Statistical Physics III

Goals of the Course

Statistical Mechanics is one of the major fields of physics: around 30% of Physics Nobel Prizes have been awarded for discoveries directly or indirectly related to Statistical Mechanics, particularly condensed matter physics, phase transitions and field theories. The principles and methods are applicable in many fields of physics, including condensed matter physics (e.g. Bose Einstein Condensation, superconductivity, materials science) and high energy physics (spontaneous symmetry breaking, lattice gauge field theories, the Higgs Mechanism (which is a type of phase transition) and so on) as well as astrophysics (neutron stars, simulations of galaxy evolution, and so on). The principles and methods are also applicable in a very wide variety of fields outside physics, such as biology, neuroscience, modelling of pandemics, network theory, machine learning and artificial intelligence. G30 students in the past have shown that students in chemistry, chemical engineering and materials science with a thorough grounding in the principles of Statistical Mechanics have a significant advantage over their peers who do not possess this grounding.

Course Overview

This is the second part of a full-year, advanced Year 3 course in statistical mechanics and thermodynamics. Exchange students should only consider taking this course if they average 80% or above in their university's quantum mechanics, statistical physics and mathematics units, and if they are prepared to work hard.

At the end of this course, students will have mastered basic aspects of quantum statistics of ideal gases, statistical mechanics of systems of interacting particles, and the theory of phase transitions and critical phenomena, including modern topics such as the scaling hypothesis, an introduction to renormalization group theory (the spatial renormalization group), and the Bogolyubov Variational Theorem and its application to constructing an optimal Mean Field Theory. One overarching goal is to develop the intuition for why quantum field theory can be applied to condensed matter systems, without going into quantum field theory: the course is a precursor to Year 4 quantum field theory.

In this course, students learn quantum statistics of ideal gases, introductory statistical mechanics of systems of interacting particles, introductory theory of phase transitions and critical phenomena, Mean Field Theory, and some modern theory such as the scaling hypothesis, an introduction to renormalization group theory (the spatial renormalization group), and the Bogolyubov Variational Theorem and its application to constructing an optimal Mean Field Theory. Students will encounter the ideas of spontaneous symmetry breaking, universality, critical exponents, transformation between -- proof of equivalence of -- various models such as the Two-state Ising Model, Lattice Gas Model, Binary Alloy Model and Two-state Potts Model.

Course Content

Some topics are covered in assignments. The precise order and content of the lectures might vary slightly.

Lecture 1. Revision of quantum statistical mechanics. Quantum states of a single particle. Reflecting boundary conditions, periodic boundary conditions. Density of states in 3, 2 and 1 dimensions, for linear and quadratic dispersion relations. Turning sums into integrals. Example: EM radiation. The quantum distribution functions: Fermi-Dirac, Bose-Einstein distributions. Photon statistics: Planck distribution. Systems with varying number of particles: the Grand Canonical ensemble and partition function. Occupation number formalism: mean occupation number and dispersion. Role of the chemical potential.

Lecture 2. Examples. Vapour pressure of a solid. Diatomic molecules. Grand Canonical partition function and probability of a many-body state at temperature T. Example: adsorption of a gas onto a 2D surface.

Lecture 3. The ideal Fermi fluid: conduction electrons in metals. Specific heat and ground state energy in 3D, 2D, 1D. Sommerfeld expansion.

Lecture 4. The ideal Bose fluid: Bose-Einstein condensation in 3D. What about in 2D or 1D? Critical temperature. Mean energy, specific heat.

Lecture 5. Relativistic Quantum Gas: the Photon Gas (Black body radiation). Stefan-Boltzmann Law; Wien’s Displacement Law; radiation pressure; mechanical equation of state. (If time allows:) Emergent phenomena in many-body systems. A particle in a many-body system behaves differently to when it is by itself -- Classical theory of screening: the Debye-Hueckel Model.

Lecture 6. Introduction to Non-Ideal Systems (1): The Debye Model of solids. The Harmonic Approximation. Classical Theory. Quantized Theory. Normal modes. Phonons. The Debye Approximation. Specific heat. Rundown of main points in Ashcroft and Mermin Chapts 22,23 placing Debye Theory in perspective: Classical Theory of the Harmonic Crystal; Quantum Theory of the Harmonic Crystal.

Lecture 7. Introduction to Non-Ideal Systems (2). Weakly nonideal gases: virial expansion; 2nd virial coefficient and resulting equation of state. Derivation of the Van der Waals equation of state for a weakly non-ideal gas; derivation for a fluid using a self-consistent mean field approach. Derivation of 2nd virial coefficient and van der Waals Equation again, this time using Mayer f function. The Cluster Expansion.

Lecture 8. Stability of thermodynamic systems. Concavity/convexity of thermodynamic potentials. Le Chatelier’s Principle. First Order phase transitions, features of the free energy. Discontinuity in the entropy: latent heat. Slope of the coexistence curves: Clausius-Clapeyron Equation. A Clausius-Clapeyron Equation for Magnetic Systems: Coexistence Curve of Superconducting and Normal Phases in a metal.

Lecture 9. Van der Waals fluid: unstable isotherms, physical isotherm, Maxwell equal-area rule. Multicomponent systems: Gibbs phase rule. Why does the phase diagram of water not have more than three phases coexisting at the same point?

Lecture 10. The Fluctuation-Dissipation Theorem. Response functions and correlations. Quantitative explanation of critical opalescence.

Lecture 11. Examples of phase transitions (order-disorder transition, which is a structural phase transition). Why do fluctuations get out of control near the critical point? Alben’s Model. Landau Theory: classical theory in the critical region. Order Parameter. Continuous phase transition. Spontaneous symmetry breaking. The critical exponents α,β,γ,δ and their classical values.

Lecture 12. Introduction to interacting magnetic systems: ferromagnetism and models for it. Ising model. 1D Ising chain with free ends. Mean field theory treatment of the 1D Ising chain. Effective field. Critical exponents.

Lecture 13. 1D Ising chain continued. No phase transition in the 1D Ising chain: proof by a simple argument; and by solving the model exactly. Exact solution of 1D Ising chain in zero field. Exact solution of 1D Ising ring with field switched on: transfer matrix. Spin correlation function: exact calculation for the 1D Ising chain. 2D Ising model on a square lattice (just mention): Exact critical exponents, behaviour of the specific heat. Phase diagram of ferromagnetic systems in 3D.

Lecture 14. Breakdown of the classical theory and advent of the modern theory. Cause of the breakdown (qualitative). Derivation of an inequality involving critical exponents – but all experiments suggest equality holds. Scaling hypothesis: ad hoc argument. Justification of the scaling hypothesis using Kadanoff’s block spins. Spatial renormalization group theory and sample calculation for the 1D Ising chain.

Lecture 15. Bogolyubov Variational Theorem. Order-Disorder Transition: constructing the Hamiltonian and deriving the optimal Mean Field Theory for its solution. Mean Field Theory for 1D Ising Model revisited. Transformation between Models and Universality classes: many problems that appear completely different are in fact manifestations of the same problem. Broken Symmetry, Universality Classes, and Goldstone’s Theorem (qualitative).

Lecture 16. 2D Ising model on a square lattice: Low-T solution -- Peierls Droplets; High-T solution; Kramers-Wannier Duality. Critical temperature for 2D Ising Model on a Square Lattice. Lee-Yang Zeroes and Phase Transitions.

Prerequisites

Statistical Physics II; or Consent of Instructor

Students must have passed Statistical Physics II to take Statistical Physics III.

Textbook

1. Callen, Herbert, Thermodynamics and an Introduction to Thermostatistics, 2nd Ed., Wiley. (The Japanese translation has fewer misprints.)

2. Reif, F., Fundamentals of Statistical and Thermal Physics, McGraw-Hill, 1965.

3. Plischke, M. & Bergersen, B., Equilibrium Statistical Mechanics, 3rd Ed., World Scientific, 2006.

Reference Books

1. Ashcroft & Mermin, Solid State Physics (Chapters 22,23 only).

2. Yeomans, J.M., Statistical Mechanics of Phase Transitions, Oxford Science Publications, 1992. (Simple, clear overview relevant to the second half of this course.)

3. Cardy, J., Scaling and renormalization in statistical physics, Cambridge Univ. Press, 1996. (Certain sections only.)

4. Shankar, R., Quantum Field Theory and Condensed Matter: An Introduction, Cambridge Univ. Press, 2017. (Certain sections only.)

5. Altland, A. & Simons, B., Condensed Matter Field Theory (2nd Ed.), Cambridge Univ. Press, 2010. (Mainly Chapter 1 only.)

6. Kittel, C. and Kroemer, H., Thermal Physics, W.H. Freeman. (Try as alternative to textbooks.)

7. Landau, L.D. and Lifshitz, E.M., Statistical Physics, Part I, by E.M. Lifshitz and L.P. Pitaevskii, Pergamon Press. (A classic book: thorough, advanced treatment. Highly recommended.)

Study Load (Self-directed Learning Outside Course Hours)

This course is part of your training to be a professional researcher. You are expected to revise the lecture notes, read and work through the textbook, and solve assignment problems outside lecture hours. You cannot learn physics by only attending lectures. The exams will consist of questions covering both lecture notes and assignments.

Students must be willing to work hard if they want to achieve a good, internationally competitive level.

Message from the Lecturer

Do not respond faster than you can think.

Course Evaluation Method and Criteria

Attendance, class performance and attitude: 5%; Weekly quizzes or other written assessment: 30%; Mid-term exam: 32.5%; Final Exam: 32.5%

This lecture is provided under Creative Commons Attribution-Non Commercial-ShareAlike 4.0 International.

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