Statistical physics : theory of phase transitions
Summary
Phase transitions are ubiquitous, from the first instants of the universe to living matter. Despite the vast difference in microscopic details, some features of phase transitions are universal and can be explained by the careful use of statistical mechanics, leading up to the renormalisation group.
Content
1: Introduction and Overview
What is a phase transition?
First-order vs. continuous transitions
Order parameters and broken symmetry
Examples: magnetization, liquid-gas density difference
Notions of criticality and universality
Importance of the thermodynamic limit
2: The Ising Model in 1D
Model definition and transfer matrix solution
Free energy, magnetization, correlation length
No phase transition in 1D with short-range interactions
3: Ising Model in 2D
Phase diagram and symmetry breaking
Numerical and heuristic insights into critical behavior
Notion of long-range order
4: Mean-Field Theory
Variational approach for the Ising model
Critical temperature, spontaneous magnetization
Mean-field critical exponents
Limitations of mean-field theory
5: Correlation Functions, Fluctuations and thermodynamic quantities
Connected correlation functions
Susceptibility and structure factor
Correlation length and scaling near criticality
6: Landau Theory of Continuous Transitions
Free energy expansions in terms of order parameters
Symmetry-based construction
Prediction of second-order behavior and exponents
Role of anisotropy and higher-order terms
7: Ginzburg Criterion and Fluctuations
Ginzburg-Landau field theory
Breakdown of mean-field near lower dimensions
Upper critical dimension
Gaussian approximation
8: Scaling Hypothesis
Homogeneity of the free energy
Scaling forms for thermodynamic observables
Critical exponents and their relations
Widom and hyperscaling relations
9: Renormalization Group (RG): Basics
Coarse-graining and scale transformations
Fixed points, flows, and criticality
Relevant, irrelevant, and marginal operators
Universality classes and dimensionality
10: RG for the Ising Model
Real-space RG: decimation, Migdal-Kadanoff
ε-expansion near 4D
Identification of universality classes
11: First-Order Phase Transitions
Discontinuity in the order parameter
Metastability and nucleation
Maxwell construction
Clausius-Clapeyron relation
Spinodal decomposition and hysteresis
12: Geometrical Critical Phenomena
Percolation: threshold, exponents, mean field-solution, mapping onto Potts models
O(n) model and the limit n->0 : self-avoiding walks
Keywords
Phase transitions
Critical exponents
Universality
Renormalisation group
Learning Prerequisites
Required courses
Basic Statistical Mechanics
Learning Outcomes
By the end of the course, the student must be able to:
- Use mean-field to solve models
- Use computational methods to solve models
- Demonstrate formal derivation
- Identify strengths and weaknesses of different solution methods
- Solve models using the renormalisation group
Assessment methods
Written exam
In the programs
- Semester: Fall
- Exam form: Written (winter session)
- Subject examined: Statistical physics : theory of phase transitions
- Courses: 2 Hour(s) per week x 14 weeks
- Exercises: 2 Hour(s) per week x 14 weeks
- Type: optional
- Semester: Fall
- Exam form: Written (winter session)
- Subject examined: Statistical physics : theory of phase transitions
- Courses: 2 Hour(s) per week x 14 weeks
- Exercises: 2 Hour(s) per week x 14 weeks
- Type: optional
- Semester: Fall
- Exam form: Written (winter session)
- Subject examined: Statistical physics : theory of phase transitions
- Courses: 2 Hour(s) per week x 14 weeks
- Exercises: 2 Hour(s) per week x 14 weeks
- Type: optional
- Semester: Fall
- Exam form: Written (winter session)
- Subject examined: Statistical physics : theory of phase transitions
- Courses: 2 Hour(s) per week x 14 weeks
- Exercises: 2 Hour(s) per week x 14 weeks
- Type: optional