## PHY 303: Statistical Mechanics I

3 credits | Prerequisites: MAT 212, PHY 104, 223

### Course rationale

Statistical Mechanics is an integral element of physics that gives a microscopic understanding of
thermodynamics, and critical phenomena. The concepts and techniques of Statistical Mechanics are relevant in other branches of physics, and an interplay between Statistical Mechanics and Quantum Field Theory emerges in Statistical Field Theory. This core course in Physics will build the foundation of Statistical Mechanics.

### Course content

Review of Thermodynamics: laws of thermodynamics, postulates of equilibrium thermodynamics, intensive parameter of thermodynamics, equilibrium, Euler and Gibbs-Duhem relations, thermodynamic potential, Maxwell relations, variational principle; Kinetic Theory of Gas: general definitions, Liouville’s theorem, BBGKY hierarchy, Boltzmann equation, H-theorem and irreversibility, equilibrium properties, Maxwell-Boltzmann distribution, conservation laws, hydrodynamics; Microcanonical Ensemble: postulates of classical statistical mechanics, the definition of the microcanonical ensemble, thermodynamics, equipartition theorem, classical ideal gas, Gibbs paradox; Canonical Ensemble: equilibrium between a system and a heat reservoir, definition of canonical ensemble, partition function, various statistical quantities, classical systems, energy fluctuation, equipartition and viral theorem, harmonic oscillator, paramagnetism; Grand Canonical Ensemble: equilibrium between a system and a particle–energy reservoir, chemical potential, definition of grand canonical ensemble, various statistical quantities, density and energy fluctuation, phase diagram, Clausius-Clapeyron equation; Quantum Statistical Mechanics: postulates of quantum statistical mechanics, density matrix, and ensembles in quantum statistical mechanics, the third law of thermodynamics, ideal gases: microcanonical and canonical ensemble; Ideal Bose System: thermodynamic behavior, Bose-Einstein statistics, photon, thermodynamics of the blackbody radiation, phonon in solid, Bose-Einstein condensation; Ideal Fermi System: thermodynamic and magnetic behavior of an ideal Fermi gas, Fermi-Dirac statistics, electron gas in metals, Landau diamagnetism, Pauli paramagnetism, ultracold Fermi gas, statistical equilibrium of white dwarf stars; Properties of Partition Function: DarwinFowler method, classical limit, singularities and phase transition, Lee-Yang circle theorem, cumulant expansion, cluster expansion, second viral coefficient and van der Walls equation, mean field theory, variational methods; The Ising Model: definition, equivalence to other models, spontaneous magnetization, Bragg-Williams approximation, Bethe-Peierls approximation, 2-dimensional Ising model, the Onsager solution; Critical Phenomena: order parameter, correlation function, fluctuation-dissipation theorem, critical exponents, scaling hypothesis, scale invariance, Goldstone excitation, dimensionality, Landau free energy, mean-field theory, tricritical point, Gaussian model, Ginzburg criterion, anomalous dimension; Renormalization Group: block spins, one-dimensional Ising model, renormalization group transformation, fixed point and scaling of fields, momentum space formulation, Gaussian model, Landau-Wilson model.

### Course objectives

1. Understand the basic concepts of Statistical Mechanics.
2. Understand and distinguish between different ensembles used in Statistical Mechanics.
3. Use appropriate mathematical formalism to solve different problems of interest.
4. Understand different statistics used for Bosons and Fermions.
5. Familiarize with phase transition and critical phenomena.
6. Understand the concept of the Renormalization group and calculate the renormalization transformation for simple systems.

### References

1. Statistical Mechanics (2nd edition) by Kerson Huang
2. Statistical Mechanics (4th edition) by R. K. Pathria, and Paul D. Beale
3. Introduction to Statistical Physics (1st edition) by Silvio Salinas
4. Statistical Physics of Particles (1st edition) by Mehran Kardar
5. Statistical Physics of Fields (1st edition) by Mehran Kardar
6. Statistical Physics (3rd edition) by L. D. Landau, and E. M. Lifshitz
7. Scaling and Renormalization in Statistical Physics by John Cardy