Lectures on Statistical Physics
Background: During the pandemic, I recorded my online lectures and shared the videos with my students. I had students living in various parts of India and some of them had very poor connectivity. The size of the videos was kept small through post-processing. These 35 lectures on Statistical Physics were given during January-May 2022. We had three lectures per week. While the lectures are not perfect, I think they should be useful to students interested in learning Statistical Physics at the M.Sc. level.
Pre-requisites: You should have done a course in classical mechanics and at least one course in quantum mechanics. It is imperative that you do all the problem sets as they are an integral part of the course. Solutions to the problem sets are not shared as it is expected that the student work them out on their own.
Tips on watching the lectures: Watch the videos at 1.25X slowing it down when necessary. Many modules work as standalone sets of lectures with some weak dependence on earlier lectures.
The Lectures
| Module 1: Introduction to Statistical Physics |
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| Lecture Number | Content of the Lecture | Additional Info | Problem Sets/Other Links |
|---|---|---|---|
| Lecture 1 | What can the answer be? An introduction to probabilistic methods | Slides for Lecture 1 | |
| Lecture 2 | The state of the system (Part 1 of 2) | Why do cells in phase space have area h? | Slides for Lecture 2 Problem Set 1 |
| Lecture 3 | The state of the system (Part 2 of 2) | Slides for Lecture 3 |
- A chapter from an unpublished book of mine with Prof. Balakrishnan and Prof. Lakshmi Bala that discusses the same material.
| Module 2: The microcanonical and canonical ensembles |
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| Lecture Number | Content of the Lecture | Additional Info | Problem Sets/Other Links |
|---|---|---|---|
| Lecture 4 | The ideal gas (Part 1 of 2) | Slides for Lecture 4 | |
| Lecture 5 | The ideal gas (Part 2 of 2) | The Sackur-Tetrode Formula | Slides for Lecture 5 Problem Set 2 |
| Lecture 6 | The Canonical Ensemble | Slides for Lecture 6 | |
| Lecture 7 | The ideal gas in the canonical ensemble | Slides for Lecture 7 | |
| Lecture 8 | The pressure of an ideal gas and the ideal gas law. | Problem Set 3 | Slides for Lecture 8 |
| Module 3: Connecting up with Thermodynamics |
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| Lecture Number | Content of the Lecture | Additional Info | Problem Sets/Other Links |
|---|---|---|---|
| Lecture 9 | Equivalence of Ensembles, Formula for entropy in the canonical ensemble | Slides for Lecture 9 | |
| Lecture 10 | A recap of Thermodynamics | Problem Set 4 | Slides for Lecture 10 |
| Module 4: Magnetic Systems and Phase Transitions |
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| Lecture Number | Content of the Lecture | Additional Info | Problem Sets/Other Links |
|---|---|---|---|
| Lecture 11 | An introduction to magnetic systems | Slides for Lecture 11 | |
| Lecture 12 | The Heisenberg and Ising models. Mean Field Theory solution. | Slides for Lecture 12 | |
| Lecture 13 | The mean field solution to the Ising model (1/2) | Slides for Lecture 13 | |
| Lecture 14 | The mean field solution to the Ising model (2/2) | Slides for Lecture 14 | |
| Lecture 15 | Ising model in a magnetic field | Slides for Lecture 15 | |
| Lecture 16 | Free Energy for the Ising model, Order parameters, Universality Classes of phase transitions | Slides for Lecture 16 | |
| Tutorial cum lecture | Critical exponents from the mean field solution | Problem Set 5 | Slides for Tutorial |
| Lecture 17 | Exact solution to the d=1 Ising model | Slides for Lecture 17 |
- Boris Kastening's discusses the computation of the partition function of the 2D-Ising model in his article titled Pedestrian Solution of the Two-Dimensional Ising Model. Of course, this is based on Onsager's computation but with simplifications.
- As I mention in my lecture, the computation of the partition function of the 2d-Ising model with magnetic field is as hard as the partition function of the 3d-Ising model. However, CN Yang, in a remarkable paper, titled The Spontaneous Magnetization of a Two-Dimensional Ising Model , realised that the computation of the spontaneous magnetisation was possible.
- In 2000, Sorin Istrail in a paper titled "Statistical mechanics, three-dimensionality and NP-completeness. I. Universality of intractability for the partition function of the Ising model across non-planar surfaces" showed that the computation of the partition function of the non-planar Ising model (such as the 3d-Ising model) is NP-complete. Nevertheless, all critical exponents of the 3d-Ising model can be determined numerically from Monte-Carlo simulations.
| Module 5: Real Gases |
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| Lecture Number | Content of the Lecture | Additional Info | Problem Sets/Other Links |
|---|---|---|---|
| Lecture 18 | Real gases — an introduction | Slides for Lecture 18 | |
| Lecture 19 | The Helmholtz free energy for the van der Waals gas and the virial expansion | Slides for Lecture 19 | |
| Lecture 20 | The slope of the coexistence curve, the tie line for the van der Waals gas. | Slides for Lecture 20 | |
| Lecture 21 | Critical exponents for the van der Waals gas | Problem Set 6 | Slides for Lecture 21 |
- The 1910 Nobel Prize in Physics was awarded to Johannes Diderik van der Waals "for his work on the equation of state for gases and liquids".
| Module 6: On the specific heat of solids |
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| Lecture Number | Content of the Lecture | Additional Info | Problem Sets/Other Links |
|---|---|---|---|
| Lecture 22 | Part (1/2) | Problem Set 7 | Slides for Lecture 22 |
| Lecture 23 | Part (2/2) | Some experimental data | Slides for Lecture 23 |
- My notes from these lectures.
| Module 7: The Grand Canonical Ensemble |
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| Lecture Number | Content of the Lecture | Additional Info | Problem Sets/Other Links |
|---|---|---|---|
| Lecture 24 | Part (1/3) | Slides for Lecture 24 | |
| Lecture 25 | Part (2/3) | Problem Set 8 | Slides for Lecture 25 |
| Lecture 26 | Part (3/3) | Slides for Lecture 26 |
- A quiz to test your understanding of the material.
| Module 8: Ideal Quantum Gases |
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| Lecture Number | Content of the Lecture | Additional Info | Problem Sets/Other Links |
|---|---|---|---|
| Lecture 27 | Photon Gas Part (1/2) | Slides for Lecture 27 | |
| Lecture 28 | Photon Gas Part (2/2) | Problem Set 10 | Slides for Lecture 28 |
| Lecture 29 | Ideal Quantum Gases Part (1/4) | Slides for Lecture 29 | |
| Lecture 30 | Ideal Quantum Gases - bosons Part (2/4) | Bose-Einstein Condensation Problem Set 11 | Slides for Lecture 30 |
| Lecture 31 | Ideal Quantum Gases - fermions Part (3/4) | Slides for Lecture 31 | |
| Lecture 32 | Ideal Quantum Gases - specific heat of electrons Part (4/4) | Slides for Lecture 32 |
- The experimental observation of Bose-Einstein condensation required a very low temperature. This was attained through two separate innovations that lead to Nobel Prizes. The first idea was that of laser cooling and the The 1997 Nobel Prize in Physics was awarded to Steven Chu, Claude Cohen-Tannoudji and William D. Phillips for their developments of methods to cool and trap atoms with laser light. Temperatures of around a few $\mu K$ were obtained in this process. The 2001 Nobel Prize in Physics was awarded jointly to Eric A. Cornell, Wolfgang Ketterle and Carl E. Wieman "for the achievement of Bose-Einstein condensation in dilute gases of alkali atoms, and for early fundamental studies of the properties of the condensates". They achieved temperatures of the order of 10 nK which made it possible for them to see BEC in Rubidium and Sodium atoms.
- At low temperatures, the specific heat of an alkali metal obtains contributions from the positive ions (via phonons as discussed in Lecture 23) and from the gas of valence electrons (as discussed in Lecture 32). This is indeed a connection with the real world of the formalism developed in these lectures. Courses in Condensed Matter Physics will see plenty of other applications.
- The derivation of the Chandrasekhar limit on the mass of a white dwarf and the study of magnetic properties of the Fermi gas will constitute two other examples that will be covered in the next three lectures.
| Module 9: Special Applications |
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Magnetic properties of electron gases: Landau diamagnetism, de Haas - van Halphen effect and Pauli paramagnetism,
| Lecture Number | Content of the Lecture | Additional Info | Problem Sets/Other Links |
|---|---|---|---|
| Lecture 33 | The Chandrasekhar limit on the mass of white dwarfs, | Problem Set 12 | Slides for Lecture 33 |
| Lecture 34 | Magnetic properties of electron gases: Landau diamagnetism | Slides for Lecture 34 | |
| Lecture 35 | Magnetic properties of electron gases: de Has - van Halphen effect, Pauli paramagnetism | Slides for Lecture 35 |
- Chandrasekhar's Nobel lecture titled On stars, their evolution and their stabilty
- A fun exercise is to use the calculations discussed in Lecture 33 to obtain a similar bound the mass of neutron stars. Here the gas of neutrons replaces the gas of electrons.
- In our discussion of the magnetic properties of a fermi gas, we make use of the spectrum of a charged particle in a constant magnetic field. Here is a derivation of the spectrum in the form of a problem set that I had given in one of my lecture on quantum mechanics.
- * In lecture 35, we computed the spin contribution to magnetization (aka Pauli paramagnetism) in the canonical ensemble. In my 2023 lecture, I redid this computation using the grand canonical ensemble. Here are my notes from this lecture.
Additional material
- Let $F_{class}$ denote the semi-classical Helmholtz free-energy of a particle in a potential $V(x_i)$ and $F_q$ denote quantum Helmholtz free energy of the same particle. Then there is a formula due to Wigner
\begin{align} F_q = F_{class} + \frac{\hbar^2}{24m(k_BT)^2} \sum_{i} \left\langle\frac{\partial^2 V}{\partial x_i^2} \right\rangle + O(\hbar^3)\ . \end{align}
There is a nice article by M. Deserno and OT Turgut in the American Journal of Physics titled Leading quantum correction to the classical free energy that derives this cool formula. Why is there no term at $O(\hbar)$? Read their derivation to find the reason. What about multi-particle systems? How do corrections from statistics arise? This is discussed in the supplementary material!