Statistical Mechanics
Project III with Dr. Nabil Iqbal and Dr. Stefano Cremonesi (2023-2024)
Project III with Dr. Nabil Iqbal (2021-2022)
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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.
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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.
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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: ​​
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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.
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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.
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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.
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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.
Prerequisites:
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MATH2071 Mathematical Physics II.
Resources:
Some textbooks that we might follow include:
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David Tong's lectures on statistical physics give a fantastic overview to the field in general.
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Mehran Kardar's Statistical Physics of Particles.
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Reif's Thermal and Statistical Physics.
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Many more detailed and specialized readings will be provided in due course.