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.
[1] Widom scaling is an example of universality.
The critical exponents
, β , γ ,
γ ′
are defined in terms of the behaviour of the order parameters and response functions near the critical point as follows where Near the critical point, Widom's scaling relation reads where
being Wegner's exponent governing the approach to scaling.
The scaling hypothesis is that near the critical point, the free energy
dimensions, can be written as the sum of a slowly varying regular part
and a singular part
, with the singular part being a scaling function, i.e., a homogeneous function, so that Then taking the partial derivative with respect to H and the form of M(t,H) gives Setting
in the preceding equation yields Comparing this with the definition of
yields its value, Similarly, putting
into the scaling relation for M yields Hence
Applying the expression for the isothermal susceptibility
in terms of M to the scaling relation yields Setting H=0 and
) yields Similarly for the expression for specific heat
in terms of M to the scaling relation yields Taking H=0 and
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 The relations are experimentally well verified for magnetic systems and fluids.