2.2.2 Static Scaling Hypothesis (continued)

In 1965, Widom postulated the homogeneity postulate, also called the scaling hypothesis. It states that the singularities of various measured quantities near critical points can be represented as variables of generalized homogeneous functions. If t denotes the reduced temperature and F the singular part of the free energy, then:

If the parameter λ scales the reduced temperature t, the function is scaled accordingly. If another scaling variable μ is introduced, Eq. (2.17) gives:

Comparing the factors multiplying F(t), the relation g(λ)g(μ) = g(λμ) is satisfied when g is a power function. Equation (2.17) therefore becomes:

Here a is the degree of homogeneity. Since derivatives and integrals of homogeneous functions are themselves homogeneous, it is plausible that thermodynamic quantities derived from the free energy follow power laws. This provides a strategy for treating the singular contributions to thermodynamic quantities.

The concept can also be generalized to functions of several variables. Adding, for example, the order parameter o gives:

The static scaling hypothesis reveals not only the universality of critical exponents in different systems, but also relations among those exponents: scaling laws. For binary systems, some of the exponents listed in Table 2.1 obey such relations.

Original footnote: A function f on D ⊆ ℝⁿ is positively homogeneous of degree r if f(λx) = λʳ f(x) for every x ∈ D and λ > 0 for which λx ∈ D.