More is Different — The Mathematics of Equilibrium Statistical Mechanics (4th year project)
Philip W. Anderson (Nobel Prize in Physics, 1977) famously wrote the ‘more is different’ to summarise how large collections of interacting particles can have behaviours that are not at all straightforward to predict from the microscopic behaviour of single (or small numbers of) particles. New phenomena emerge when large numbers of things interact. This project is about the mathematically precise formulation of these ideas, particularly as they occur in equilibrium statistical mechanics. There are a range of mathematical subjects that are used in these investigations, but this project will focus primarily on probabilistic methods.
To give a brief flavour of the type of question that is of interest, consider a closed container filled with water. The water is made up of an enormous number of water molecules — around 10^23 of them. What does the water look like? The answer depends a lot on the temperature and pressure of the container: it might contain ice, liquid, or water vapour. This happens even though the physical laws that govern how the molecules interact are the same at all temperatures and pressures! The transitions between liquid and gas, and liquid and solid, are called phase transitions, and they arise from the collective interactions of the molecules with one another. A precise mathematical understanding of water turns out to be hugely complicated, but for other phase transitions like the Curie point of magnetism we can formulate rather simple and satisfying explanatory mathematical models, e.g., the Ising model and its relatives the O(N) vector models.
This project will start by exploring the Ising model to gain some intuition and to prepare you to explore the variety of interesting mathematics that enter into a rigorous understanding of equilibrium statistical mechanics. Further theoretical topics could include:
Continuous symmetries and phase transitions (Mermin--Wagner theorem; reflection positivity).
Discrete spin systems and their phenomena (Potts models; nematic phases; striping and spotting phenomena).
Lee--Yang theory (absence of phase transitions), theory of Gibbs measures (general mathematical framework).
Continuum particle models (hard spheres; crystallisation conjecture). Potts models and random cluster models.
There is also the possibility of carrying out computations and simulations, e.g.:
Simulation of discrete spin phenomena (striping; nematic phases)
Computation of Lee—Yang zeros and investigation of related phenomena
Simulation of fully-packed models like dimers, and their associated frozen regions.
Prerequisites: Probability II. Discrete Maths / Complex Analysis would be useful, but is not essential.
Resources:
Statistical Mechanics of Lattice Systems by Friedli and Velenik.
Reflection Positivity and Phase Transitions in Lattice Spin Models by Biskup.
Lecture Notes on the Ising and Potts Models on the Hypercubic Lattice by Duminil-Copin.
Further resources may be suggested once the project is underway.
Motivation:
More is Different by Philip Anderson.
Supervisors: Michaelmas term will be supervised by Prof. Tyler Helmuth, and Epiphany term by Prof. Ostap Hryniv.