Duke University Physics Department Course Plan
Physics 303 -- Statistical Mechanics
Objectives and Goals
Statistical Mechanics is the physics of systems containing a large number
of particles. The main subject is to connect macroscopic observable properties
to microscopic properties of matter. The goals of this course are, first,
to explain the foundations of statistical mechanics and, second, to work
through most of the classic examples of statistical mechanics, as well
as some current ones, so that the student develops familiarity and facility
with the topic. At the end of the course, the student will be able to tackle
the statistical mechanics questions that come up in all areas of experimental
and theoretical physics and have a good foundation for further study in
statistical physics should she so choose.
The course is basically divided into 3 parts:
(1) Foundations and Fundmentals of Statistical Mechanics (preceded
by a review of prerequisite material)
(2) Classic Examples no educated physicist can do without (the core
of the course)
(3) Advanced Topics and Other Examples
The separation of the examples from the presentation of the foundations
is intentional. Such a separation would be a poor way to present a first
course on thermal physics - the examples are essential to understand the
rather abstract concepts involved. But for a second course, this structure
is better: the student gets a clearly organized and comprehensive view
of the foundations without the distractions of details of particular systems.
The course provides exposure to the following topics:
laws of thermodynamics and simple applications
density matrices in quantum mechanics
ergodic approach to statistical mechanics and its failure
ensemble approach to statistical mechanics - principles for choosing ensembles
derivation of microcanonical, canonical and grand canonical ensembles
interpretation of entropy
fluctuations in the different ensembles and the correspondence between
the irreversibility "paradox" and Maxwell's demon
ideal classical gas, including rotational and vibrational internal structure
ideal quantum gas - occupation numbers
phonons in crystals
electrons in metals - specific heat and spin magnetization
non-ideal gases and virial coeficients
phase transitions - van der Waals equation of state and mean field theory
selected advanced topics as time permits
Methods and Approach
Prerequisites: Undergraduate courses in classical mechanics, quantum
mechanics, and thermal physics are prerequisites. Some graduate quantum
mechanics would be helpful.
In classical mechanics, the main topics needed are (1) the concept of
phase space and (2) Liouville's Theorem. Though they are briely reviewed
in this course, for real comprehension the student should have seen them
before. In addition, some knowledge of normal modes for small oscillations
is assumed in treating phonons.
In quantum mechanics, the examples used in this course depend on the
student knowing the solution to certain simple quantum mechanical problems.
These are the harmonic oscillator, an arbitrary spin in a magnetic field,
the rigid rotator, and a single particle in a magnetic field (Landau levels).
In addition, the concept of a mixed state and density matrices are important
in statistical mechanics - these will be developed in this course in order
to explain quantum statistical mechanics, but any previous exposure would
certainly be helpful.
Format: This course is taught through 50-minute lectures (3 per
week) and weekly homework sets. Lectures generally involve blackboard presentations
by the professor, but student participation is encouraged. Homework sets
generally consist of about 6 problems designed to take 10-12 hours of concentrated
effort to complete. Students are encouraged to discuss the homework with
their peers, but are required to write solutions independently. Students
are responsible for all material covered in the lectures and in the homework
problems. Some concepts and applications that are important are covered
only in the homework.
Texts: There is no good textbook for a graduate course in this
subject, and as a result there is no consensus about what text to use in
this course. Currently the required texts are
R. K. Pathria, Statistical Mechanics, 2nd ed.
A. B. Pippard, Classical Thermodynamics.
Pathria is the general text; he does a reasonable job on many of the topics.
Certain topics are presented differently in lecture than in the text.
Pippard is used for the thermodyanmics section at the beginning of the
course; it is a lovely,
clear and short presentation. In addition, two supplemental texts
C. Kittel and H. Kroemer, Thermal Physics, 2nd ed.
Landau and Lifshitz, Statistical Physics, 3rd ed. part 1.
Kittel and Kroemer is used mainly for its many excellent problems. Landau
and Lifshitz is excellent on certain topics for which Pathria is poor and
is used mainly as a source of lecture material.
Exams and Grades: There is a take-home midterm and a final exam
in this course. Exams are designed to test each student's grasp of the
fundamental concepts and ability to solve problems. Grades for the course
are determined by homework (30%), midterm (30%), and the final exam (40%).
(Each bullet represents one week.)
Lecture 1: Introduction
Lecture 2: Laws of thermodynamics
Lecture 3: Thermodynamic relations
Lecture 4: Simple applications of thermodynamics - 5
Lecture 5: Classical mechanics - review
Lecture 6: Density matrices in quantum mechanics
Lecture 7: Ergodic approach to equilibrium statistical
Lecture 8: Ensemble approach - presentation of the
three main ensembles
Lecture 9: Canonical ensemble
Lecture 10: Interpretation of entropy
Lecture 11: Energy fluctuations in the canonical ensemble and
the equipartition theorem
Lecture 12: Grand canonical ensemble and the equivalence of
the canonical and grand canonical ensembles
Lecture 13: Approach to equilibrium and two paradoxes - irreversibility
and Maxwell's demon
Lecture 14: Paramagnets
Lecture 15: Classical ideal gas: translation, mixing, and effusion
Lecture 16: Classical ideal gas: internal structure
Lecture 17: Ideal gas: role of symmetry of nuclear wavefunction
- ortho- and para- hydrogen
Lecture 18: Quantum ideal gas: occupation numbers
Lecture 19: Bose-Einstein condensation - thermodynamics
Lecture 20: Bose-Einstein condensation - thermodynamics and
atom trapping experiments
Lecture 21: Black-body radiation
Lecture 22: Phonons in crystals - Einstein and Debye models
Lecture 23: Ideal Fermi gas - electrons in metals - specific
Lecture 24: Magnetization of the electron gas
Lecture 25: Non-ideal gases - virial and cluster expansions
Lecture 26: Evaluation of the virial coefficient
Lecture 27: White dwarf stars
Lecture 28: Thomas-Fermi model of atoms, solids, and white dwarf
Lecture 29: Chemical equilibria and solutions
Lecture 30: Phase transitions: Introduction via van der Waals
equation of state
Lecture 31: Phase transitions: Mean field theory
Lecture 32: Phase transitions: Beyond mean field theory
Lecture 33: Phase transitions: Landau theory
Lecture 34: Brownian motion: Random walk and Langevin approaches
Lecture 35: Brownian motion: Fokker-Planck equation