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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 the 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.
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We consider the asymmetric simple exclusion process (ASEP) on the one-dimensional lattice. The particles can be created/annihilated at the boundaries with time-dependent rate. These boundary dynamics are properly accelerated. We prove the hydrodynamic limit of the particle density profile, under the hyperbolic space-time rescaling, evolves with the entropy solution to Burgers equation with Dirichlet boundary conditions. The boundary conditions are characterised by boundary entropy flux pair.
We analyze the mixing behavior of the biased exclusion process on a path of length $n$ as the bias $beta_n$ tends to $0$ as $n to infty$. We show that the sequence of chains has a pre-cutoff, and interpolates between the unbiased exclusion and the process with constant bias. As the bias increases, the mixing time undergoes two phase transitions: one when $beta_n$ is of order $1/n$, and the other when $beta_n$ is order $log n/n$.
Consider a system of particles performing nearest neighbor random walks on the lattice $ZZ$ under hard--core interaction. The rate for a jump over a given bond is direction--independent and the inverse of the jump rates are i.i.d. random variables belonging to the domain of attraction of an $a$--stable law, $0<a<1$. This exclusion process models conduction in strongly disordered one-dimensional media. We prove that, when varying over the disorder and for a suitable slowly varying function $L$, under the super-diffusive time scaling $N^{1 + 1/alpha}L(N)$, the density profile evolves as the solution of the random equation $partial_t rho = mf L_W rho$, where $mf L_W$ is the generalized second-order differential operator $frac d{du} frac d{dW}$ in which $W$ is a double sided $a$--stable subordinator. This result follows from a quenched hydrodynamic limit in the case that the i.i.d. jump rates are replaced by a suitable array ${xi_{N,x} : xinbb Z}$ having same distribution and fulfilling an a.s. invariance principle. We also prove a law of large numbers for a tagged particle.
We consider a one-dimensional, weakly asymmetric, boundary driven exclusion process on the interval $[0,N]cap Z$ in the super-diffusive time scale $N^2 epsilon^{-1}_N$, where $1ll epsilon^{-1}_N ll N^{1/4}$. We assume that the external field and the chemical potentials, which fix the density at the boundaries, evolve smoothly in the macroscopic time scale. We derive an equation which describes the evolution of the density up to the order $epsilon_N$.
We consider the simple exclusion process on Z x {0, 1}, that is, an horizontal ladder composed of 2 lanes. Particles can jump according to a lane-dependent translation-invariant nearest neighbour jump kernel, i.e. horizontally along each lane, and vertically along the scales of the ladder. We prove that generically, the set of extremal invariant measures consists of (i) translation-invariant product Bernoulli measures; and, modulo translations along Z: (ii) at most two shock measures (i.e. asymptotic to Bernoulli measures at $pm$$infty$) with asymptotic densities 0 and 2; (iii) at most three shock measures with a density jump of magnitude 1. We fully determine this set for certain parameter values. In fact, outside degenerate cases, there is at most one shock measure of type (iii). The result can be partially generalized to vertically cyclic ladders with arbitrarily many lanes. For the latter, we answer an open question of [5] about rotational invariance of stationary measures.
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