Spatial-segregation limit for exclusion processes with two components under unbalanced reaction

Abstract

We consider exclusion processes with two types of particles which compete strongly with each other. In particular, we focus on the case where one species does not diffuse at all and killing rates of two species are given by monomials with distinct exponents. We study limiting behavior of interfaces which appear by such a strong competition. Consequently, three kinds of limiting behavior of interfaces (vanishing, moving and immovable interfaces as in [9]) are derived directly from our interacting particle system taking advantage of hydrodynamic limit procedure with singular limit for annihilation dynamics.

A one-phase Stefan problem with non-linear diffusion from highly competing two-species particle systems

“…Remark 2.2. As discussed in [8] there exists non-trivial macroscopic initial functions u 0 and v 0 and microscopic approximating sequences {u N 0 } N ∈N and {v N 0 } N ∈N satisfying the above assumptions (A1), (A2) and (A3) (though [8] imposed a restriction only for the first derivative growth, we can construct an example in a similar way).…”

“…In this subsection, we prove Lemma 4.4. The key idea of the proof is similar to previous researches ( [7], [5] and [8]) so we only give a sketch here. Define…”

“…They considered two-species exclusion processes where the same kind of particles can not stay at the same site and two different kinds of particles coexisting at the same site are killed with rate K(N ), which diverges as the scaling parameter N tends to infinity. For the case where competition between two species is unbalanced, [8] considered two-species exclusion processes which macroscopically correspond to the reaction-diffusion systems studied in [11] and derived the three kinds of interface behavior directly from the particle systems through the hydrodynamic limit. In the model of [8], one species has no diffusion dynamics even microscopically, which asserts the diffusion coefficient of macroscopic density, say v, of that kind of particles exactly equals to zero.…”

“…For the case where competition between two species is unbalanced, [8] considered two-species exclusion processes which macroscopically correspond to the reaction-diffusion systems studied in [11] and derived the three kinds of interface behavior directly from the particle systems through the hydrodynamic limit. In the model of [8], one species has no diffusion dynamics even microscopically, which asserts the diffusion coefficient of macroscopic density, say v, of that kind of particles exactly equals to zero. For another kind of research concerning singular limit for interacting particle systems, [7] considered a one-species exclusion process with creation and annihilation dynamics whose hydrodynamic limit equation becomes the Cahn-Hilliard equation with a double-well potential (Allen-Cahn type equation) and they studied a fast-reaction limit problem when the reaction rate diverges depending on the scaling parameter of the microscopic model.…”

Motion by mean curvature from Glauber-Kawasaki dynamics with speed change

“…Our aim builds further on the literature on understanding curvature flows (or PDEs in general) as hydrodynamic limits of particle systems; see e.g. [15,1,9,3,12,4]. In particular, two of these papers are close to our aim.…”