setting H=0 and λ = (t) − 1 / p for t \downarrow 0 (resp. λ = ( − t) − 1 / p for t \uparrow 0) yields

\gamma = \gamma' = \frac{2q -1}{p}

Similarly for the expression for specific heat cH in terms of M to the scaling relation yields

λ2pcHptqH) = λcH(t,H)

Taking H=0 and λ = (t) − 1 / p for t \downarrow 0 (or λ = ( − t) − 1 / p for t \uparrow 0) yields

\alpha = \alpha' = 2 -\frac{1}{p}

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

\alpha = \alpha' = 2 - \beta(\delta +1) = 2 - \frac{1}{p}
γ = γ' = β(δ − 1)

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

References

H.E. Stanley, Introduction to Phase Transitions and Critical Phenomena

 
This article is licensed under the GNU Free Documentation License. It uses material from the Wikipedia article "Widom_scaling". A list of authors is available in Wikipedia.
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