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Statistical Physics (MATH327), Spring 2020

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:


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. We will provide the input needed from those disciplines. Resources on Maple and MATLAB are provided through VITAL.

These gapped lecture notes are our main learning resource. Compared to the version of the notes uploaded to VITAL and handed out during the first lecture, this linked pdf file is a living document (last modified 3 May 2020) that contains a large number of typo fixes and other improvements. My office hours are 15:00--16:00 on Tuesdays and 11:00--12:00 on Fridays (following our lectures and tutorials) in room 124 of the theoretical physics wing of the Mathematical Sciences Building. 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 reproduce the derivations and examples presented in lecture, complete the tutorial problems in the lecture notes (only some of which will be discussed during tutorials) and solve the homework problems in the lecture notes (only some of which will be assigned for assessment). A sample exam will also soon be available to allow additional practice. Should you ask me questions about these problems in tutorials and office hours, I will endeavour to avoid telling you how to solve them; 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:

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


Lectures took place on the following Tuesdays (13:00–14:50) and Fridays (9:00–9:50):

28 January: Lecture notes
Course overview; Statistical physics overview; Probability essentials; Law of Large Numbers; Central Limit Theorem; Diffusion in one dimension

31 January: Lecture notes
Diffusion in one dimension

4 February: Lecture notes
Micro-canonical ensemble and First Law of Thermodynamics; Entropy and its properties; Thermodynamic equilibrium; Second Law of Thermodynamics; Temperature

7 February: Lecture notes
Temperature; Heat exchange and energy flow

11 February: Lecture notes
Canonical ensemble; Partition function; Temperature; Heat capacity; Helmholtz Free Energy

14 February: Lecture notes
Observable effects of knowledge/information: Distinguishable degrees of freedom (spin chain)

18 February: Lecture notes
Observable effects of knowledge/information: Indistinguishable degrees of freedom (spin gas); Canonical ensemble application: Classical ideal gas

21 February: Lecture notes
Gibbs 'paradox'; Mixing entropy

25 February: Lecture notes
Definition of pressure; Ideal gas law (equation of state); Work; Heat; Thermodynamical cycles and pV diagrams

28 February: Lecture notes
Carnot cycle and its pV diagram

3 March: Lecture notes
Carnot cycle: pV diagram, work, heat, efficiency; Grand-canonical ensemble; Grand-canonical partition function and entropy

6 March: Lecture notes
Grand-canonical partition function and entropy; Chemical potential

10 March: Lecture notes; Recording
Chemical potential; Grand potential; Quantum gases; Bose gas

13 March: Lecture notes; Recording
More about quantum statistics; Bose gas becomes classical at high temperature

17 March: Lecture notes; Recording
Classical occupation numbers; Fermi gas; Fermi gas becomes classical at high temperature; Photon gas; Planck spectrum; Modelling sunlight and cosmic microwave background

20 March: Lecture notes; Recording
Virtual computer lab: Ordinary diffusion; Pseudo-random numbers; Inverse transform sampling; MATLAB access; MATLAB basics demo

24 March: Lecture notes; Recording
Quantum gas of fermions and its grand-canonical equation of state; Degeneracy pressure; Look ahead to interacting systems

27 March: Slides; Recording
Virtual computer lab: Anomalous diffusion; Cauchy–Lorentz distribution; Generalized diffusion length; MATLAB power-law fitting demo

21 April: Slides; Recording
Interacting systems; Lattice spin system; Ising model; Ordered and disordered phases; Crossover vs. phase transition; Look ahead to mean-field approximation

24 April: Slides; Recording
Virtual computer lab: Code optimization; Correlated data; Review fitting in MATLAB; Higher-dimensional diffusion

28 April: Slides; Recording
Ising model mean-field approximation; Magnetization self-consistency condition; Predicted phase transition vs. exact solution in one dimension

1 May: Slides; Recording
Loose ends: Monte Carlo importance sampling; Universality; Broad applications of interacting statistical systems

5 May: Slides; Recording
Module overview for exam revision

11 May: Slides; Recording
Computer project feedback; Requests regarding exam revision

Last modified 12 May 2020

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