Lecture Notes on Equilibrium Statistical Mechanics (Physics 211)

UNIVERSITY OF CALIFORNIA, BERKELEY DEPARTMENT OF PHYSICS

Instructor: Prof. Oskar Hallatschek (he/his), Office: 274D Stanley Hall, Email: ohallats@berkeley.edu

The following lecture notes are designed to complement the Berkeley Physics course 211 material covered in the spring of 2024. Special thanks to Mike Zalatel and Dunghai Lee for their invaluable lecture notes, which have partially influenced my own.

The course structure is adapted and reordered from Kardar’s textbook.

• Probability & information entropy

  • Kardar 2, Arovas 1

• Fundamentals of thermodynamics via molecular perspective - ensemble theory

  • Kardar 4, Arovas 4

• Dynamics of interacting particles

  • Kardar 5, Arovas 6

• Monte Carlo, Debye-Hueckel, Diffusion

  • lecture notes

• Study of Hamiltonian dynamics and ergodicity

  • Arovas 3, Kardar 3.1, 3.2

• Quantum statistical mechanics

  • Kardar 6+7, Arovas 5

• Exploration of mean field and Landau theory

  • Arovas 7

Important Notes:

• Undergraduate level thermodynamics, a prerequisite for this course, will not be covered. Students are encouraged to review this material independently if needed.

• A brief overview of probabilistic reasoning will be provided (in class and discussions); however, extensive time cannot be dedicated to this. Students are advised to actively revise these concepts on their own.

• The syllabus is subject to modifications, aiming to incorporate advanced topics relevant to current research in active matter, (quantum) information theory, and machine learning. As this is my first time teaching this course, the extent of these topics’ inclusion is yet to be determined.

Syllabus Updates: The syllabus will be updated throughout the course.

• Course Information
• Probability and information entropy
• Entropy and information
• Citations
• Classical statistical mechanics
• Meaning of equilibrium
• Equipartition
• When a Macrostate is a Microstate
• Gibbs entropy / Partition Function / Free energy
• Trading conserved quantities between reservoir and system // External forces
• Spontaneous Reactions
• Non-equilibrium fluctuation theorems
• Chemical reactions
• Statistical mechanics with interactions - Ising models
• Finite dimensional Ising models
• Existence of a phase transition in the 2D Ising model
• Variational Method
• Interacting Gases
• Variational treatment of a gas and the van der Waals equation of state
• Monte Carlo and the Metropolis Algorithm
• Debye-Hückel theory - charge screening
• Diffusion as a free energy minimizing process
• Coarse-graining and Ergodicity theory
• When does quantum mechanics make a difference?
• Gapless quantum matter: phonons + Debye model Coldstone’s Theorem
• Debye Model.
• Formalism of quantum statistical mechanics
• The Many-Body Hilbert space
• Hilbert space of Bosons and Fermions
• Pressure of a dilute ideal quantum gase
• Ideal quantum gases - Degenerate Fermi gas
• Ideal quantum gases - Bose-Einstein Condensate
• Landau theory
• Breakdown of Landau theory