# Physics 203 -- Introduction to 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.

The course provides exposure to the following topics:

• laws of thermodynamics and simple applications
• density matrices in quantum mechanics
• 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
• paramagnets
• ideal classical gas, including rotational and vibrational internal structure
• ideal quantum gas - occupation numbers
• Bose-Einstein condensation
• 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 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 75-minute lectures (2 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 3-5 problems designed to take 6-8 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 recommended texts are:
• Thermodynamics and Statistical Mechanics, 1st edition, by W. Greiner, L. Neise and H. Stoecker (Springer, New York, 1995)
• Statistical Mechanics, 2nd Edition, by R.K. Pathria (Butterworth and Heinemann).
• 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 for the more advanced topics and for theoretically inclined students.
There is one midterm exam taken in class 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 (45%), midterm (25%), and the final exam (30%).

## Sample Syllabus

• Introduction
• Thermodynamic quantities and relations
• Statistical Mechanics: microstates and entropy
• Ideal gas: density of states and Gibbs factor
• Classic ensembles and Liouville's theorem
• Microcanonical ensemble
• Canonical ensemble
• Boltzmann factor
• Partition function
• Helmholtz free energy
• Ideal gas and equipartition theorem
• Paramagnetism
• Grand canonical ensemble
• Chemical potential
• Grand partition function
• Quantum statistics
• Density matrix in the canonical ensemble
• Density matrix in the coordinate basis
• Thermal wavelength
• Bosons and Fermions
• Multiparticle wavefunctions
• Ideal gas in the GCE
• Ideal Bose systems
• Thermodynamics
• Bose condensation
• Ideal Fermi systems
• Occupation numbers
• Fermi pressure
• Electron gas in metals
• White Dwarfs
• Interacting systems
• Cluster expansion
• Virial expansion
• Phase transitions
• van der Waals gas
• Ising model
• Critical Indices
• Phase Transitions in Nuclear Physics
• Kinetic theory
• Maxwell velocity distribution and pressure
• Transport coefficients
• H-theorem