Part XI: ASEP + Langmuir
The richest exactly solvable non-equilibrium model — general ASEP with Langmuir on/off kinetics producing five phases, quantum group symmetry, domain wall localization, and Tracy-Widom KPZ universality.
Part Overview
Combining ASEP with Langmuir kinetics produces the most complete exactly solvable model of driven diffusive systems. The coupling ratio \(\Omega\) controls the competition between boundary-driven currents and bulk attachment/detachment, yielding a five-phase diagram. Bethe ansatz and quantum group \(U_q(\mathfrak{sl}_2)\) symmetry provide exact solutions, while domain wall theory localizes shocks and Tracy-Widom statistics connect to KPZ universality.
Key Topics
- • General ASEP master equation
- • Langmuir coupling ratio \(\Omega\)
- • Five-phase diagram
- • Bethe ansatz
- • \(U_q(\mathfrak{sl}_2)\) quantum group
- • Domain wall theory
- • Tracy-Widom GUE distribution
3 chapters | Exact solutions & universality | From master equations to Tracy-Widom
Chapters
Chapter 1: General ASEP
The general asymmetric simple exclusion process with both forward and backward hopping. Master equation formulation, Bethe ansatz solution, and the \(U_q(\mathfrak{sl}_2)\) quantum group symmetry underlying integrability.
Chapter 2: Five-Phase Diagram
Coupling ASEP to Langmuir kinetics with ratio \(\Omega\) produces five distinct phases beyond the standard three. Domain wall theory predicts shock localization and phase boundaries. Mean-field analysis with exact corrections from the matrix product ansatz.
Chapter 3: Tracy-Widom & KPZ
Current fluctuations in ASEP obey Tracy-Widom GUE statistics: \(P(J \leq s) \to F_2(s)\) as \(t \to \infty\). This connects particle transport to KPZ universality, random matrix theory, and the longest increasing subsequence problem.