PHYS427
PHYS427 (Thermal and Statistical Physics) is a 4-credit hour course that counts as a technical elective for both EEs and CEs. It is offered in the fall and spring semesters.
Content Covered
- Probability and multiplicity
- Entropy and temperature
- Canonical ensemble
- Thermodynamics
- Grand canonical ensemble
- Quantum statistics
- Phase transformations
- Thermal radiation and conduction
PHYS427 is the undergraduate course in statistical mechanics. The course begins with key mathematical concepts relevant to statistical mechanics. In this section, probability, multiplicity, the binomial distribution, macrostates and microstates, and Stirling’s approximation are covered. These concepts are applied to physical systems, including Einstein solids, paramagnets, and ideal gases, in the microcanonical ensemble, and are used to define entropy, temperature, and the laws of thermodynamics. The canonical ensemble is then introduced, along with the Boltzmann distribution, partition functions, and the quantum density.
PHYS427 then moves to thermodynamics. Heat and work, along with heat engines, are covered. The thermodynamic potentials--internal energy, Helmholtz free energy, enthalpy, and the Gibbs free energy--are defined, along with the Maxwell relations, and used to describe complex macroscopic systems that exchange energy. To describe systems that exchange particles, the concept of a chemical potential is also defined. The grand canonical ensemble is then discussed. The statistical mechanics of photon gases, electromagnetic waves in a cavity, is covered. The concept of the density of states is introduced and used to derive the Planck radiation law. The statistical mechanics of elastic waves in a solid is then discussed, along with the Debye model.
The course then covers quantum statistics. Fermions, bosons, and the statistics that describe them, Fermi-Dirac statistics and Bose-Einstein statistics, are discussed. Degenerate Fermi gases are examined and the Fermi energy is defined. Bose gases are examined along with the phenomenon of Bose-Einstein condensation.
PHYS427 then moves on to discussing systems with interacting particles. Phase transitions are introduced, including the Clausius-Clapeyron equation and the concept of latent heat. Mean field theory is used to define the van der Waals equation of state and explain liquid-gas transitions. To describe ferromagnets and paramagnet-ferromagnet phase transitions, the Ising model and Curie-Weiss law are introduced. The more general Landau theory of phase transitions is also discussed. Kinetic theory is discussed to describe transitions between states. The course concludes with an explanation of thermal radiation and thermal conduction.
Prerequisites
The official prerequisites to PHYS427 are PHYS213, PHYS214, and either PHYS435 or ECE329. PHYS213 serves as an introduction to thermal physics and statistical mechanics, and will be useful for understanding some of the concepts covered in PHYS427, such as probability and multiplicity, entropy, thermodynamics, and phase transitions. Many of the derivations in PHYS427 arise from quantum mechanics, so PHYS214 is a necessary prerequisite to this course. Students who have taken PHYS486 will be even better prepared for this course, but taking PHYS486 is not strictly necessary to do well in this class. A course in electromagnetism, either PHYS435 or ECE329, is also an important prerequisite, as students are expected to understand electromagnetic wave propagation, paramagnetism, and ferromagnetism. That being said, PHYS427 is relatively self-contained.
When to Take It
PHYS427 is generally taken by seniors and first-year graduate students in physics. Take this course if you have an interest in condensed matter physics, solid-state physics, astrophysics and cosmology, or biophysics.
Course Structure
PHYS427 holds two lectures a week, which generally consist of derivations of important concepts and theories in statistical mechanics. In addition to the lectures, PHYS427 also has required discussion sections, which meet weekly. Discussions are a chance for students to go through practice problems relevant to the material covered in lecture that week, which can be helpful for completing the problem sets.
Problem sets are assigned every week, except during exam weeks. They generally consist of a mixture of derivations and computational problems.
PHYS427 also has two midterm exams, held in class, and a cumulative final exam. Formula sheets are provided.
Instructors
In recent semesters, this course has been taught by Profs. Draper, Shelton, and Dahmen. Instructors generally specialize in condensed matter physics, astrophysics and cosmology, and/or high-energy physics.
Course Tips
The problem sets can be difficult, so it is recommended to read through the lecture notes and textbook, as well as attend office hours, to complete them. Finding a group to work with is recommended.
The exams can be difficult to finish on time, so a strong understanding of the underlying concepts and experience with practice problems is important. For the exams, it is recommended to look through the typeset lecture notes and go through problems from past discussions and problem sets deliberately.
Life After
Students who enjoyed this course should consider PHYS486 - Quantum Physics I, PHYS487 - Quantum Physics II, and PHYS460 - Condensed Matter Physics.
Infamous Topics
- Density of states: It is easy to just apply a random formula from lecture when calculating the density of states and do these problems completely wrong. Approach these problems as discussed in lecture, by approximating a sum over wavevectors as an integral. Understand Debye’s trick and when to use it.
- Mean field theory: Problems on this topic can be difficult. It is recommended to become familiar with mean field theory by going to office hours, as it is difficult to understand by just reading the textbook.