Part X: Traffic as Lattice Gas
Traffic flow modeled as particles on a lattice — from Langmuir adsorption kinetics to TASEP exact solutions, LWR conservation laws, and Cahn-Hilliard spinodal decomposition of jams.
Part Overview
Traffic is the quintessential non-equilibrium lattice gas. This part builds the theory from three angles: Langmuir adsorption maps directly onto the fundamental diagram, TASEP provides exact solutions for boundary-driven flow with three phases, and the LWR PDE framework connects to Cahn-Hilliard spinodal decomposition where traffic jams emerge as phase-separated domains.
Key Topics
- • Langmuir\(\to\)fundamental diagram mapping
- • Langmuir-Hinshelwood intersections
- • TASEP three phases (LD/HD/MC)
- • Matrix Ansatz: \(DE - ED = D + E\)
- • LWR PDE
- • Rankine-Hugoniot shocks
- • Cahn-Hilliard jam spacing
3 chapters | Particles, phases & shocks | From adsorption to jam formation
Chapters
Chapter 1: Langmuir \(\to\) Fundamental Diagram
The exact mapping between Langmuir adsorption kinetics and traffic flow. Adsorption rate \(\to\) inflow, desorption \(\to\) outflow, surface coverage \(\theta \to \rho\). Langmuir-Hinshelwood intersections identify critical density thresholds.
Chapter 2: TASEP Phase Diagram
The totally asymmetric simple exclusion process with exact solution via the Matrix Ansatz \(DE - ED = D + E\). Three phases emerge: low density (LD), high density (HD), and maximal current (MC), with first- and second-order phase boundaries.
Chapter 3: LWR & Spinodal Decomposition
The Lighthill-Whitham-Richards conservation law \(\partial_t \rho + \partial_x[\rho v(\rho)] = 0\) and Rankine-Hugoniot shock conditions. Cahn-Hilliard dynamics explain spontaneous jam formation as spinodal decomposition, predicting characteristic jam spacing.