Lecture Notes

Collective Behavior, from Particles to Fields (cont.)

Phase transitions, critical behavior

The Landau-Ginzburg Approach

Introduction, saddle point approximation, and mean-field theory

The Landau-Ginzburg Approach (cont.)

Scattering and fluctuations, correlation functions and susceptibilities, comparison to experiments

The Scaling Hypothesis

The homogeneity assumption, divergence of the correlation length, critical correlation functions and self-similarity

The Scaling Hypothesis (cont.)

The Gaussian model (direct solution), the Gaussian model (renormalization group)

Perturbative Renormalization Group (cont.)

Perturbative RG (first order)

Perturbative Renormalization Group (cont.)

Irrelevance of other interactions, comments on the ε-expansion

Position Space Renormalization Group (cont.)

The Niemeijer-van Leeuwen cumulant approximation, the Migdal-Kadanoff bond moving approximation

Series Expansions (cont.)

Self-duality in the two dimensional Ising model, dual of the three dimensional Ising model

Series Expansions (cont.)

Exact free energy of the square lattice Ising model

Continuous Spins at Low Temperatures

The non-linear σ-model

Continuous Spins at Low Temperatures (cont.)

Renormalization group for the coulomb gas

Dissipative Dynamics

Brownian motion of a particle

Continuous Spins at Low Temperatures (cont.)

Generic scale invariance in equilibrium systems, non-equilibrium dynamics of open systems, dynamics of a growing surface