Systematic Application of Generalized Homogeneous Functions to Static Scaling, Dynamic Scaling, and Universality

Abstract

A function $f({x}_{1}, {x}_{2},\dots{}, {x}_{n})$ is a generalized homogeneous function (GHF) if we can find numbers ${a}_{1}, {a}_{2}, \dots{}, {a}_{n}$ such that for all values of the positive number $\ensuremath{\lambda}$, $f({\ensuremath{\lambda}}^{{a}_{1}}{x}_{1}, {\ensuremath{\lambda}}^{{a}_{2}}{x}_{2}, \dots{}, {\ensuremath{\lambda}}^{{a}_{n}}{x}_{n})={\ensuremath{\lambda}}^{{a}_{f}}f({x}_{1}, {x}_{2}, \dots{}, {x}_{n})$. We organize the properties of GHFs in four theorems. These are used to systematically examine the consequences of various scaling hypotheses. An advantage of this approach is that the same formalism may be used to treat thermodynamic functions, static correlation functions, dynamic correlation functions, and "universality." The simple case of thermodynamic scaling (two independent variables) is first generalized to static and dynamic correlation functions (three and four variables), and then to scaling with a parameter (for which the critical subspace becomes higher dimensional). In this last case, where a second GHF hypothesis is made, the necessity of crossover lines is demonstrated. The assumption of homogeneity is clearly separated from any extra assumptions that may also be called scaling (or "strong scaling"), but are independent of and different from that of homogeneity. One practical insight gained from the present approach is that all experimentally measured exponents are expressible as the ratio of two scaling powers, ${a}_{f}$ (which refers to the function) and ${a}_{j}$ (which refers to the path of approach to the critical point). A second practical advantage is that, since a GHF can be scaled with respect to any of its arguments, one can immediately write a variety of scaling functions for each type of scaling hypothesis. The GHF approach thereby permits data to be plotted in a variety of convenient fashions, and is found to facilitate computation of the relevant scaling functions (in particular, the GHF approach led directly to the recent calculation of the Heisenberg model scaling function by Milos\ifmmode \breve{}\else \u{}\fi{}evi\ifmmode \acute{c}\else \'{c}\fi{} and Stanley).