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We investigate the macroscopic behavior of asymmetric attractive zero-range processes on $mathbb{Z}$ where particles are destroyed at the origin at a rate of order $N^beta$, where $beta in mathbb{R}$ and $Ninmathbb{N}$ is the scaling parameter. We prove that the hydrodynamic limit of this particle system is described by the unique entropy solution of a hyperbolic conservation law, supplemented by a boundary condition depending on the range of $beta$. Namely, if $beta geqslant 0$, then the boundary condition prescribes the particle current through the origin, whereas if $beta<0$, the destruction of particles at the origin has no macroscopic effect on the system and no boundary condition is imposed at the hydrodynamic limit.
We propose a definition o meta-stability and obtain sufficient conditions for a sequence of Markov processes on finite state spaces to be meta-stable. In the reversible case, these conditions reduce to estimates of the capacity and the measure of cer
The diffusivity $D(t)$ of finite-range asymmetric exclusion processes on $mathbb Z$ with non-zero drift is expected to be of order $t^{1/3}$. Sepp{a}lainen and Balazs recently proved this conjecture for the nearest neighbor case. We extend their resu
We prove almost sure Euler hydrodynamics for a large class of attractive particle systems on $Z$ starting from an arbitrary initial profile. We generalize earlier works by Seppalainen (1999) and Andjel et al. (2004). Our constructive approach require
We introduce and solve exactly a class of interacting particle systems in one dimension where particles hop asymmetrically. In its simplest form, namely asymmetric zero range process (AZRP), particles hop on a one dimensional periodic lattice with as
We consider a long-range version of self-avoiding walk in dimension $d > 2(alpha wedge 2)$, where $d$ denotes dimension and $alpha$ the power-law decay exponent of the coupling function. Under appropriate scaling we prove convergence to Brownian moti