## 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.

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.

Original language | English |
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Article number | 23 |

Number of pages | 29 |

Journal | Electronic Journal of Probability |

Volume | 27 |

Early online date | 11 Feb 2022 |

DOIs | |

Publication status | E-pub ahead of print - 11 Feb 2022 |

### Bibliographical note

Funding Information:*Felix Maxey-Hawkins was supported by an EPSRC studentship and Márton Balázs was partially supported by the EPSRC EP/R021449/1 Standard Grant of the UK. This study did not involve any underlying data. †University of Bristol, UK. E-mail: [email protected],[email protected] https://www.maths.bris.ac.uk/~mb13434/

Publisher Copyright:

© 2022, Institute of Mathematical Statistics. All rights reserved.

## Keywords

- Zero range process ; Hydrodynamic limit ; Random environment ; Relative entropy