Statistical Mechanics

A rigorous graduate-level treatment of statistical mechanics—from ensembles and partition functions through quantum gases, phase transitions, critical phenomena, and the renormalization group.

Course Overview

Statistical mechanics provides the bridge between the microscopic laws of physics and the macroscopic behavior of matter. From Boltzmann's entropy formula to the modern renormalization group, this course develops the complete theoretical framework for understanding thermodynamic systems, quantum gases, phase transitions, and far-from-equilibrium phenomena.

What You'll Learn

  • • Microcanonical, canonical, and grand canonical ensembles
  • • Partition functions and thermodynamic potentials
  • • Quantum statistics: Bose-Einstein and Fermi-Dirac
  • • Bose-Einstein condensation and Fermi systems
  • • Phase transitions and critical phenomena
  • • Ising model and mean-field theory
  • • Renormalization group and universality
  • • Non-equilibrium methods and fluctuation theorems

Prerequisites

• Classical mechanics• Thermodynamics• Multivariable calculus• Probability theory• Quantum mechanics fundamentals

• Complex analysis (helpful)

References

  • • R. K. Pathria & P. D. Beale, Statistical Mechanics (4th ed.)
  • • M. Kardar, Statistical Physics of Particles & Fields
  • • K. Huang, Statistical Mechanics (2nd ed.)
  • • D. Chandler, Introduction to Modern Statistical Mechanics

Course Structure

🎲

Part I: Ensembles

Microcanonical, canonical, and grand canonical ensembles. Quantum statistics foundations.

🌡️

Part II: Quantum Gases

Ideal quantum gases, Bose-Einstein condensation, Fermi systems, and the classical limit.

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Part III: Phase Transitions

Phase transitions, Ising model, mean-field theory, and critical phenomena.

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Part IV: Advanced Topics

Renormalization group, universality, non-equilibrium stat mech, and fluctuation theorems.

Key Equations

Boltzmann Entropy

$$S = k_B \ln \Omega$$

Entropy as a function of the number of microstates

Canonical Partition Function

$$Z = \sum_i e^{-\beta E_i}, \quad \beta = \frac{1}{k_B T}$$

Sum over all microstates weighted by Boltzmann factors

Grand Potential

$$\Phi_G = -k_B T \ln \mathcal{Z}, \quad \mathcal{Z} = \sum_{N=0}^{\infty} z^N Z_N$$

Grand canonical potential from the grand partition function

Bose-Einstein & Fermi-Dirac

$$\langle n_i \rangle = \frac{1}{e^{\beta(\epsilon_i - \mu)} \mp 1}$$

Mean occupation numbers: − for bosons, + for fermions

Ising Hamiltonian

$$H = -J \sum_{\langle i,j \rangle} s_i s_j - h \sum_i s_i$$

Nearest-neighbor spin interaction with external field

RG Transformation

$$\mathbf{K}' = \mathcal{R}(\mathbf{K}), \quad \xi' = \xi / b$$

Coupling constants flow under coarse-graining by factor b