Graduate course, Department of Physics, Gyeongsang National University

Phase Transitions and Critical Phenomena

A companion to Sang Hoon Lee's lecture notes on N. Goldenfeld's Lectures on Phase Transitions and the Renormalization Group. Each chapter pairs the lecture notes with simulations and exact calculations that run live in your browser.

Related graduate course: Special Topics in Complex Systems, following Christensen and Moloney, applies the same ideas to percolation on lattices, Bethe lattices, and networks, and carries the Ising model on to the real-space renormalization group.

spins

coarse-grained field

A 128 × 128 Ising model just above its critical temperature, and the same configuration averaged over blocks: the coarse graining at the heart of Landau theory (Chapter 5).

  1. Chapter 1Introduction

    Scaling, power laws, and universality; dimensional analysis, critical exponents, and why a sum of smooth terms can turn singular.

    Demos: Water waves, coexistence curves, the 2D Ising model, self-avoiding walks, Widom scaling.

  2. Chapter 2How phase transitions occur in principle, Part 1

    The thermodynamic limit, phase boundaries, the Ising model and its analytic properties, symmetry, and the absence of order in one dimension.

    Demos: Exact enumeration, convexity, frustration, level crossings, domain walls.

  3. Chapter 2How phase transitions occur in principle, Part 2: symmetry breaking, ergodicity, and lattice gases

    The Peierls argument, spontaneous symmetry breaking, ergodicity breaking, spin glasses and replicas, and the lattice gas.

    Demos: Non-commuting limits, reversal times, exact P(q), quenched vs annealed.

  4. Chapter 3How phase transitions occur in practice

    Transfer matrices, correlations, the low-temperature expansion, and Weiss mean-field theory, with an appendix on networks.

    Demos: Brute-force checks, Onsager's free energy, mean-field exponents, scale-free networks.

  5. Chapter 4Critical phenomena in fluids

    Thermodynamics of coexistence, the van der Waals equation as mean-field theory, density fluctuations, and measuring exponents.

    Demos: Maxwell's construction, critical opalescence, a 2D Lennard-Jones fluid, fitting α.

  6. Chapter 5Landau theory, Part 1

    Order parameters, the common structure of mean-field theories, and the Landau free energy for continuous and first-order transitions.

    Demos: Weiss/van der Waals collapse, the Landau explorer, network Landau terms.

  7. Chapter 5Landau theory, Part 2: coarse graining and correlations

    Coarse graining, the Landau free energy as a constrained free energy, functional derivatives, linear response, and Ornstein–Zernike correlations.

    Demos: Convexity restored, domain walls and Green functions, G(r) in d dimensions.

All pages work on phones, tablets, and desktops, in light and dark mode. Animations are deliberately slow and can be paused.

Created by Claude Opus 5.5 (Anthropic) based on Sang Hoon Lee's lecture notes, for the graduate course of the Department of Physics, Gyeongsang National University.