**Statistical Physics 2021**

### Statistical Physics (MATH327), Spring 2021

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 outcomes

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. Any necessary information on those 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 that should be needed for the computer project. All of these are kept under version control at GitHub, providing an easy way to monitor any typo fixes or other improvements. Reports of any issues, and pull requests to address them, are also welcome. There will also be roughly 20 minutes of video recordings per week, to enhance the human element especially for those unable to join the twice-weekly synchronous sessions. Here is an introductory video summarizing the module logistics.

Synchronous sessions are timetabled at 9:00 on Mondays and 13:00 on Thursdays. Office hours will take place immediately afterwards, at 10:00 on Mondays and 14:00 on Thursdays. You can also make an appointment to speak 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 fill in the gaps in the lecture notes and solve the homework problems yourself (though I encourage you to discuss these topics with each other). Sample exams are also available, and I expect to design additional 'tutorial' exercises in response to reported difficulties, to offer further opportunities for practice and reinforcement. Should you ask me questions about these exercises and problems, I will endeavour to avoid doing the work for you; 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:

- David Tong,
*Lectures on Statistical Physics*(2012) - Daniel V. Schroeder,
*An Introduction to Thermal Physics*(first edition, 1999) - C. Kittel and H. Kroemer,
*Thermal Physics*(second edition, 1980) - 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) - Weinan E, Tiejun Li and Eric Vanden-Eijnden,
*Applied Stochastic Analysis*(first edition, 2019) - 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)

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 (8–14 February):** Central limit theorem and diffusion

Intro and Wrap-up: Probability foundations; Law of large numbers; Probability distributions; Central limit theorem; Diffusion

**Week 2 (15–21 February):** Micro-canonical ensemble

Intro and Wrap-up: Statistical ensembles and thermodynamic equilibrium; Entropy and its properties; Temperature; Heat exchange

**Week 3 (22–28 February):** Canonical ensemble

Intro and Wrap-up: The thermal reservoir; Replicas and occupation numbers; Partition function; Internal energy, heat capacity, and entropy; Helmholtz free energy; The physics of information (distinguishable vs. indistinguishable spins)

**Week 4 (1–7 March):** Ideal gases

Intro and Wrap-up: Volume, energy levels, and partition function; Internal energy, and entropy; The mixing entropy and the 'Gibbs paradox'; Pressure, ideal gas law, and equations of state

**Week 5 (8–14 March):** Thermodynamic cycles

Part 1: Work, pressure, and force; Heat and entropy

Part 2: PV diagrams; The Carnot cycle; Efficiency

**Week 6 (15–21 March):** Grand-canonical ensemble

Part 1: Particle reservoir and chemical potential; Grand-canonical partition function

Part 2: Grand-canonical potential, internal energy, entropy and particle number

**Week 7 (12–18 April):** Quantum statistics

Part 1: Quantized energy levels and their micro-states; Bosons and fermions

Part 2: Bose–Einstein statistics; Fermi–Dirac statistics; The classical limit

**Week 8 (19–25 April):** Quantum gases of bosons

Part 1: The photon gas

Part 2: The sun and the void; Equation of state for the photon gas

**Week 9 (26 April – 2 May):** Quantum gases of fermions

Part 1: Non-relativistic ideal fermion gas; Low-temperature equation of state

Part 2: Low-temperature equation of state; Type-Ia supernovas; Relativistic ideal fermion gas

**Week 10 (3–9 May):** Interacting systems

Part 1: The Ising model; Ordered and disordered phases; Phase transitions

Part 2: The mean-field approximation; Second-order Ising model phase transition; Critical exponents

**Week 11 (10–16 May):** Synthesis and broader applications

Part 1: Numerical computations; Monte Carlo methods; Importance sampling

Part 2: Universality; Broader applications; Module overview

Last modified 8 May 2021