Project III with Dr. Nabil Iqbal and Dr. Stefano Cremonesi (2023-2024)
Project III with Dr. Nabil Iqbal (2021-2022)
The field of statistical mechanics describes how many small microscopic constituents (such as molecules bouncing around in a box, or spins on a lattice) conspire to give macroscopic behavior (such as the wetness of water, the fact that a hot object radiates with a calculable frequency spectrum, or magnetism). The fact that we can understand the collective behavior of 10^23 molecules (despite not understanding any individual one of them all that well) is an amazing triumph of 20th century science.
In this project, in the first term we will learn the basic principles of statistical mechanics: the microcanonical and canonical ensembles, and quantum statistical mechanics. In the second term each student will specialize, picking a particular phenomenon of interest to understand.
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.
These topics include, but are not at all limited to:
Superconductivity and superfluidity: when many metals are cooled down to very low temperatures, they suddenly enter a new phase of matter where the resistivity is strictly zero; this is a superconductor. A related phenomena is that of superfluidity, a low temperature phase of matter with (in a sense) dissipationless fluid flows. These counterintuitive behaviors can be understood from basic principles of statistical mechanics and the breaking of symmetries.
Lattice and effective models of spontaneous symmetry breaking: there are simple models (such as the Ising model above) which capture very rich physics -- e.g. that of phase transitions in magnetism -- starting from a simple microscopic description. Some of them can even be solved exactly.
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 far-reaching consequences in very different-seeming areas of science such as quantum field theory and the renormalization group.
Some of these projects are open to numerical simulation using Monte Carlo methods if students are so inclined, but this is certainly not necessary. If this forms a major part of your motivation for this project please contact the supervisors Nabil Iqbal and Stefano Cremonesi ahead of time.
MATH2071 Mathematical Physics II.
Some textbooks that we might follow include:
David Tong's lectures on statistical physics give a fantastic overview to the field in general.
Mehran Kardar's Statistical Physics of Particles.
Reif's Thermal and Statistical Physics.
Many more detailed and specialized readings will be provided in due course.