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:
-
Foundations and Fundmentals of Statistical Mechanics (preceded
by a review of prerequisite material)
-
Classic Examples no educated physicist can do without (the
core of the course)
-
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 ensembles
- the irreversibility "paradox" and Maxwell's demon
- paramagnets
- ideal classical gas, including rotational and vibrational
internal structure
- ideal quantum gas - occupation numbers
- Bose-Einstein condensation
- black-body radiation
- 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 are used:
- 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%).
Sample Syllabus
(Each bullet represents one week.)
- Lecture 1: Introduction
Lecture 2: Laws of thermodynamics
Lecture 3: Thermodynamic relations- Lecture 4: Simple applications of thermodynamics -
5 examples
Lecture 5: Classical mechanics - review
Lecture 6: Density matrices in quantum mechanics- Lecture 7: Ergodic approach to equilibrium
statistical mechanics
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 heat
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 stars
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