Widom scaling (after Benjamin Widom) 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. The hypothesis can be seen to arise as a natural consequence of the block-spin renormalization procedure, when the block size is chosen to be of the same size as the correlation length.
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Widom scaling is an example of universality.
Definitions
The critical exponents
where
Near the critical point, Widom's scaling relation reads
where
with
Derivation
The scaling hypothesis is that near the critical point, the free energy
Then taking the partial derivative with respect to H and the form of M(t,H) gives
Setting
Comparing this with the definition of
Similarly, putting
Hence
Applying the expression for the isothermal susceptibility
Setting H=0 and
Similarly for the expression for specific heat
Taking H=0 and
As a consequence of Widom scaling, not all critical exponents are independent but they can be parameterized by two numbers
The relations are experimentally well verified for magnetic systems and fluids.