## PHY 412: Statistical Mechanics II

3 credits | Prerequisites: PHY 303

### Course rationale

This course is a continuation of Statistical Mechanics I. While the first course on Statistical Mechanics mainly focused on the equilibrium basis of thermodynamics phenomena, this course will introduce basics of nonequilibrium phenomena, and phase transition. One important part of this course will be the continuum description, known as statistical field theory. The idea of renormalization will be a central focus of this course.

### Course content

Elements of Non-equilibrium Statistical Mechanics: random walk, Markov process, Brownian motion, Langevin equation, Fokker-Planck equation, Boltzmann equation, H-theorem and irreversibility, linearized Boltzmann equation, collision integral, relaxation time approximation, irreversibility and approach to the equilibrium; Statistical Fields: collective behavior, phonon and elasticity, from bits to fields, continuum limit, Hubbard-Stratonovich transformation, Landau-Ginzburg Hamiltonian, saddle point approximation, mean field theory, continuous symmetry breaking, Goldstone modes, discrete symmetry breaking, domain walls; Fluctuations: scattering and fluctuation, correlation functions and susceptibilities, lower critical dimension, gaussian integrals, fluctuation corrections to the saddle point, Ginzburg criterion; Phase Transitions and Critical Phenomena: phase, phase diagram, phase transition, critical phenomena, scale transformation, meanfield approximation, critical exponents, Landau theory, variational method, Bethe approximation, correlation function; Renormalization Group And Scaling: coarse graining and scale transformation, parameter space, renormalization group equation, scaling law, anomalous dimensions; Perturbative Renormalization Group: critical Hamiltonian, Feynman diagram, fixed point and critical behavior, models with O(N) symmetry, renormalization group near dimension 4, renormalization group for N-components fields, gradient flow; Conformal Field Theory: introduction, conformal invariance, quasi-primary fields, Ward identity, central charge, Virasoro algebra, representation theory, Hamiltonian on a cylinder, Thermodynamic Bethe ansatz.

### Course objectives

- Familiarize yourself with the basic concepts of non-equilibrium statistical mechanics.
- Understand the concept of phase transition and critical phenomena.
- Use techniques from field theory to statistical mechanics.
- Familiarize with spontaneous symmetry breaking and Nambu-Goldstone theorem.
- Understand and calculate different quantities in the renormalization group.

### References

- Statistical Mechanics (2nd edition) by Franz Schwabl
- Statistical Physics of Fields (1st edition) by Mehran Kardar
- Elements of Phase Transitions and Critical Phenomena by Hidetoshi Nishimori and Gerardo Ortiz
- Phase Transitions and Renormalization Group by Jean Zinn-Justin
- Statistical Field Theory: An Introduction to Exactly Solved Models in Statistical by Giuseppe Mussardo
- Quantum Field Theory and Critical Phenomena (5th edition) by Jean Zinn-Justin