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:
- Demonstrate understanding of the micro-canonical, canonical and grand-canonical ensembles, their relation and the derived concepts of entropy, temperature and particle number density.
- Understand the derivation of the equation of state for non-interacting classical and quantum gases.
- Demonstrate numerical skills to understand 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.
Resources
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:
- Charles Kittel and Herbert Kroemer, Thermal Physics (second edition, 1980)
- David Tong, Lectures on Statistical Physics (2012)
- Daniel V. Schroeder, An Introduction to Thermal Physics (first edition, 1999)
- F. Reif, Fundamentals of Statistical and Thermal Physics (first edition, 1965)
There are also many textbooks at a higher level than this module, including the following:
- R. K. Pathria, Statistical Mechanics (second edition, 1996)
- Sidney Redner, A Guide to First-Passage Processes (first edition, 2007)
- Pavel L. Krapivsky, Sidney Redner and Eli Ben-Naim, A Kinetic View of Statistical Physics (first edition, 2010)
- Kerson Huang, Statistical Mechanics (second edition, 1987)
- Michael Plischke and Birger Bergersen, Equilibrium Statistical Physics (third edition, 2005)
- L. D. Landau and E. M. Lifshitz, Statistical Physics, Part 1 (third edition, 1980)
Schedule
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
