## Physics 203 — 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 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
• 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.
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 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
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