Scale-independent mixing laws

Mixing laws have been introduced in effective medium physics to calculate a bulk parameter of mixtures of several phases as a function of the parameter values and volume fractions for each phase. They have been successfully applied to derive mixture models for dielectric constant, thermal conductivity, electrical conductivity, etc. Studied here are mixing laws that can be written in the form f(x)=a1.f(x1)+...+ak.f(xk) with a1+a2+...+ak=1 where f() is a continuous function applied to a bulk parameter x of a mixture of several phases with bulk parameters x1,...,xk and volume fractions a1,...,ak. In the case of scale-independent mixing laws, i.e. such that for all t>0 t.f(x)=a1.f(t.x1)+...+ak.f(t.xk) it is shown that f() can take only two forms: f(x)=A.lnx+B or f(x)=Ax^p+B. Therefore scale-independent mixing laws can only be the geometric mean x=x1^a1....xk^ak or the 'power mean' x^p=a1.x1^p+...+ak.xk^p.