Hydrodynamics and Hydrostatics for a Class of Asymmetric Particle Systems with Open Boundaries

Abstract

We consider attractive particle systems in Z d with product invariant measures. We prove that when particles are restricted to a subset of Z d , with birth and death dynamics at the boundaries, the hydrodynamic limit is given by the unique entropy solution of a conservation law, with boundary conditions in the sense of Bardos et al. ([7]). For the hydrostatic limit between parallel hyperplanes, we prove a multidimensional version of the phase diagram conjectured in [38], and show that it is robust with respect to perturbations of the boundaries.AMS 2000 subject classifications. 60K35, 82C22, 82C26; 35L65, 35L67, 35L50.

Macroscopic fluctuation theory

“…In this case the hydrodynamics is given by hyperbolic conservation laws and not by driven diffusive equations. We refer to (Kipnis and Landim, 1999) for periodic boundary conditions and to (Bahadoran, 2012a) for the case of models with reservoirs.…”

The Key Parameters that Govern Translation Efficiency

“…The purpose of the hydrodynamic limit is to describe the local evolution of the macroscopic particle density in the large system limit. As such, it does not explicitly rely on the precise formalism by which particles enter and exit the lattice at the boundaries (which will only later be needed to impose boundary conditions on the resulting PDE (Bahadoran, 2012)). In particular, we are free to choose periodic boundary conditions for our limiting procedure without changing the resulting PDE (Schönherr and Schütz, 2004).…”

On the scaling limits of weakly asymmetric bridges

“…3. If α ∈ (0, 1), m is the solution of the following Hamilton-Jacobi equation 2 , m(t, 0) = m(t, 1) = 0 , m(0, ·) = m 0 (·) .…”

“…towards the deterministic process ρ(t, dx) = η(t, x)dx where η is the entropy solution of the Burgers equation with the above Dirichlet conditions. This convergence result is taken from [29], it is in the flavour of the works of Rezakhanlou [36] and Bahadoran [2].…”