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

TL;DR

This work derives rigorous hydrodynamic and relaxation limits for a multilane exclusion process with decoupled longitudinal and interlane dynamics, revealing that the global density obeys a scalar conservation law with a flux $G$ obtained by projecting lane-wise fluxes onto an equilibrium manifold. Lane densities relax to a Maxwellian-type closure on the manifold $\mathcal{F}$, yielding a hyperbolic system with weak coupling that captures interlane interactions. A detailed analysis of the two-lane flux exposes rich phase behavior, including nonmonotone flux transitions and non-differentiable features arising from lane competition, while the many-lane limit connects to a Burgers-type equation with a macroscopic flux $F$. The proofs blend entropy methods, macroscopic stability refinements, and block estimates adapted to multilane geometry, providing a comprehensive framework for hyperbolic relaxation in coupled exclusion processes with potential applications to traffic-flow and multi-species transport. The results illuminate how interlane coupling shapes macroscopic transport, with implications for modeling overtaking and lane-structured traffic dynamics under strongly coupled, yet weakly interacting, regimes.

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.

Figures (12)

• Figure 1: Phase diagram 1:$\tilde{d}_1$, $\tilde{d}_0$, $\bar{d}_1$ are defined in \ref{['critical_d']}. The rightmost dashed line is $d=1$. In the white region, the flux function is strictly concave; in the yellow (resp. pink) region, it has one (resp. two) inflexion points. Intervals $(-\infty;\tilde{d}_1$), $(\tilde{d}_1;\tilde{d}_0)$, $(\tilde{d}_0;1)$ and $(1;+\infty)$ respectively correspond to cases (i), (ii), (iii) and (iv)--(v) in Theorem \ref{['thm:phase_trans']}. Curves $r_3(d)$ and $r_4(d)$ are the lower and upper branches of the solution $r$ of $g(r)=0$ (green curve), cf. \ref{['ineq:G2-2']}, with $r_4(d)\to+\infty$ as $d\to 1-$. Curve $\bar{r}_1(d)$ is defined by \ref{['def:bar-d1']}--\ref{['ineq:G2-rho_0-2']}. The blue line at the bottom is $r=1$.
• Figure 2: Phase diagram 2: zooming around the intersection of $r_4(d)$ with $\bar{r}(d)$ reveals the anomalous region between $d=\tilde{d}_1$ and $d=\tilde{d}_0$.
• Figure 3: Phase diagram 3: zooming around the anomalous region.
• Figure 4: Flux function 1:$d=0.5$, $\gamma_0=\gamma_1=0.5$, symmetric graph. Top concave curve: $r=1$; bottom double-bump curve $r=+\infty$. Unique phase transition from $0$ to $2$ inflexion points occurs at $r=\bar{r}_1(0.5)=(2+\sqrt{3})^2\simeq 14$ (cf. fig 1) inside the pink layer.
• Figure 5: Flux function 2:$d=0.8$, $\gamma_0=0.8$, $\gamma_1=0.2$. Top concave curve: $r=1$; bottom double-bump curve $r=+\infty$. Unique phase transition from $0$ to $2$ inflexion points occurs at $r=\bar{r}_1(0.8)\simeq 9.4$ (pink layer).

Theorems & Definitions (75)

• Definition 2.1
• Remark 2.1
• Definition 2.2
• Remark 2.2
• Proposition 2.1
• Theorem 2.1
• Remark 2.3
• Remark 3.1
• Theorem 3.1
• Definition 3.1