More is the Same; Phase Transitions and Mean Field Theories

TL;DR

This paper reviews early theories of phase transitions, focusing on concepts like singularity, order parameter, and mean field theory, to explain how matter supports diverse phases such as ice, water, and vapor.

Contribution

It provides a historical and conceptual overview of phase transition theory, emphasizing the role of mean field approaches and related ideas in understanding matter's phase diversity.

Findings

Clarifies the concept of order parameters in phase transitions.

Explains the role of mean field theory in statistical mechanics.

Describes the nature of first and continuous phase transitions.

Abstract

This paper looks at the early theory of phase transitions. It considers a group of related concepts derived from condensed matter and statistical physics. The key technical ideas here go under the names of "singularity", "order parameter", "mean field theory", and "variational method". In a less technical vein, the question here is how can matter, ordinary matter, support a diversity of forms. We see this diversity each time we observe ice in contact with liquid water or see water vapor, "steam", come up from a pot of heated water. Different phases can be qualitatively different in that walking on ice is well within human capacity, but walking on liquid water is proverbially forbidden to ordinary humans. These differences have been apparent to humankind for millennia, but only brought within the domain of scientific understanding since the 1880s. A phase transition is a change from one behavior to another. A first order phase transition involves a discontinuous jump in a some statistical variable of the system. The discontinuous property is called the order parameter. Each phase transitions has its own order parameter that range over a tremendous variety of physical properties. These properties include the density of a liquid gas transition, the magnetization in a ferromagnet, the size of a connected cluster in a percolation transition, and a condensate wave function in a superfluid or superconductor. A continuous transition occurs when that jump approaches zero. This note is about statistical mechanics and the development of mean field theory as a basis for a partial understanding of this phenomenon.