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

, for

, for

where

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^pt,λ^qH) = λf_s(t,H)

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

λ^qM(λ^pt,λ^qH) = λM(t,H)

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

for

Comparing this with the definition of β yields its value,

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

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

λ^2qχ_T(λ^pt,λ^qH) = λχ_T(t,H)

Setting H=0 and λ = (t) ^{− 1 / p} for (resp. λ = ( − t) ^{− 1 / p} for ) yields

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

λ^2pc_H(λ^pt,λ^qH) = λc_H(t,H)

Taking H=0 and λ = (t) ^{− 1 / p} for (or λ = ( − t) ^{− 1 / p} for yields

As a consequence of Widom scaling, not all critical exponents are independent but they can be parameterized by two numbers with the relations expressed as

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