Scaling, universality and renormalization group theory.

*(English)*Zbl 0589.76004
Critical phenomena, Proc. Summer Sch., Stellenbosch/South Afr. 1982, Lect. Notes Phys. 186, 1-139 (1983).

[For the entire collection see Zbl 0516.00028.]

These lectures are mainly concerned with the discussion of some rather subtle ideas which underlie renormalization group theory and its application to critical phenomena. First a phenomenological introduction to the more important and interesting aspects of critical phenomena in magnetic and fluid systems is given. The concepts of scaling and critical exponents are explained and the idea of universality is emphasized.

Following this various microscopic models are reviewed and certain series expansion methods which are applicable to critical phenomena are considered. From the statistical mechanics of these models some introductory concepts of renormalization group theory are derived. Then the general renormalization group ideas are described. They are essentially topological in nature, the renormalization group being considered as a group of transformations in a manifold of Hamiltonians. Finally some of the practical successes of the theory are discussed. These are based on the so called epsilon expansions which are generated in terms of the parameter \(\epsilon =4-d\), where d is the spatial dimensionality of the physical system.

The attitude adopted in these lectures is that the task of theory is not in the first place to be able to calculate the results of any experiment one can do, but rather to understand the universal aspects of the natural world. A major step may consist in finding a way of looking at things, a language for thinking about things, which need not necessarily be a computational scheme. This aspect is believed to be underplayed in a number of articles and books on renormalization group theory.

These lectures are mainly concerned with the discussion of some rather subtle ideas which underlie renormalization group theory and its application to critical phenomena. First a phenomenological introduction to the more important and interesting aspects of critical phenomena in magnetic and fluid systems is given. The concepts of scaling and critical exponents are explained and the idea of universality is emphasized.

Following this various microscopic models are reviewed and certain series expansion methods which are applicable to critical phenomena are considered. From the statistical mechanics of these models some introductory concepts of renormalization group theory are derived. Then the general renormalization group ideas are described. They are essentially topological in nature, the renormalization group being considered as a group of transformations in a manifold of Hamiltonians. Finally some of the practical successes of the theory are discussed. These are based on the so called epsilon expansions which are generated in terms of the parameter \(\epsilon =4-d\), where d is the spatial dimensionality of the physical system.

The attitude adopted in these lectures is that the task of theory is not in the first place to be able to calculate the results of any experiment one can do, but rather to understand the universal aspects of the natural world. A major step may consist in finding a way of looking at things, a language for thinking about things, which need not necessarily be a computational scheme. This aspect is believed to be underplayed in a number of articles and books on renormalization group theory.

Reviewer: U.Uhlhorn

##### MSC:

76A02 | Foundations of fluid mechanics |

76E99 | Hydrodynamic stability |

76T99 | Multiphase and multicomponent flows |

82B05 | Classical equilibrium statistical mechanics (general) |

80A05 | Foundations of thermodynamics and heat transfer |