[Paper Review] Hydrodynamics of a driven lattice gas with open boundaries: the asymmetric simple exclusion

O. Benois, R. Esposito+2 more|ArXiv.org|Feb 8, 2002

Stochastic processes and statistical mechanics9 references3 citations

TL;DR

This paper establishes the hydrodynamic limit of the asymmetric simple exclusion process (ASEP) in d≥3 dimensions with open boundaries, showing that the rescaled empirical density field converges to the solution of the viscous Burgers equation with boundary conditions. The key result is the convergence under an incompressible scaling where initial and boundary densities differ from 1/2 by order ε, capturing finite dissipation effects on the diffusive time scale ε⁻².

ABSTRACT

We consider the asymmetric simple exclusion process in $d\ge 3$ with open boundaries. The particle reservoirs of constant densities are modeled by birth and death processes at the boundary. We prove that, if the initial density and the densities of the boundary reservoirs differ for order of $ε$ from 1/2, the density empirical field, rescaled as $ε^{-1}$, converges to the solution of the initial-boundary value problem for the viscous Burgers equation in a finite domain with given density on the boundary.

Motivation & Objective

• To establish the hydrodynamic limit of the asymmetric simple exclusion process (ASEP) in d≥3 with open boundaries.
• To analyze the system's behavior under the incompressible scaling where initial and boundary densities are perturbations of 1/2 by order ε.
• To derive the macroscopic evolution equation for the density field in the presence of both bulk asymmetry and boundary reservoirs.
• To overcome challenges from non-reversibility in both bulk dynamics and boundary conditions, which prevent the use of standard gradient or reversible system techniques.
• To extend the hydrodynamic limit to general convex domains using a boundary generator that fixes the density profile via non-reversible jump rates.

Proposed method

• The system is defined on a finite cylinder Λε with periodic boundary conditions in all but the x₁ direction, and open boundaries at x₁ = ±ε⁻¹.
• Boundary dynamics are modeled via birth and death processes with rates proportional to δ₁(1/2 ± εb(εx)), ensuring reservoir densities of 1/2 + εb±.
• The initial measure is local equilibrium with a density profile ρ₀(εx) = 1/2 + εm₀(εx), a perturbation of 1/2 by order ε.
• The rescaled empirical field is defined as ε^{d−1}∑(η_{ε⁻²t}(x) − 1/2)δ(εx), which converges weakly to the solution of the viscous Burgers equation.
• The proof relies on relative entropy methods and estimates of Dirichlet forms, with careful control of boundary contributions via integration by parts and Taylor expansion.
• Generalization to smooth convex domains is achieved by defining a boundary generator that depends on the normal component δ·n, ensuring correct macroscopic density at the boundary.

Experimental results

Research questions

• RQ1 How does the asymmetric simple exclusion process (ASEP) with open boundaries converge to a macroscopic PDE in d≥3 dimensions?
• RQ2 What is the effect of a small perturbation (order ε) in initial and boundary densities on the hydrodynamic limit?
• RQ3 Can the hydrodynamic limit be established for non-reversible systems with both bulk asymmetry and non-reversible boundary dynamics?
• RQ4 How does the viscous Burgers equation emerge as the macroscopic equation in the incompressible limit?
• RQ5 Can the hydrodynamic limit be extended from a cylindrical geometry to general convex domains?

Key findings

• The rescaled empirical density field converges weakly to the solution of the viscous Burgers equation with initial condition m₀ and boundary condition b(u) on the macroscopic boundary.
• The macroscopic equation is ∂ₜm = δ·∇(m²) + ∑Dᵢⱼ∂²ᵤᵢ,ᵤⱼm, where D is a positive definite diffusion matrix derived from the Green-Kubo formula.
• The convergence holds under the incompressible scaling, where initial and boundary density deviations from 1/2 are of order ε, ensuring finite dissipation effects on the diffusive time scale ε⁻².
• The proof establishes a law of large numbers for the empirical field using relative entropy and Dirichlet form estimates, even in the presence of non-reversible bulk and boundary dynamics.
• For general convex domains, a boundary generator is constructed using |δ·n| and the sign of δ·n to fix the density profile 1/2 + εb(x) at the boundary.
• The method controls boundary contributions via cancellation and integration by parts, avoiding reliance on reversibility of the boundary generator.