**Statistical Physics 2021**

### Statistical Physics (MATH327), Spring 2021

Probabilistic processes provide often-outstanding mathematical descriptions of systems within the domain of statistical physics. This course introduces the foundations of statistical physics, including the concepts of statistical ensembles, the laws of thermodynamics, and derived quantities such as entropy. These foundations will be applied to investigate diffusion, the behaviour of idealized physical systems such as classical and quantum gases, thermodynamical cycles, and phase transitions. In particular, computer simulations will be used to numerically investigate diffusion and transport in terms of stochastic processes.

### Learning outcome

Upon completing this module, students are able to:

- Demonstrate understanding of the micro-canonical, canonical and grand-canonical ensembles, their relation and the derived concepts of entropy, temperature and particle number density.
- Understand the derivation of the equation of state for non-interacting classical and quantum gases.
- Demonstrate numerical skills to understand diffusion from an underlying stochastic process.
- Know the laws of thermodynamics and demonstrate their application to thermodynamic cycles.
- Be aware of the effects of interactions, including an understanding of the origin of phase transitions.

### Resources

This module is intended for third- and fourth-year undergraduates as well as students in the one-year taught MSc programme. Familiarity with combinatorics and quantum mechanics is **not** assumed. Any necessary information from those disciplines will be provided. All resources for the module will be gathered at its Canvas site.

These gapped lecture notes are the main learning resource. They are kept under version control at GitHub, providing an easy way to monitor any typo fixes or other improvements. Reports of any issues, and pull requests to address them, are also welcome.

My office hours are still to be set. You can also make an appointment to meet with me at other times.

As you know by this point in your studies, the best way to learn mathematics is by doing mathematics. This includes (but is not limited to!) making sure you can fill in the gaps in the lecture notes, complete the tutorial exercise and solve the homework problems (yourself, though I encourage you to discuss these problems with each other). Sample exams are also available to allow additional practice. When you ask me questions about these exercises and problems, I will endeavour to avoid doing this work for you; instead I will have you explain to me what you have tried so far, and will ask leading questions to suggest where I see problems or potential next steps.

In addition to the gapped lecture notes, potentially useful statistical physics references at roughly the level of this module include:

- David Tong,
*Lectures on Statistical Physics*(2012) - Daniel V. Schroeder,
*An Introduction to Thermal Physics*(first edition, 1999) - C. Kittel and H. Kroemer,
*Thermal Physics*(second edition, 1980) - F. Reif,
*Fundamentals of Statistical and Thermal Physics*(first edition, 1965)

There are also many textbooks at a higher level than this module, including the following:

- R. K. Pathria,
*Statistical Mechanics*(second edition, 1996) - Sidney Redner,
*A Guide to First-Passage Processes*(first edition, 2007) - Pavel L. Krapivsky, Sidney Redner and Eli Ben-Naim,
*A Kinetic View of Statistical Physics*(first edition, 2010) - Kerson Huang,
*Statistical Mechanics*(second edition, 1987) - Michael Plischke and Birger Bergersen,
*Equilibrium Statistical Physics*(third edition, 2005) - L. D. Landau and E. M. Lifshitz,
*Statistical Physics, Part 1*(third edition, 1980)

### Schedule

**Week 1 (8–14 February):** Central limit theorem and diffusion

**Week 2 (15–21 February):** Micro-canonical ensemble

**Week 3 (22–28 February):** Canonical ensemble

**Week 4 (1–7 March):** Ideal gases

**Week 5 (8–14 March):** Thermodynamic cycles

**Week 6 (15–21 March):** Grand-canonical ensemble

**Week 7 (12–18 April):** Quantum statistics

**Week 8 (19–25 April):** Quantum gases of bosons

**Week 9 (26 April – 2 May):** Quantum gases of fermions

**Week 10 (3–9 May):** Interacting systems

**Week 11 (10–16 May):** Synthesis and broader applications

Last modified 7 January 2021