Neglect of irrelevant variables and corrections to scaling due to nonlinear scaling fields

William Scott McCullough, Purdue University

Abstract

The implications of the renormalization group (RG) theory of scaling for the structure of various thermodynamic functions are explored under the hypothesis that irrelevant variables can be neglected. It is shown that this hypothesis makes specific predictions for the structures of various thermodynamic functions and predicts certain relations among them. These results are discussed and used in two situations; (1) as a phenomenology for real ferromagnetic systems and (2) as a calculational tool for the two-dimensional Ising model. In the first case, the scaling form predicted by the RG when irrelevant variables are neglected is compared with an earlier proposed phenomenology for ferromagnetic systems, and the two approaches are shown to be equivalent. As an example of the ability of this approach to describe a system in an extended region around its critical point, the results are used to fit experimental data for the susceptibility of Nickel in the temperature range $T\sb{c} \leq T \leq 3T\sb{c}$. In the second application, the theory is used as a predictive tool by computing several new terms for the quadratic Ising lattice susceptibility. The structure and coefficients of the predicted terms are also computed by means of a differential approximant analysis of an extended high-temperature series, and the results are found to be in complete agreement with the predictions of the RG theory. To achieve these results, a new method for analyzing functions with logarithmic singularities by means of second-order homogeneous differential approximants has been proposed and used. It is concluded that the hypothesis of neglecting irrelevant variables is useful for extending the region over which a scaling description is applicable for both real and model systems.

Degree

Ph.D.

Advisors

Gartenhaus, Purdue University.

Subject Area

Condensation

COinS