Exclusion process with slow boundary

Abstract

We study the hydrodynamic and the hydrostatic behavior of the Simple Symmetric Exclusion Process with \emph{slow boundary}. The term \emph{slow boundary} means that particles can be born or die at the boundary sites, at a rate proportional to $N^{-\theta }$, where $\theta >0$ and $N$ is the scaling parameter. In the bulk, the particles exchange rate is equal to $1$. In the hydrostatic scenario, we obtain three different linear profiles, depending on the value of the parameter $\theta$; in the hydrodynamic scenario, we obtain that the time evolution of the spatial density of particles, in the diffusive scaling, is given by the weak solution of the heat equation, with boundary conditions that depend on $\theta$. If $\theta \in (0,1)$, we get Dirichlet boundary conditions, (which is the same behavior if $\theta =0$, see \cite{f}); if $\theta =1$, we get Robin boundary conditions; and, if $\theta \in (1,\infty )$, we get Neumann boundary conditions.

25 pages, 2 figures

References in corpus (3)

• Non equilibrium steady states: fluctuations and large deviations of the density and of the current
• Stationary and Nonequilibrium Fluctuations in Boundary Driven Exclusion Processes
• Hydrodynamic limit of gradient exclusion processes with conductances

Cited by in corpus (28)

• Slow to fast infinitely extended reservoirs for the symmetric exclusion process with long jumps
• Large deviations in the symmetric simple exclusion process with slow boundaries
• Hydrodynamics of Porous Medium Model with slow reservoirs
• Symmetric inclusion process with slow boundary: hydrodynamics and hydrostatics
• Hydrodynamic limit of boundary driven exclusion processes with nonreversible boundary dynamics
• Hydrostatic limit for exclusion process with slow boundary revisited
• Hydrodynamics for SSEP with non-reversible slow boundary dynamics: Part I, the critical regime and beyond
• Sharp Convergence to Equilibrium for the SSEP with Reservoirs
• Current fluctuations in a semi-infinite line
• Large Deviations in the Symmetric Simple Exclusion Process with Slow Boundaries: A Hydrodynamic Perspective
• Hydrodynamics for one-dimensional ASEP in contact with a class of reservoirs
• The boundary driven zero-range process
• Thermodynamics of nonequilibrium driven diffusive systems in mild contact with boundary reservoirs
• Moderate Deviations for the SSEP with a Slow Bond
• Hydrodynamical behavior for the generalized symmetric exclusion with open boundary
• Diffusive to super-diffusive behavior in boundary driven exclusion
• Steady state large deviations for one-dimensional, symmetric exclusion processes in weak contact with reservoirs
• Large deviations of current for the symmetric simple exclusion process on a semi-infinite line, and on an infinite line with a slow bond
• Scaling Limits of Random Walks, Harmonic Profiles, and Stationary Non-Equilibrium States in Lipschitz Domains
• Hydrodynamic limit for asymmetric simple exclusion with accelerated boundaries
• Asymptotic behavior of density in the boundary-driven exclusion process on the Sierpinski gasket
• On energy solutions to stochastic Burgers equation
• Energy estimates and convergence of weak solutions of the porous medium equation
• Hydrodynamic limit for a boundary driven super-diffusive symmetric exclusion
• Local ergodicity in the exclusion process on an infinite weighted graph
• Hydrodynamics for asymmetric simple exclusion on a finite segment with Glauber-type source
• Cutoffs for exclusion processes on graphs with open boundaries
• The cutoff profile for exclusion processes in any dimension