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
- , for
- , for
where
- 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 fr and a singular part fs, with the singular part being a scaling function, ie, a homogeneous function, so that
- fs(λpt,λqH) = λfs(t,H)
Then taking the partial derivative with respect to H and the form of M(t,H) gives
- λqM(λpt,λqH) = 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 χT in terms of M to the scaling relation yields
- λ2qχT(λpt,λqH) = λχT(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 cH in terms of M to the scaling relation yields
- λ2pcH(λpt,λqH) = λcH(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.
[edit] Reference
H.E. Stanley, Introduction to Phase Transitions and Critical Phenomena