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Register a new userSupercritical behavior of asymmetric zero-range process with sitewise disorder
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We establish necessary and sufficient conditions for weak convergence to the upper invariant measure for asymmetric nearest neighbour zero range processes with non homogeneous jump rates. The class of environments considered is close to that considered by Andjel, Ferrari, Guiol and Landim, while our class of processes is broader. We also give a simpler proof of a result of Ferrari and Sisko with weaker assumptions.
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We discuss necessary and sufficient conditions for the convergence of disordered asymmetric zero-range process to the critical invariant measures.
We study asymmetric zero-range processes on Z with nearest-neighbour jumps and site disorder. The jump rate of particles is an arbitrary but bounded nondecreasing function of the number of particles. For any given environment satisfying suitable averaging properties, we establish a hydrodynamic limit given by a scalar conservation law including the domain above critical density, where the flux is shown to be constant.
We survey our recent articles dealing with one dimensional attractive zero range processes moving under site disorder. We suppose that the underlying random walks are biased to the right and so hyperbolic scaling is expected. Under the conditions of our model the process admits a maximal invariant measure. The initial focus of the project was to find conditions on the initial law to entail convergence in distribution to this maximal distribution, when it has a finite density. Somewhat surprisingly, necessary and sufficient conditions were found. In this part hydrody-namic results were employed chiefly as a tool to show distributional convergence but subsequently we developed a theory for hydrodynamic limits treating profiles possessing densities that did not admit corresponding equilibria. Finally we derived strong local equilibrium results.
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 calculate the exact stationary distribution of the one-dimensional zero-range process with open boundaries for arbitrary bulk and boundary hopping rates. When such a distribution exists, the steady state has no correlations between sites and is uniquely characterized by a space-dependent fugacity which is a function of the boundary rates and the hopping asymmetry. For strong boundary drive the system has no stationary distribution. In systems which on a ring geometry allow for a condensation transition, a condensate develops at one or both boundary sites. On all other sites the particle distribution approaches a product measure with the finite critical density rho_c. In systems which do not support condensation on a ring, strong boundary drive leads to a condensate at the boundary. However, in this case the local particle density in the interior exhibits a complex algebraic growth in time. We calculate the bulk and boundary growth exponents as a function of the system parameters.
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