Communications in Mathematical Sciences

Volume 17 (2019)

Number 3

A variational structure for interacting particle systems and their hydrodynamic scaling limits

Pages: 739 – 780



Marcus Kaiser (Department of Mathematical Sciences, University of Bath, United Kingdom)

Robert L. Jack (Dept. of Applied Mathematics & Theoretical Physics, University of Cambridge, United Kingdom; Dept. of Chemistry, University of Cambridge,United Kingdom; and Dept. of Physics, University of Bath, United Kingdom)

Johannes Zimmer (Department of Mathematical Sciences, University of Bath, United Kingdom)


We consider hydrodynamic scaling limits for a class of reversible interacting particle systems, which includes the symmetric simple exclusion process and certain zero-range processes. We study a (non-quadratic) microscopic action functional for these systems. We analyse the behaviour of this functional in the hydrodynamic limit and we establish conditions under which it converges to the (quadratic) action functional of Macroscopic Fluctuation Theory. We discuss the implications of these results for rigorous analysis of hydrodynamic limits.


interacting particle systems, macroscopic fluctuation theory, large deviations, action functionals, $\Gamma$-convergence

2010 Mathematics Subject Classification

35Q35, 76M28, 82C22

Copyright © 2019 M. Kaiser, R. L. Jack and J. Zimmer

M.K. is supported by a scholarship from the EPSRC Centre for Doctoral Training in Statistical Applied Mathematics at Bath (SAMBa), under the project EP/L015684/1.

J.Z. gratefully acknowledges funding by the EPSRC through project EP/K027743/1, the Leverhulme Trust (RPG-2013-261) and a Royal Society Wolfson Research Merit Award.

Received 10 July 2018

Accepted 22 January 2019

Published 30 August 2019