**Statistical Physics 2020**

### 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