**Statistical Physics 2024**

### Statistical Physics (MATH327), Spring Term 2024

Along with quantum mechanics and relativity, statistical physics is a central pillar of modern physics. It uses stochastic (i.e., probabilistic) techniques to predict the large-scale behaviour that emerges from the microscopic dynamics of *many* underlying objects --- far more than can possibly be analyzed in complete detail. Statistical physics find applications from nuclear physics and cosmology to climate science and biophysics, often with outstanding success.

This module covers both foundations and applications of statistical physics. Foundational topics include the concepts of statistical ensembles, the laws of thermodynamics, and derived quantities such as entropy. We will apply these foundations to investigate diffusion, the behaviour of idealized physical systems such as classical and quantum gases, thermodynamic cycles, and phase transitions. In particular, we will use numerical computer programming to investigate diffusion in terms of stochastic processes. **No** prior exposure to quantum mechanics or computer programming is required --- all necessary information on these topics will be provided.

### Learning outcomes

Upon completing this module, students are able to:

- Use the central limit theorem to analyze macroscopic diffusion emerging from stochastic microscopic dynamics.
- Numerically analyze macroscopic behaviour emerging from an underlying stochastic process.
- Apply the micro-canonical, canonical, and grand-canonical ensembles to analyze statistical systems subject to the corresponding constraints.
- Use the concepts of work, heat, and the laws of thermodynamics to analyze thermodynamic processes and thermodynamic cycles, determining the efficiency of the latter.
- Derive the equation of state for classical and quantum ideal gases.
- Carry out simple calculations for interacting statistical systems, including the use of order parameters to distinguish phases separated by a phase transition.

### 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 9:00–10:00 on Tuesdays, 14:00–15:00 on Thursdays, and 11:00–13:00 on Fridays. During weeks 2, 3 and 4 (starting 5, 12 and 19 February) we will instead meet at 9:00–11:00 on Tuesdays for a computer lab, along with the usual Friday time. 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). Two sample exams will also be 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 is a tutorial activity on probabilities, 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 two homework assignments.

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

- David Tong,
*Lectures on Statistical Physics*(2012) - MIT OpenCourseWare for undergraduate Statistical Physics I (2013) and Statistical Physics II (2005)
- 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 resources at a higher level than this module, including the following:

- MIT OpenCourseWare for postgraduate Statistical Mechanics I (2013) and Statistical Mechanics II (2014)
- 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) - Sacha Friedli and Yvan Velenik,
*Statistical Mechanics of Lattice Systems*(2018) - 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

**1 February (tutorial):** Probability foundations and activity

**2 February:** Law of large numbers; Probability distributions; Central limit theorem; Random walks

**Week 2**

**6 February (computer lab):** Pseudo-random numbers; Inverse transform sampling; Random walks

**9 February:** Random walks; Law of diffusion; Statistical ensembles

**Week 3**

**13 February (computer lab):** Cauchy--Lorentz distribution; Anomalous diffusion.

**16 February:** Micro-canonical ensemble; Thermodynamic equilibrium; Entropy; Second law of thermodynamics

**Week 4**

**20 February (computer lab):** Project wrap-up

**23 February:** Micro-canonical temperature and heat exchange; Canonical ensemble; Replica trick

**Week 5**

**27 February:** Partition function and Gibbs distribution; Heat capacity; Helmholtz free energy

**29 February (tutorial):** Entropy bounds and Stirling's formula activities

**1 March:** Physical effects of information content (distinguishable vs. indistinguishable spins); Classical ideal gas

**Week 6**

**5 March:** Ideal gas regularization; Distinguishable vs. indistinguishable partition functions, energies and entropies

**7 March (tutorial):** Comments on entropy bounds and Stirling's formula; Mixing entropy activity

**8 March:** Ideal gas energies and entropies; Mixing entropy; Pressure; Ideal gas law and equations of state; Work and heat

**Week 7**

**12 March:** Work and heat; First law of thermodynamics; PV diagrams, isotherms and adiabats

**14 March (tutorial):** Comments on mixing entropy and Maxwell's demon; Otto cycle activity

**15 March:** Carnot cycle; Heat engine efficiency; Grand-canonical ensemble; Chemical potential

**Week 8**

**19 March:** Grand-canonical partition function; Grand-canonical potential

**21 March (tutorial):** Comments on computer project, Otto cycle and Diesel cycle; Einstein solid activity

**22 March:** Generalized thermodynamic identity; Maxwell–Boltzmann statistics; Quantum statistics; Bosons vs. fermions

**Week 9**

**16 April:** Post-break review; Quantum statistics; Bose–Einstein statistics; Fermi–Dirac statistics; Classical limit

**18 April (tutorial):** Comments on homeworks; Einstein solid activity

**19 April:** Occupation number expectation values; The high-temperature classical limit; Grand-canonical ideal quantum gases; Ultra-relativistic photon energies, wavelengths and frequencies; Photon gas internal energy density; Planck spectrum

**Week 10**

**23 April:** Planck spectrum vs. Rayleigh–Jeans ultraviolet catastrophe; Solar radiation and cosmic microwave background; Photon gas internal energy

**25 April (tutorial):** Photon polarization; Dark matter and the cosmic microwave background; Comments on Einstein solid heat capacity vs. experiment; Phonons; Debye solid activity

**26 April:** Photon gas particle number; Radiation pressure and equation of state; Non-relativistic fermion gas; Low-temperature Fermi function; Fermi energy; Internal energy; Positive chemical potential; Degeneracy pressure

**Week 11**

**30 April:** White dwarf stars and Type-Ia supernovas; Phase transitions to motivate interacting systems

**2 May (tutorial):** Comments on Debye solid and electron gas heat capacities; Ising model; Lattice structures; Complete graph; Lattice structures activity

**3 May:** Interacting systems; Ising model magnetization in ordered and disordered phases; Order parameters and phase transitions; Magnetization as Ising model order parameter

**Week 12**

**7 May:** Ising model mean-field approximation; Mean-field self-consistency condition

**9 May (tutorial):** Comments on lattice structures; Anti-ferromagnetism and frustration; Exam prep and module recap

**10 May:** Mean-field critical temperature, critical exponent and reliability in various dimensions; Monte Carlo importance sampling and MRRTT algorithm; Markov chains, cluster algorithms and lattice quantum field theory

Last modified 10 May 2024