Part VIII: Fractals & Quantum
Fractal growth, anomalous diffusion with memory, and quantum-inspired game theory for urban coordination.
Part Overview
Cities exhibit fractal morphology, non-Markovian memory in their evolution, and strategic coordination problems that admit quantum-inspired solutions. This part develops the mathematical machinery for all three: diffusion-limited aggregation, fractional kinetics, and quantum game theory.
Key Topics
- • DLA algorithm
- • Box-counting fractal dimension
- • Fractional Fokker-Planck: \(\partial_t^\alpha p = \nabla \cdot [D \nabla p - \mathbf{F} p]\)
- • Urban memory effects
- • Quantum game theory
- • Urban coordination
3 chapters | Geometry, memory & strategy | From fractal branches to quantum moves
Chapters
Chapter 1: DLA Fractals
Diffusion-limited aggregation as a model for urban sprawl. Random walkers stick on contact, producing fractal clusters with dimension \(d_f \approx 1.71\). Box-counting measurement and comparison with real city boundaries.
Chapter 2: Fractional Fokker-Planck
Anomalous diffusion with memory: the fractional Fokker-Planck equation \(\partial_t^\alpha p = \nabla \cdot [D \nabla p - \mathbf{F} p]\) captures subdiffusive urban dynamics where past states influence future evolution through power-law waiting times.
Chapter 3: Quantum Game Theory
Quantum-inspired strategies for urban coordination games. Entangled strategy spaces resolve classical dilemmas, yielding Pareto-optimal equilibria for infrastructure investment and land-use coordination.