Statistisk mekanik

Aim

The course aims to cover the core notions of statistical physics for anyone interested in theoretical physics. The main focus is on statistical properties of interacting systems. The course introduces and discusses the notions of order parameters, spontaneously broken symmetries, Landau theory of phase transitions, effective models and basic aspects of renormalization group methods. The discussed methods are of cornerstone importance in various branches of physics including classical and quantum condensed matter physics, biophysics, neural networks, econophysics and high-energy physics.

Content

The main topics of the course are the following:

• core statistical models: Izing, XY, Heisenberg
• phase transitions and critical phenomena
• order parameters, spontaneous symmetry breaking
• mean-field approach and Landau theory of phase transitions
• scaling
• renormalization group, universality
• applications: superfluids, liquid crystals, superconductors, polymers.

Teachers

Egor Babaev (lecturer) babaevegor@gmail.com

Daniel Weston (substitute lecturer for first few lectures) weston@kth.se

Per Moosavi (teaching assistant) pmoosavi@kth.se

Literature

Course book: Equilibrium Statistical Physics (3rd edition) by Michael Plischke and Birger Bergersen

Supplementary reading: We will try to follow the course book closely. However, if you would like additional reading the following additional literature is recommended:

• Statistical Mechanics in a Nutshell by Luca Peliti
• Statistical Mechanics (3rd edition) by R K Pathria (very clear description of core concepts, very detailed step-by-step calculations, but less discussion of modern topics)
• Statistical Physics, Part 1 (3rd edition) (Course of theoretical physics, volume 5) by L D Landau and E M Lifshitz (one of the best books on statistical physics ever, very deep, but is harder to read, requires more time to get to core concepts)
• Statistical Mechanics: Entropy, Order Parameters and Complexity (Oxford Master Series in Physics) by James P Sethna  (very modern book, but of the "do it yourself" type: to really learn from it requires to do large part of exercises there)
• Fundamentals of Statistical and Thermal Physics by Frederick Reif (although an undergraduate book, it provides clear discussion of many relevant points)
• Lectures on Phase Transitions and the Renormalization Group by Nigel Goldenfeld

Lectures

Preliminary plan:

1. Thermodynamics and statistical ensembles, brief review (1-2)
2. Magnetic systems, mean-field theory of Ising model (3.1)
3. Bragg-Williams approximation, critical behaviour (3.2-3.5)
4. Exact solution of Ising chain (3.6)
5. Landau theory of phase transitions (3.7-3.8)
6. Fluctuations in Landau theory (3.10)
7. Applications of mean-field theory (4.1 4.4)
8. Scaling (6.3 6.5)
9. Renormalization group theory (7.1-7.2)
10. Cumulant method (7.4)
11. ϵ-expansion (7.5-7.7: I will try to cover this if time permits; it was not part of the course earlier.) 
12. Percolation (13.2)
13. Percolation (13.2)

Numbers in parentheses refer to sections of the course book.

Tutorials

Preliminary plan for problems to consider:

1. 1.4, 1.3, Clausius-Clapeyron (1.2, 1.9, 1.10, 1.11)
2. 2.3, 2.4, 2.6 (2.1, 2.2, 2.7)
3. 2.5, 2.9, 2.12
4. 3.1, 3.4, MFT for d-dim. AF Ising model (3.2, 3.3)
5. 3.6, 3.14 (3.15)
6. 3.9, 3.10
7. Fluctuations in Landau-Ginzburg theory
8. VdW eq., Critical exponents of a VdW fluid
9. 4.1, 4.7 (4.2, 4.3)
10. 6.5, Scaling laws and critical exponents from equation of state
11. Cumulant expansion RG (7.1, 7.2)
12. Bond percolation RG, 13.4, Percolation on Bethe lattice
13. 7.5

The problems in parentheses are additional problems: likely there will not be time to solve these during the tutorials; instead they are suggested to be worked out at home.

Examination

Old exams and midterms

Collection of equations allowed on the exam