Hydrodynamic limit of the zero range process on a randomly oriented graph

Marton Balazs, Felix M Maxey-Hawkins

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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.
Original languageEnglish
Article number23
Number of pages29
JournalElectronic Journal of Probability
Volume27
Early online date11 Feb 2022
DOIs
Publication statusE-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

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