An introduction to the Ginzburg-Landau theory of phase transitions and nonequilibrium patterns

TL;DR

This paper introduces the Ginzburg-Landau theory as a unified framework for understanding phase transitions and nonequilibrium pattern formation, covering mean-field theory, renormalization, and applications to dynamical systems.

Contribution

It provides a comprehensive overview of applying Ginzburg-Landau theory to both equilibrium critical phenomena and nonequilibrium pattern formation, bridging static and dynamic systems.

Findings

Mean-field theory validity tested and extended below four dimensions.

Renormalization group justifies universality in critical phenomena.

Derived Ginzburg-Landau equations describe pattern formation near instabilities.

Abstract

This paper presents an introduction to phase transitions and critical phenomena on the one hand, and nonequilibrium patterns on the other, using the Ginzburg-Landau theory as a unified language. In the first part, mean-field theory is presented, for both statics and dynamics, and its validity tested self-consistently. As is well known, the mean-field approximation breaks down below four spatial dimensions, where it can be replaced by a scaling phenomenology. The Ginzburg-Landau formalism can then be used to justify the phenomenological theory using the renormalization group, which elucidates the physical and mathematical mechanism for universality. In the second part of the paper it is shown how near pattern forming linear instabilities of dynamical systems, a formally similar Ginzburg-Landau theory can be derived for nonequilibrium macroscopic phenomena. The real and complex Ginzburg-Landau equations thus obtained yield nontrivial solutions of the original dynamical system, valid near the linear instability. Examples of such solutions are plane waves, defects such as dislocations or spirals, and states of temporal or spatiotemporal (extensive) chaos.