Self-Organization and Pattern Formation¶

These are self-study notes for the course Self-Organization and Pattern Formation, Prof. Erwin Frey, LMU Munich, Winter Semester 2025/2026. Prof. Frey prefers chalkboard lectures. To document my learning process, I've structured my notes into articles and written Python code to deepen my understanding.

Course Playlist: YouTube Playlist

Official Course Link: LMU Munich - Self-Organization and Pattern Formation

Course Overview¶

Can we build predictive theories for self-organizing systems—spanning fluids, reaction–diffusion systems, soft/active matter, and living systems—without tracking every microscopic detail? The answer is yes: by using universality, symmetry, and conservation laws, and by adopting coarse-graining methods, we can reveal instability mechanisms, mesoscale physical laws, and scaling behavior.

The course covers spatially extended nonequilibrium systems that spontaneously generate order. It focuses on the onset of instabilities, pattern selection, dynamical processes, and the emergence of mesoscale physical laws. Methods include dynamical systems, continuum field theory, statistical physics, weakly nonlinear analysis, and differential geometry.

Course Topics¶

Dynamical systems & bifurcations¶

Linear/nonlinear stability, fixed points/attractors, excitability, synchronization, chaos; normal forms and amplitude equations.

1. Self-Organization: From Bird Flocks to Living Cells
2. Dynamical Systems: Phase Space, Stability, Ginzburg-Landau, Evolutionary Game Theory
3. Dynamical Systems: Fixed Points, Saddle-Node & Pitchfork Bifurcations
4. Dynamical Systems: Cusp Bifurcation, Hysteresis, Infection Model & Transcritical Bifurcation
5. Dynamical Systems: Mass Conserving 2D, Turing Patterns, Stability Analysis, Jacobian Matrix
6. Dynamical Systems: 2D Non-Conservative Systems & Oscillators
7. Dynamical Systems: Poincaré-Bendixson Theorem, RhoGTPase Oscillator & Rock-Paper-Scissors
8. Dynamical Systems: Replicator Dynamics & Spatial Extension, Lyapunov Function, Diffusion Equation

Pattern formation¶

Phase separation and coarsening; interface dynamics; reaction–diffusion patterns and waves; excitable media; fronts, pulses, dendrites; thin-film dewetting and flow.

9. Relaxational Dynamics: Ginzburg-Landau Model, Model A, Interface Dynamics & Surface Tension
10. Relaxational Dynamics: Interface Dynamics & Curvature-Driven Flow
11. Thermodynamics and Phase Separation in Liquid Mixtures, Maxwell Construction, Phase Diagrams
12. Dynamics of Liquid Mixtures, Osmotic Pressure, Cahn-Hilliard Equation
13. The Lattice Gas Model & Mean-Field Theory, Mixing Entropy, Flory-Huggins Theory
14. Cahn-Hilliard Equation & Phase Separation Dynamics, Dispersion Relation, Fastest Growing Mode
15. Nucleation Theory & Interface Dynamics, Gibbs-Thomson, Growth and Shrinkage of a Single Droplet
16. Ostwald Ripening & Coarsening Dynamics, Lifshitz-Slyozov-Wagner (LSW) Theory

Active field theories & broken detailed balance¶

Non-variational couplings and stresses; absence of free energy/Lyapunov functional; deterministic currents; universality and scaling near nonequilibrium phase transitions.

17. Chemotaxis & Active Motion of Bacteria, Run-and-Tumble Model
18. Keller-Segel Model, Chemotactic Instability, Finite-Time Blowup
19. Scalar Active Matter, Motility-Induced Phase Separation, Active Model B
20. Active Brownian Particles, Active Model B+, Bubbling Phase

Instability mechanisms¶

Turing and Hopf instabilities; interface instabilities (Mullins–Sekerka); hydrodynamic instabilities (Rayleigh–Bénard, shear/viscous, Marangoni); morphoelastic buckling and wrinkling; Saffman–Taylor fingering.

21. Turing Patterns, Swift-Hohenberg Equation, Amplitude Equation
22. From Perturbation to Amplitude Equation, NWS Equation, Multiple Scale Analysis
23. Newell-Whitehead-Segel Equation, Secondary Instabilities
24. Eckhaus & Zigzag Instabilities, Phase Winding Solution
25. The Complex Ginzburg-Landau Equation
26. Stability of the Complex Ginzburg-Landau Equation
27. Phase Diagram of Complex Ginzburg-Landau Equation
28. Application of CGLE, May-Leonard Model, Evolutionary Games
29. Reaction-Diffusion Waves, Front Spreading Velocity
30. The Kuramoto Model & Synchronization Transition

Geometry & confinement¶

Curvature and weakly non-flat geometry; boundary conditions; selection by confinement and heterogeneity.

31. Mass-Conserving Reaction-Diffusion Systems, E. coli Min Oscillations
32. Mass-Redistribution Instability, Adiabatic Elimination
33. Geometric Construction of Stationary Patterns
34. Steady-State Phase Diagram & Bulk Gradient Effects
35. Bulk-Boundary Coupling & Nucleotide Exchange
36. Stationary Solutions & Emergent Saturation Effect
37. Linear Stability Analysis in Box Geometry
38. Modeling the E. coli Min System: The Skeleton Model
39. Linear Stability Analysis for Min Skeleton Model
40. The Min Switch Model & Robustness

Coarsening dynamics¶

Advanced topics on coarsening, phase field models, and interfacial dynamics.

41. Model A/B/C, Solidification Dynamics & Phase Field Model
42. Coarsening Dynamics in Mass-Conserving Systems
43. Universal Growth Laws & Logarithmic Coarsening
44. Scaling Analysis of the Coarsening Law
45. From Peaks to Mesas: Interaction via Exponential Tails
46. Arrested Coarsening via Weakly Broken Mass Conservation
47. Linear Stability Analysis of Weakly Broken Mass Conservation
48. The Mechanism of Mesa Splitting: Source-Driven Lateral Instability
49. Interrupted Coarsening of Mesas & The Gibbs-Thomson Relation
50. Surface Tension in Multicomponent Reaction-Diffusion Systems
51. The Gibbs-Thomson Relation in Reaction-Diffusion Systems

Usage¶

Each Python file corresponds to specific topics covered in the lecture series. The code serves as practical implementations of the theoretical concepts presented in the YouTube videos, developed as part of self-study and learning notes.

Here are some code output demonstrations:

Lecture 2: Dynamical Systems - Phase Transitions to Evolutionary Game Theory

Landau Free Energy & Phase Transition	Evolutionary Game Dynamics	Replicator Equation Flow

Lecture 3: Fixed Points, Saddle-Node & Pitchfork Bifurcations

Bifurcation Diagram	Phase Portrait

Lecture 4: Cusp Bifurcation, Infection Model & Transcritical Bifurcation

Cusp Bifurcation Surface	Infection Dynamics

Lecture 5: Two-Dimensional Mass-Conserving Systems

Jacobian Analysis & Phase Space

Lecture 6: Two-Dimensional Non-Conserving Systems & Oscillators

Hopf Bifurcation	Limit Cycle	Van der Pol Oscillator

Lecture 7: Poincaré-Bendixson Theorem, RhoGTPase & Rock-Paper-Scissors

RhoGTPase Oscillator	Rock-Paper-Scissors Dynamics	Heteroclinic Orbit

Lecture 8: Replicator Dynamics & Lyapunov Functions

Lyapunov Function	Diffusion Equation

Lecture 9: Ginzburg-Landau Theory & Allen-Cahn Equation (Model A)

Model A (Allen-Cahn) dynamics: Spinodal decomposition, domain formation, and coarsening

Lecture 10: Interface Dynamics & Curvature-Driven Flow

Curvature-driven shrinkage of a circular droplet following Allen-Cahn dynamics

Lecture 11: Thermodynamics & Phase Separation in Liquid Mixtures

Binary Phase Diagram

Lecture 12: Cahn-Hilliard Equation & Osmotic Pressure

Comparison: Cahn-Hilliard (Model B) vs Mass-Conserving Reaction-Diffusion dynamics

Lecture 13: Lattice Gas Model & Free Energy

Monte Carlo simulation of lattice gas model showing phase separation and free energy evolution

Lecture 14: Phase Separation Dynamics & Dispersion Relation

Crowd Oscillation Dynamics	Spectral Heatmap

Lecture 15: Nucleation Theory & Interface Dynamics

3D Cahn-Hilliard Droplets	Evolution Slices

Concentration Profile	Nucleation Barrier

Prerequisites¶

Statistical mechanics and thermodynamics
Dynamical systems theory
Differential equations
Basic knowledge of field theory (helpful but not required)

Citation¶

If you find these notes or code useful, you may cite this repository using the following BibTeX entry:

@misc{liu2025selforganization,
  author = {Liu, Zhihang},
  title  = {Self-Organization and Pattern Formation: Course Notes and Code},
  year   = {2025},
  url    = {https://zhihangliu.cn/Self-Organization-and-Pattern-Formation/},
  note   = {Course by Prof. Erwin Frey, LMU Munich, Winter 2025/2026}
}