Widom scaling - Citizendia

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 parameterized in terms of two values. Statistical mechanics is the application of Probability theory, which includes mathematical tools for dealing with large populations to the field of Mechanics In Thermodynamics, the term thermodynamic free energy refers to the amount of work that can be extracted from a System, and is helpful in Engineering Critical exponents describe the behaviour of physical quantities near continuous Phase transitions.

Widom scaling is an example of universality. In Statistical mechanics, universality is the observation that there are properties for a large class of systems that are independent of the dynamical details of the

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 $f r$ 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. In Mathematics, a homogeneous function is a function with multiplicative scaling behaviour if the argument is multiplied by a factor then the result is multiplied by some power In Mathematics, a partial derivative of a function of several variables is its Derivative with respect to one of those variables with the others held constant $λ = ( - t) - 1 / p$ 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

$\alpha = \alpha' = 2 - \beta(\delta +1) = 2 - \frac{1}{p}$

γ = γ' = β(δ - 1)

The relations are experimentally well verified for magnetic systems and fluids. Specific heat capacity, also known simply as specific heat, is the measure of the heat energy required to increase the Temperature of a unit quantity

References

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

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