**Statistical Physics 2023**

### 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:

- Demonstrate understanding of the micro-canonical, canonical and grand-canonical ensembles, their relation and derived concepts such as entropy, temperature and chemical potential.
- Understand the derivation of the equation of state for non-interacting classical and quantum gases.
- Program numerical computer simulations to analyze 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.

### 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:

- David Tong,
*Lectures on Statistical Physics*(2012) - Daniel V. Schroeder,
*An Introduction to Thermal Physics*(2021) - C. Kittel and H. Kroemer,
*Thermal Physics*(1980) - F. Reif,
*Fundamentals of Statistical and Thermal Physics*(1965)

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

- 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) - 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*(first edition, 2014). A short summary video is also available

### Schedule

**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