This lecture will be given by Professor Helias.
This lecture systematically develops the theory of phase transitions in equilibrium and non-equilibrium statistical mechanics with a focus on the renormalization group. We present the development starting from Landau theory, via Widom's scaling hypothesis, to the seminal idea of successive coarse-graining by Kadanoff that finally lead to the formulation of Wilson's renormalization group. We will here for the most parts stay in the realm of statistical physics but at appropriate places also expose the tight link to the renormalizaton in field-theory, including the more modern views based on recent reviews. The role of symmetries for Goldstone modes and Ward-Takahashi identities will be explained. Modern developments, such as the functional renormalization group, will be covered in later chapters on the examples of the Ising model. Both main streams, the vertex expansion and the derivative expansion, will be covered. The last part of the notes provides an entry point into dynamic critical phenomena following the seminal review by Hohenberg & Halperin (1977), and using the Kardar-Parisi-Zhang model as an important application. The lecture notes, in parallel, develop the necessary machinery of field theory formulated in terms of its various generating functionals and presenting also the systematic fluctuation expansion, formulated with the help of the effective action. We here employ toy models to illustrate the concepts detached from any mathematical complications and then apply them to real-world applications in the exercises. We dedicate specific chapters to the development of the diagrammatic formulation of these computational tools, both for equilibrium statistical mechanics, starting with the partition function as the fundamental generating functional, but also for the dynamical formulation, following the Martin-Siggia-Rose-DeDominicis-Janssen formulation.
Problems at the end of each section are intended to familiarize with the concepts. The material presented here is based in large parts on the books by Nigel Goldenfeld, Daniel Amit, A.N. Vasiliev, and Jean Zinn-Justin. In some parts and for many exercises also original literature was used, such as Landau 1937, Wilson 1975, Wilson & Kogut 1974, Hohenberg & Halperin 1977, Kardar et al. 1984, Canet et al. 2012, Delamotte 2012. We cite these works at appropriate places to provide entry points for further reading.
Statistical physics (bachelor)
- Ability to analyze systems with phase transitions
- Ability to employ diagrammatic methods to treat problems perturbatively
- Obtain understading of critical phenomena and their treatment by the renormalization group
Mon 8.30am - 10am
|Online||12.04.2021 - 19.07.2021|
|Wed 8.30am - 10am||Online||14.04.2021 - 21.07.2021|
|Wed 10.30am - 12pm||Online||14.04.2021 - 21.07.2021|