On the Asymmetric Zero-Range in the Rarefaction Fan

Abstract

ABSTRACT. We consider one-dimensional asymmetric zero-range processes starting from a step decreasing profile leading, in the hydrodynamic limit, to the rarefaction fan of the associated hydrodynamic equation. Under that initial condition, and for totally asymmetric jumps, we show that the weighted sum of joint probabilities for second class particles sharing the same site is convergent and we compute its limit. For partially asymmetric jumps, we derive the Law of Large Numbers for a second class particle, under the initial configuration in which all positive sites are empty, all negative sites are occupied with infinitely many first class particles and there is a single second class particle at the origin. Moreover, we prove that among the infinite characteristics emanating from the position of the second class particle it picks randomly one of them. The randomness is given in terms of the weak solution of the hydrodynamic equation, through some sort of renormalization function. By coupling the constant-rate totally asymmetric zero-range with the totally asymmetric simple exclusion, we derive limiting laws for more general initial conditions.

Exclusion and Zero-Range in the Rarefaction Fan

“…As a consequence of Theorem 3 and a simple modification of the coupling described in the previous section (see [3] for details) the following result holds. Corollary 1.…”

“…In this section we present a coupling between the TASEP and the TAZEP in the presence of one second class particle. It uses the particle to particle coupling introduced in [3] and it relates the TAZRP and TASEP in such a way that the position of the second class particle in the TAZRP corresponds to the flux of holes that crossover the second class particle in the TASEP. Now we explain the relation between the configurations of the two processes.…”

Well-posedness of the Dean--Kawasaki and the nonlinear Dawson--Watanabe equation with correlated noise

“…A concrete example is given by the asymmetric zero range process. In this case, we have that Φ is the mean-local jump rate, a 2 (ρ) = Φ(ρ), and ν(ρ) = Φ(ρ), see, for example, Gonçalves [40]. That is,…”

The boundary driven zero-range process

“…In this model, a particle leaves a site according to a jump rate g(k) that only depends on the number of particles, k, in that site. The zero-range process has been mostly studied in infinite lattices (see [2,3,18,19,26]) and in discrete torus (see [20,21,4,9,22] and the references therein). In the present work we consider the process defined in the finite lattice I N = {1, .…”