Critical phenomena and singularities in two complex variables

Koo-Chul Lee

Olle Stormark, Clas Blomberg

By an extension of the Yang-Lee picture of phase transitions to the case of two complex variables, we have been able to study the properties of a ferromagnetic system in the vicinity of the critical point. A linear approximation is used, which yields a satisfactory result close to that point. Especially, we see how a critical point and a phase boundary can arise from statistical mechanical laws. It is shown that scaling laws are generally valid, although there may be special cases when they are not.

M. P. Kawatra, J. I. Budnick

Michael Ellis Fisher

The droplet or cluster theory of condensation is reviewed critically and extended. It is shown to imply that the condensation point is marked by a singularity of the thermodynamic potential as conjectured by Mayer. The singularity turns out to be an essential singularity at which all derivatives of the thermodynamic variables remain finite. The theory also yields an understanding of the uniqueness of the critical point (in contrast to an extended critical region or Derby-hat type of behaviour) and leads to relations between the various critical point singularities.A one-dimensional model is described with a Hamiltonian containing short-range many-body potentials. The exact solution of the model is sketched and shown to exhibit condensation and critical phenomena for suitable (fixed) potentials. The analysis confirms the conclusions of the cluster theory and thereby lends support to the validity of its underlying assumptions.

John F. Nagle

Akira Shudo, Kensuke S. Ikeda

It is a strict requirement that physical observables should be real-valued; the position and momentum of a particle obeying the Newtonian equations of motion are both real, and any dynamical variables derived from them are also real. The wavefunction in quantum mechanics may be complex-valued, but observables obtained averaged over the square modulus of the wavefunction turn out to be real. Occasionally, one examines the nature of singularities in the complex plane in order to judge whether the system under consideration is completely integrable or not, but in almost all cases complex variables appear only implicitly in the description of physical phenomena. Indeed, there is no serious demand or even motivation to investigate dynamics whose variables are complex.