Widom scaling

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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.

[edit] 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.

[edit] 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 $f r$ and a singular part $f s$ , with the singular part being a scaling function, ie, a homogeneous function, so that

f s (λ p t,λ q H) = λ f s (t, H)

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

λ q M (λ p t,λ q H) = λ M (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 $χ T$ in terms of M to the scaling relation yields

λ 2 q χ T (λ p t,λ q H) = λχ T (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 $c H$ in terms of M to the scaling relation yields

λ 2 p c H (λ p t,λ q H) = λ c H (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