We construct an exclusion process with Bernoulli product invariant measure and having, in the diffusive hydrodynamic scaling, a non symmetric diffusion matrix, that can be explicitly computed. The antisymmetric part does not affect the evolution of t
he density but it is relevant for the evolution of the current. In particular because of that, the Ficks law is violated in the diffusive limit. Switching on a weakly external field we obtain a symmetric mobility matrix that is related just to the symmetric part of the diffusion matrix by the Einstein relation. We show that this fact is typical within a class of generalized gradient models. We consider for simplicity the model in dimension $d=2$, but a similar behavior can be also obtained in higher dimensions.
In this review we discuss the weak KPZ universality conjecture for a class of 1-d systems whose dynamics conserves one or more quantities. As a prototype example for the former case, we will focus on weakly asymmetric simple exclusion processes, for
which the density is preserved and the equilibrium fluctuations are shown to cross from the Edwards-Wilkinson universality class to the KPZ universality class. The crossover depends on the strength of the asymmetry. For the latter case, we will present an exclusion process with three species of particles, known as the ABC model, for which we aim to prove the convergence to a system of coupled stochastic Burgers equations, i.e. gradien
In [AAV] Amir, Angel and Valk{o} studied a multi-type version of the totally asymmetric simple exclusion process (TASEP) and introduced the TASEP speed process, which allowed them to answer delicate questions about the joint distribution of the speed
of several second-class particles in the TASEP rarefaction fan. In this paper we introduce the analogue of the TASEP speed process for the totally asymmetric zero-range process (TAZRP), and use it to obtain new results on the joint distribution of the speed of several second-class particles in the TAZRP with a reservoir. There is a close link from the speed process to questions about stationary distributions of multi-ty
We prove the hydrodynamic limit for the symmetric exclusion process with long jumps given by a mean zero probability transition rate with infinite variance and in contact with infinitely many reservoirs with density $alpha$ at the left of the system
and $beta$ at the right of the system. The strength of the reservoirs is ruled by $kappa$N --$theta$ > 0. Here N is the size of the system, $kappa$ > 0 and $theta$ $in$. Our results are valid for $theta$ $le$ 0. For $theta$ = 0, we obtain a collection of fractional reaction-diffusion equations indexed by the parameter $kappa$ and with Dirichlet boundary conditions. Their solutions also depend on $kappa$. For $theta$ < 0, the hydrodynamic equation corresponds to a reaction equation with Dirichlet boundary conditions. The case $theta$ > 0 is still open. For that reason we also analyze the convergence of the unique weak solution of the equation in the case $theta$ = 0 when we send the parameter $kappa$ to zero. Indeed, we conjecture that the limiting profile when $kappa$ $rightarrow$ 0 is the one that we should obtain when taking small values of $theta$ > 0.
We consider a harmonic chain perturbed by an energy conserving noise and show that after a space-time rescaling the energy-energy correlation function is given by the solution of a skew-fractional heat equation with exponent 3/4.
