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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.

## 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