Statistical Physics: Phase Transitions and Simulations
Project III with Dr. Nabil Iqbal (20212022)
Project III with Dr. Nabil Iqbal (20212022)
The field of statistical mechanics describes how many small microscopic constituents (such as atoms, or spins on a lattice) conspire to give macroscopic behavior (such as the wetness of water, or magnetism). Some of the most dramatic manifestations of this macroscopic behavior are phase transitions, where the longdistance behavior of the system dramatically changes at a particular point. In this project we will develop basic ideas of statistical physics in a handson manner through computer simulation using Monte Carlo methods.
In particular, we will build our own simulations of simple spin systems such as the Ising model and understand how ideas such as symmetries, spontaneous symmetry breaking, and phase transitions can be understood both theoretically from statistical physics and numerically from computer simulations.
This is a snapshot of a simulation of the 2d Ising model. Click on it to find an interactive version that you can yell at.
Students will then specialize to a more specific topic, possibly (but not at all limited to)

Critical behavior at phase transitions: remarkably, many very different systems behave in a similar manner close to a phase transition; this observation  called universality  has farreaching consequences in very differentseeming areas of science such as quantum field theory and the renormalization group. We can study this in simple numerical models.

Percolation: consider filling a surface randomly with tiles which are either black or white. How big is the largest black cluster? There is a surprisingly rich theory associated with this question of percolation, which again can be probed through numerical simulations.

Gauge theories on the lattice: gauge theories can be understood as systems which describe extended objects, which wiggle about when heated. They are very important for elementary particle physics, and certain toy versions of them have important connections to statistical physics, which we will explore.
Given the rich variety of systems which can be simulated, there are many different options for things to study; students will be expected to construct their own simulations and explain their findings in the framework of statistical physics.
Prerequisites:

MATH2071 Mathematical Physics II.

Programming will form a major component of this project, so you should be comfortable writing code in Python (or your language of choice).
Resources:
There are some scattered links to Wikipedia and popular science articles in the description above. Some textbooks that we will follow include:

David Tong's lectures on statistical physics give a fantastic overview to the field in general.

Monte Carlo methods in statistical physics, Newman and Barkema (an introduction to the numerical methods we will use; the first four chapters are online here.)
Many more detailed and specialized readings will be provided in due course.