More Is Different: Statistical Mechanics, Thermodynamics, and All That (MATH327), Spring Term 2026
"More Is Different" is the title of a famous 1972 essay that established the concept of emergent phenomena --- the idea that large, complex physical systems generally can't be understood by extrapolating the properties of small, simple systems. Instead, we have to apply the stochastic (i.e., probabilistic) techniques of statistical mechanics --- one of the central pillars of modern physics, along with quantum mechanics and relativity. While statistical mechanics was originally developed in the context of thermodynamics in the nineteenth century, it is more generally applicable to any large-scale (macroscopic) behaviour that emerges from the microscopic dynamics of many underlying objects --- far more than can possibly be analysed in complete detail. It is intimately connected to quantum field theory, and has been applied to topics from nuclear physics and cosmology to climate science and biophysics, often with outstanding success.
This module covers both foundations and applications of statistical mechanics. Foundational topics include the concepts of statistical ensembles, the laws of thermodynamics, and derived quantities such as entropy. We will apply these foundations to investigate diffusion, the behaviour of idealized physical systems such as classical and quantum gases, thermodynamic cycles, and phase transitions. In particular, we will use numerical computer programming to investigate diffusion in terms of stochastic processes. No prior exposure to quantum mechanics or computer programming is required --- all necessary information on these topics will be provided.
Learning outcomes
Upon completing this module, students are able to:
- Use the central limit theorem to analyse macroscopic diffusion emerging from stochastic microscopic dynamics.
- Numerically analyse macroscopic behaviour emerging from an underlying stochastic process.
- Apply the micro-canonical, canonical, and grand-canonical ensembles to analyse statistical systems subject to the corresponding constraints.
- Use the concepts of work, heat, and the laws of thermodynamics to analyse thermodynamic processes and thermodynamic cycles, determining the efficiency of the latter.
- Derive the equation of state for classical and quantum ideal gases.
- Carry out routine calculations for interacting statistical systems, including the use of order parameters to distinguish phases separated by a phase transition.
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 assignment.
We will meet at 10:00–11:00 on Tuesdays & Wednesdays, and 11:00–13:00 on Thursdays. The tutorials in weeks 2, 3 and 4 (on 4, 11 and 18 February) will be computer lab sessions. 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 problems and solve the homework problems yourself (though I encourage you to discuss your work with each other). Two 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
In addition to the gapped lecture notes, these include tutorial problems (on probabilities), extra practice problems (on gaussian integrals), and assignments (one computer-based) and two traditional homeworks).
Potentially useful resources at roughly the level of this module include:
- David Tong, Lectures on Statistical Physics (2012)
- MIT OpenCourseWare for undergraduate Statistical Physics I (2013) and Statistical Physics II (2005)
- Daniel V. Schroeder, An Introduction to Thermal Physics (2021)
- Harvey Gould and Jan Tobochnik, Statistical and Thermal Physics with Computer Applications (2021)
- Jonathan Allday and Simon Hands, Introduction to Entropy: The Way of the World (2024)
- C. Kittel and H. Kroemer, Thermal Physics (1980)
- F. Reif, Fundamentals of Statistical and Thermal Physics (1965)
There are also many resources at a higher level than this module, including the following:
- MIT OpenCourseWare for postgraduate Statistical Mechanics I (2013) and Statistical Mechanics II (2014)
- 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)
- Sacha Friedli and Yvan Velenik, Statistical Mechanics of Lattice Systems (2018)
- 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 (2014). A short summary video is also available
Schedule
Week 1
27 January: Big-picture overview; Logistics; Probability foundations
28 January (tutorial): Probability foundations and application
29 January: Law of large numbers; Probability distributions; Central limit theorem; Random walks
Week 2
3 February: Random walks; Law of diffusion
4 February (computer lab): Pseudo-random numbers; Inverse transform sampling; Random walks; Polynomial fitting
5 February: Review probability application; Law of diffusion; Statistical ensembles; Micro-canonical ensemble; Thermodynamic equilibrium; Entropy
Last modified 5 February 2026
