Melville S. Green

The renormalization group and the universality of critical exponents

A class of renormalization groups characterized as the general Gaussian renormalization group is defined. It includes the sharp cutoff renormalization group of Wilson, the linear renormalization group of Bell and Wilson, and the incomplete‐integration renormalization group of Wilson. The renormalization group approach to critical phenomena and the expansion around four dimensions is reviewed. The fixed point of the general Gaussian renormalization group is determined to order e2, where e=4−d (d is the dimensionality of the system), and the critical exponents are determined to the same order. The main conclusion is that the exponents do not depend on the cutoff functions of the renormalization group.

A Van Der Waals Fixed Point

In recent years, the interest of both experimentalists and theoreticians in phase transitions has been centered on the non-analytic behavior of thermodynamic and other properties in the neighborhood of a critical point or continuous phase transition [1]. In pursuing this interest it has been easy to lose sight of the fact that global features of many phase transitions are satisfactorily described by a physical picture which in the context of liquid-vapor phase transitions is called the Van der Waals theory, and in the context of magnetism, the mean-field theory. This physical picture is exact for the weak long range force model in which the forces are infinite ranged but weak in such a way that the product of the potential and the volume in which the force operates is finite [2]. Recently, in the course of an attempt to apply the scaling theory of critical phenomena and its extensions to the liquid-vapor transition in SF6 by M. Ley-Koo and the author [3], it became apparent that the Van der Waals theory may be more than a qualitative guide but may rather form the basis for a quantitative representation of the data to within a few percent of the critical density. It would appear that an adequate theory of the liquid-vapor phase transition requires a model with short range forces near the critical point and the weak long range force model far from the critical point.