Abstract
We prove the hydrodynamic limit of a totally asymmetric zero range process on a torus with two lanes and randomly oriented edges. The asymmetry implies that the model is non-reversible. The random orientation of the edges is constructed in a bistochastic fashion which keeps the usual product distribution stationary for the quenched zero range model. It is also arranged to have no overall drift along the Z direction, which suggests diffusive scaling despite the asymmetry present in the dynamics. Indeed, using the relative entropy method, we prove the quenched hydrodynamic limit to be the heat equation with a diffusion coefficient depending on ergodic properties of the orientation of the edges. The zero range process on this graph turns out to be non-gradient. Our main novelty is the introduction of a local equilibrium measure which decomposes the vertices of the graph into components of constant density. A clever choice of these components eliminates the non-gradient problems that normally arise during the hydrodynamic limit procedure.
Highlights
Hydrodynamic limits of interacting particle systems have a long and rich history, starting from relatively simpler cases where the model is of gradient type and the dynamics is reversible for the stationary distribution, to more complicated setups with non-gradient models and/or non-reversible dynamics
Extra complications arise when the dynamics is run in a random environment
The graph is bistochastic, which keeps the usual product distribution stationary for zero range. It has zero drift in the Z direction, which comes with diffusive scaling
Summary
Hydrodynamic limits of interacting particle systems have a long and rich history, starting from relatively simpler cases where the model is of gradient type and the dynamics is reversible for the stationary distribution, to more complicated setups with non-gradient models and/or non-reversible dynamics. Koukkous [9] used the entropy method to prove that the heat equation is the hydrodynamic limit of a symmetric zero range process on a d-dimensional torus for which the sites have random jump rates and the model is non-gradient. Goncalves and Jara [6] obtained the hydrodynamic limit of a zero range process on the d-dimensional torus with the jump rates across edges given by a random environment, making use of the entropy method combined with a corrected empirical measure and homogenisation results. It is this random orientation that our zero range process follows: nearest neighbour jumps on vertices are permitted across each edge in the direction of its orientation only.
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