Current Reservoirs in the Simple Exclusion Process

Abstract

Abstract. We consider the symmetric simple exclusion process in the interval [−N, N ] with additional birth and death processes respectively on (N −K, N ], K > 0, and [−N, −N + K). The exclusion is speeded up by a factor N 2 , births and deaths by a factor N . Assuming propagation of chaos (a property proved in a companion paper, [3]) we prove convergence in the limit N → ∞ to the linear heat equation with Dirichlet condition on the boundaries; the boundary conditions however are not known a priori, they are obtained by solving a non linear equation. The model simulates mass transport with current reservoirs at the boundaries and the Fourier law is proved to hold.

Hydrodynamics of a particle model in contact with stochastic reservoirs

“…The equivalence of ensembles implies a one-to-one correspondence between the densities 0 ≤ ρ ≤ 1 and chemical potentials λ − = λ − ρ ∈ R (see Section 5.2 for more details). The interval (1,2) can be treated in the same way. Similarly, if we remove the interactions at x = 0, x N = 1 and look at the process in the interval (0, 1), we get its own Gibbs distributions, which converge to a probability measure ν λ,+ .…”

“…The main result of the article is Theorem 2.4 below, which states that the empirical measure π N (t) of the process {η N (t)} t≥0 defined by (1.1) converges to a measure, which density is a weak solution of the parabolic partial differential equation for t > 0, 2), ), u ∈ (0, 1),…”

“…5) were obtained in [11], where the hydrodynamic behavior of the zero range process with an asymmetry at the origin was studied. In [2] the symmetric simple exclusion process was considered in the interval [0, N ], and the reservoirs were presented in form of a birthdeath process, which was attached to the boundaries of the interval [0, N ]. The authors proved that the limiting density satisfies the linear heat equation with implicit Dirichlet boundary conditions.…”

Hydrodynamic limit in a particle system with topological interactions

“…We choose τ ζ = aζ 8 , a a positive constant whose value will be specified later. If δ > τ ζ , there is no t : t < t < t + τ ζ and the second inequality in (7.5) is automatically satisfied.…”

“…We interpret L as the generator of a system of independent walkers with "current reservoirs" which impose a positive current j at site 0 and at the edge of the configuration. See [8,9] for a comparison with the density reservoirs used in the analysis of the Fourier law. Here is a list of the main issues which are studied in this and in the other papers in this series.…”