Hydrodynamics and relaxation limit for multilane exclusion process and related hyperbolic systems

TL;DR

This paper studies the hydrodynamic limits of a multilane exclusion process, showing that the total density follows a scalar conservation law and analyzing the relaxation limits of coupled hyperbolic systems, with detailed flux computations for two lanes.

Contribution

It establishes the hydrodynamic behavior of multilane exclusion processes as relaxation limits of coupled hyperbolic systems, including detailed flux analysis for two-lane models.

Findings

Total density governed by scalar conservation law

Hydrodynamic limits as relaxation of hyperbolic systems

Detailed flux transition analysis for two-lane model

Abstract

We investigate the hydrodynamic behavior and local equilibrium of the multilane exclusion process, whose invariant measures were studied in our previous paper \cite{mlt1a}. The dynamics on each lane follows a hyperbolic time scaling, whereas the interlane dynamics has an arbitrary time scaling. We prove the following: (i) the hydrodynamic behavior of the global density (i.e. summed over all lanes) is governed by a scalar conservation law; (ii) the latter, as well as the limit of individual lanes, is the relaxation limit of a weakly coupled hyperbolic system of balance laws that approximates the particle system. For the hydrodynamic limit, to highlight new phenomena arising in our model, a precise computation of the flux function, with the transitions between different possible shapes (and a physical interpretation thereof), is given for the two-lane model.