Not running in 2023/24
Term(s): Term 2
Status for Mathematics students: List C
Commitment: 30 Lectures
Assessment: 100% by 3 hour examination
Formal registration prerequisites: None
Assumed knowledge: Basic probability theory and some combinatorics:
- Notions of events and their probability
- Conditional probabilities
- Law of large numbers
- Random variables
- Joint distributions
- Independence of random variables
- Moment generating functions
- Law of large numbers
- Basic counting
- Binomial and multinomial theorems
- Generating functions
- Basics of graph theory
- Notion of Markov process
Useful background: In addition, some notions of measure theory can be useful indeed:
- Fatou's lemma
- Monotone and dominated convergence theorems
- Fubini's theorem
- Riesz representation theory
Synergies: The models of statistical mechanics provide excellent illustrations for the following module:
Content: Statistical mechanics describes physical systems with a huge number of particles.
In physics, the goal is to describe macroscopic phenomena in terms of microscopic models and to give meaning to notions such as temperature or entropy. Mathematically, it can be viewed as the study of random variables with spatial dependence. Models of statistical mechanics form the background for recent advances in probability theory and stochastic analysis, such as Schramm-Loewner evolutions and the theory of regularity structures. So, they form an important background for understanding these topics of modern mathematics.
This module is aimed at giving a thorough mathematical introduction to equilibrium statistical mechanics, mainly focusing on concrete models of lattice spin systems. Topics will include:
- The Curie-Weiss model (thermodynamic limit and critical exponents).
- The Ising model (van Hove theorem, infinite-volume Gibbs states, correlation inequalities, phase diagram, Peierls' argument, Lee-Yang theorem, Kramers-Wannier duality).
- The Gaussian free field (infinite-volume Gibbs states, random walk representation).
- Models with continuous symmetry (Mermin-Wagner theorem).
Time permitting, we will be discussing some additional topics based on the interests of the students. Here are some possible topics:
- Exactly solvable models (dimer models and random tiling, Onsager's solution of the 2d Ising model, six vertex model and Yang-Baxter equation, etc).
- More on quantum spin systems (XY model, Heisenberg model, quantum Ising model, etc).
- Disordered systems (spin glasses, random field Ising model, etc).
Aims: To familiarise students with statistical mechanics models, phase transitions, and critical behaviour.
Objectives: By the end of the module students should be able to:
- Apply basic ideas of phase transitions and critical behaviour to lattice systems of statistical mechanics.
- Understand how large complex systems at equilibrium can be described from microscopic rules.
- Have understood basic ideas of phase transitions and critical behaviour in several concrete examples.
Books: We will mainly follow Chapters 2, 3, 8, 9 of the new introductory textbook:
Sacha Friedli and Yvan Velenik, Equilibrium Statistical Mechanics of Classical Lattice Systems: a Concrete Introduction. Available at: http://www.unige.ch/math/folks/velenik/smbook/index.html
Additional topics will be accompanied by lecture notes.
Interested students can also look into:
David Ruelle, Statistical Mechanics: Rigorous Results, World Scientific, 1999.
James Sethna: Statistical Mechanics: Entropy, Order Parameters and Complexity, Oxford Master Series in Physics, 2006.