**Statistical Physics 2023**

### Statistical Physics (MATH327), Spring Term 2023

Stochastic (probabilistic) processes provide often-outstanding mathematical descriptions of systems within the domain of statistical physics — one of the central pillars of modern physics. This module 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, thermodynamic cycles, and phase transitions. In particular, numerical computer programming will be used to investigate diffusion and transport in terms of stochastic processes.

### Learning outcomes

Upon completing this module, students are able to:

- Demonstrate understanding of the micro-canonical, canonical and grand-canonical ensembles, their relation and derived concepts such as entropy, temperature and chemical potential.
- Understand the derivation of the equation of state for non-interacting classical and quantum gases.
- Program numerical computer simulations to analyze 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.

### Background and logistics

This module is intended for third- and fourth-year undergraduates as well as students in the one-year taught MSc programme. Familiarity with quantum mechanics or computer programming is **not** assumed. All necessary information on these topics will be provided. All resources for the module will be gathered at its Canvas site.

These gapped lecture notes are the main learning resource. There is also a python programming demo illustrating all the tools needed for the computer project. All of these are kept under version control at GitHub, providing an easy way to monitor any extensions, corrections or other improvements. Reports of any issues, and pull requests to address them, are also welcome.

We will usually meet at 9:00–11:00 on Mondays, 16:00–17:00 on Tuesdays, and 10:00–11:00 on Thursdays. During weeks 2, 3 and 4 (starting 6, 13 and 20 February) we will instead meet at 9:00–11:00 on Mondays, and at 13:00–15:00 on Tuesdays for a computer lab. I will hold office hours in Room 123 of the Theoretical Physics Wing (and online) after each class meeting. You can also make an appointment to meet me at other times, or reach me by email.

### How to get the most out of this module

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, work through tutorial activities and solve the homework problems yourself (though I encourage you to discuss your work with each other). Sample exams are also available. Come to class and ask questions, even questions you think you're supposed to know the answer to. Should you ask me questions about assignments, I will have you explain to me what you have tried so far, to see what problems or potential next steps can be identified.

### Additional resources

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*(2021) - C. Kittel and H. Kroemer,
*Thermal Physics*(1980) - F. Reif,
*Fundamentals of Statistical and Thermal Physics*(1965)

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

- R. K. Pathria,
*Statistical Mechanics*(1996) - Sidney Redner,
*A Guide to First-Passage Processes*(2007) - Pavel L. Krapivsky, Sidney Redner and Eli Ben-Naim,
*A Kinetic View of Statistical Physics*(2010) - Kerson Huang,
*Statistical Mechanics*(1987) - Andreas Wipf,
*Statistical Approach to Quantum Field Theory*(2021) - Weinan E, Tiejun Li and Eric Vanden-Eijnden,
*Applied Stochastic Analysis*(2019) - Michael Plischke and Birger Bergersen,
*Equilibrium Statistical Physics*(2005) - L. D. Landau and E. M. Lifshitz,
*Statistical Physics, Part 1*(1980)

Finally, this general book about learning provides useful information about what strategies are most effective, for example retrieval practice compared to re-reading lecture notes or re-watching videos:

- Peter C. Brown, Henry L. Roediger III and Mark A. McDaniel,
*Make it Stick: The Science of Successful Learning*(first edition, 2014). A short summary video is also available

### Schedule

**Week 1**

**30 January:** Big-picture overview; Logistics; Probability foundations

**31 January:** Law of large numbers; Probability distributions; Central limit theorem

**2 February (tutorial):** Central limit theorem activity

**Week 2**

**6 February:** Random walks; Law of diffusion...

**7 February (computer lab):** Pseudo-random numbers; Inverse transform sampling...

**Week 3**

**13 February:** ...

**14 February (computer lab):** ...

**Week 4**

**20 February:** ...

**21 February (computer lab):** ...

**Week 5**

**27 February:** ...

**28 February:** ...

**2 March (tutorial):** ...

Last modified 31 January 2023