More Is Different: Statistical Mechanics, Thermodynamics, and All That (MATH327), Spring Term 2025
"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. 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 meet at 13:00–14:00 on Mondays, 11:00–13:00 on Wednesdays, and 10:00–11:00 on Thursdays. The tutorials in weeks 2, 3 and 4 (on 6, 13 and 20 February) will be computer lab sessions. I will hold office hours in Room 123 of the Theoretical Physics Wing (and online) after the Monday and Thursday class meetings. 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, entropy bounds, Stirling's formula, the mixing entropy, the Otto cycle, the Einstein solid, and phonons), extra practice problems (on gaussian integrals, random walks, and diffusion), 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)
- 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
29 January: Probability foundations; Law of large numbers; Probability distributions; Central limit theorem; Random walks
30 January (tutorial): Probabilities and central limit theorem
Week 2
3 February: Random walks; Law of diffusion
5 February: Law of diffusion; Statistical ensembles; Micro-canonical ensemble; Thermodynamic equilibrium
6 February (computer lab): Pseudo-random numbers; Inverse transform sampling; Random walks; Polynomial fitting
Week 3
10 February: Micro-canonical thermodynamic equilibrium; Entropy; Extensivity
12 February: Second law of thermodynamics; Generalized thermodynamic equilibrium; Micro-canonical temperature and heat exchange
13 February (computer lab): Cauchy--Lorentz distribution; Anomalous diffusion
Week 4
17 February: Canonical ensemble; Replica trick; Occupation numbers
19 February: Partition function and Gibbs distribution; Heat capacity; Helmholtz free energy; Physical effects of information content
20 February (computer lab): Computer assignment wrap-up
Week 5
24 February: Physical effects of information content (distinguishable vs. indistinguishable spins)
26 February: Classical ideal gas; Regularization; Distinguishable vs. indistinguishable partition functions, energies and entropies; Mixing
27 February (tutorial): Entropy bounds and Stirling's formula
Week 6
3 March: Mixing; Mixing entropy; Reversibility and irreversibility
5 March: Pressure; Ideal gas law; Work and heat; Thermodynamic cycles; PV diagrams, isotherms and adiabats
6 March (tutorial): Review entropy bounds and Stirling's formula; Mixing entropy
Week 7
10 March: Carnot cycle; Heat engine efficiency
12 March: Heat engine efficiency; Grand-canonical ensemble; Chemical potential; Grand-canonical partition function
13 March (tutorial): Review mixing entropy and Maxwell's demon; Otto cycle
Week 8
17 March: Grand-canonical potential; Generalized thermodynamic identity; Quantized energy levels
19 March: Classical Maxwell–Boltzmann statistics; Quantum statistics: Bose–Einstein and Fermi–Dirac; The high-temperature classical limit
20 March (tutorial): Review computer assignment; Review Otto cycle; Diesel cycle; Einstein solid
Week 9
24 March: Ideal quantum gases; Ultra-relativistic photon energies, wavelengths and frequencies; Photon gas internal energy density; Planck spectrum
26 March: Planck spectrum vs. Rayleigh–Jeans ultraviolet catastrophe; Solar radiation and cosmic microwave background; Photon gas internal energy, particle number, radiation pressure and equation of state; Non-relativistic fermion gas; Low-temperature Fermi function
27 March (tutorial): Photon polarizations; Dark matter and the cosmic microwave background; Review Einstein solid heat capacity vs. experiment; Phonon gas (Debye model) and electron gas
Week 10
31 March: Fermi energy; Fermion gas internal energy and chemical potential; Degeneracy pressure
2 April: White dwarf stars and Type-Ia supernovas; Ultra-relativistic neutrino gas equation of state; Density of states; Sommerfeld expansion for low-temperature fermion gas
3 April (tutorial): Review homework; Review phonon gas (Debye model); Review electron gas (Sommerfeld expansion)
Last modified 3 April 2025
