# MA4L2 Statistical Mechanics

Lecturer: Daniel Ueltschi

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:

ST111 Probability (Part A):

• Notions of events and their probability
• Conditional probabilities
• Law of large numbers

ST112 Probability (Part B):

• Random variables
• Joint distributions
• Independence of random variables
• Moment generating functions
• Law of large numbers

MA241 Combinatorics:

• Basic counting
• Binomial and multinomial theorems
• Generating functions
• Basics of graph theory

MA3H2 Markov Processes and Percolation Theory:

• Notion of Markov process

Useful background: In addition, some notions of measure theory can be useful indeed:

MA359 Measure Theory:

• 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 a 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 SLE and the theory of regularity structures. So, they form an important background for understanding these topics of modern mathematics.

The module will give a thorough mathematical introduction to the Ising model and to the gaussian free field on regular graphs, and to the theory of infinite volume Gibbs measures.

Aims: To familiarise students with statistical mechanicsmodels, 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 the theory of infinite volume Gibbs measures
• Understand how large complex systems at equilibrium can be described from microscopic rules
• Have understood basic ideas of phase transitions and critical behaviour in the case of the Ising model and the
gaussian free field; they will have mastered the theory of infinite volume Gibbs measures.

Books: We will mainly follow Chapters 3, 6, 7 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

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.