Scaling Theories and the Renormalization Group

Abstract

It is too strong to assume that the free energy of a model system in the critical region can be expanded as a power series of the order parameter. The divergent specific heat of the Onsager exact solution for the two-dimensional Ising model precludes an expansion whose coefficients are analytic functions of temperature. In the 1960s, there appeared a number of (weaker) scaling hypotheses, based on some general assumptions about the form of the thermodynamic potentials. Although these scaling hypotheses do not lead to a microscopic treatment of critical phenomena, they do provide a way of going beyond the phenomenological equations of van der Waals and Curie—Weiss. The microscopic justification of these ideas, as well as a real possibility of calculating values for the critical exponents to compare with experimental data and theoretical predictions, were provided by the advent of the modern renormalization-group techniques.