Landau theory and the statics and dynamics of phase transitions

This course is part of the elective programme of the Applied Physics Masters Programme in the Graduate School (and is provided by the department of Applied Physics).

The Landau theory of phase transitions is a general framework based on the notion that near criticality, we encounter universal physics governed by the basic symmetries the system. We treat order parameters, the role of symmetries, saddle point approximations and various types of dynamics. We apply the concepts to magnets and (liquid) crystals.

The course consists of frontal (blackboard) lecturing, alternated with supervised self-study sessions in which we work on representative assignments. For some of these, it is useful to have Mathematica installed as some of the worked problems are presented as notebooks. In practice, many of the 4 hour blocks will be composed of 2 hours of lectures and 2 hours of guided study and/or working on assignments, but this division is flexible and varies throughout the course.

The book Lectures on Phase Transitions and the Renormalization Group by Nigel Goldenfeld (Frontiers in Physics series) covers most of the materials, and is supplemented by notes or literature references from the lecturers.

At the end of this course,

1. the student can explain and apply a number of basic concepts from thermodynamics, kinetic theory and statistical mechanics: thermal fluctuations, entropy, free energy, phase space, equilibrium, order parameters, correlation functions, and phase transitions;
2. given a physical system undergoing a phase transition, the student can determine the appropriate order parameter and, based on its basic symmetries, construct a phenomenological Landau theory (homogeneous and inhomogeneous);
3. given a physical system undergoing a phase transition, the student can compute the mean-field critical exponents for the temperature and field dependence of order parameters and correlation functions;
4. the student is comfortably familiar with advanced theoretical-physical/mathematical techniques such as Fourier transforming, tensor notation, and functional derivation and integration;
5. the student is able to characterize dynamical critical phenomena both in mean field theory and beyond (Gaussian fluctuation order) in terms of correlation and response functions;
6. the student is able to understand, apply and explain advanced concepts such as the breakdown of MF, critical slowing down, and the dimension-dependence of critical phenomena: the breakdown of MF theory, the Mermin-Wagner theorem, upper- and lower critical dimensions and the anomalous properties of the XY model in 2D (Kosterlitz-Thouless phase transition)

3MN020 - Biomolecules and soft matter 3FFX0 - Statistical physics

• Review, handwritten lecture notes and tutorial articles - during class
• Lectures on Phase Transitions and the Renormalization Group
• Mathematica 9 / 10 available via TU/e software website. http://www.wolfram.com/mathematica/features/system-requirements.html (Wolfram) Edition 10/11