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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:

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

In addition to the gapped lecture notes, there are tutorial activities on the central limit theorem, entropy bounds, Stirling's formula, the mixing entropy, the Otto cycle, the Einstein and Debye models of solids, and the Ising model on various lattice structures, along with a computer project and homework assignments (one and two).

Potentially useful statistical physics references at roughly the level of this module include:

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

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:


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 & central limit theorem; Statistical ensembles;
7 February (computer lab): Pseudo-random numbers; Inverse transform sampling; Random walks

Week 3
13 February: Micro-canonical ensemble; Thermodynamic equilibrium; Entropy; Second law of thermodynamics
14 February (computer lab): Cauchy--Lorentz distribution; Anomalous diffusion.

Week 4
20 February: Temperature; Canonical ensemble; Partition function and Gibbs distribution; Heat capacity; Helmholtz free energy (supplement on micro-canonical spin system temperature)
21 February (computer lab): Project wrap-up

Week 5
27 February: Physical effects of information content (distinguishable vs. indistinguishable spins); Classical ideal gas; Regularization of continuous energies
28 February: Ideal gas regularization; Distinguishable vs. indistinguishable partition functions, energies and entropies
2 March (tutorial): Entropy bounds and Stirling's formula activities

Week 6
6 March: Ideal gas energies and entropies; Mixing entropy; Pressure; Ideal gas law and equations of state; Work
7 March: Work and heat; First law of thermodynamics; PV diagrams
9 March (tutorial): Comments on computer project, entropy bounds and Stirling's formula; Mixing entropy activity

Week 7
13 March: PV diagrams, isotherms and adiabats; Carnot cycle; Heat engine efficiency; Grand-canonical ensemble
14 March: Grand-canonical ensemble; Chemical potential; Grand-canonical partition function
16 March (tutorial): Comments on mixing entropy and Maxwell's demon; Otto cycle activity

Week 8
20 March: Grand-canonical potential; Generalized thermodynamic identity; Maxwell–Boltzmann statistics; Quantum statistics; Bosons vs. fermions; Bose–Einstein statistics
21 March: Fermi–Dirac statistics; Occupation number expectation values; The high-temperature classical limit
23 March (tutorial): Comments on homework; Otto cycle and Diesel cycle; Einstein solid activity

Week 9
17 April: Post-break review; Grand-canonical ideal quantum gases; Ultra-relativistic photon energies, wavelengths and frequencies; Photon gas internal energy density; Planck spectrum vs. Rayleigh–Jeans ultraviolet catastrophe; Solar radiation and cosmic microwave background
18 April: Radiation pressure; Photon gas equation of state; Non-relativistic quantum fermion gas
20 April (tutorial): Photon polarization; Dark matter in the cosmic microwave background; Einstein solid activity

Week 10
24 April: Low-temperature non-relativistic fermion gas and Fermi function; Fermi energy; Internal energy; Positive chemical potential; Degeneracy pressure; White dwarf stars and Type-Ia supernovas; Motivate interacting systems
25 April: Phase transitions; Interacting systems; Lattice structures; Ising model
27 April (tutorial): Comments on Einstein solid heat capacity vs. experiment; Phonons; Debye solid activity

Week 11
2 May: Ising model magnetization in ordered and disordered phases; Order parameters and phase transitions; Magnetization as Ising model order parameter
4 May (tutorial): Comments on Debye solid and electron gas heat capacities; Lattice structure activity and complete graph

Week 12
Supplement: Ultra-relativistic neutrino gas equation of state
9 May: Ising model mean-field approximation; Mean-field self-consistency condition
11 May (tutorial): Comments on lattice structures; Anti-ferromagnetism and frustration; Exam prep and module recap
Supplement: Ising model exact solution in one dimension; Ising model exact critical temperature in two dimensions

Week 13
15 May: Mean-field critical temperature, critical exponent and reliability in various dimensions; Monte Carlo importance sampling and MRRTT algorithm
Supplement: Density of states; Sommerfeld expansion for low-temperature fermion gas: chemical potential and heat capacity; Classical limit

Last modified 20 May 2023

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