Large Deviations and Gradient Flows for the Brownian One-Dimensional Hard-Rod System

Mark Peletier (Corresponding author), Nir Gavish, Pierre Nyquist

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We study a system of hard rods of finite size in one space dimension, which move by Brownian noise while avoiding overlap. We consider a scaling in which the number of particles tends to infinity while the volume fraction of the rods remains constant; in this limit the empirical measure of the rod positions converges almost surely to a deterministic limit evolution. We prove a large-deviation principle on path space for the empirical measure, by exploiting a one-to-one mapping between the hard-rod system and a system of non-interacting particles on a contracted domain. The large-deviation principle naturally identifies a gradient-flow structure for the limit evolution, with clear interpretations for both the driving functional (an ‘entropy’) and the dissipation, which in this case is the Wasserstein dissipation. This study is inspired by recent developments in the continuum modelling of multiple-species interacting particle systems with finite-size effects; for such systems many different modelling choices appear in the literature, raising the question how one can understand such choices in terms of more microscopic models. The results of this paper give a clear answer to this question, albeit for the simpler one-dimensional hard-rod system. For this specific system this result provides a clear understanding of the value and interpretation of different modelling choices, while giving hints for more general systems.

Original languageEnglish
Pages (from-to)71-121
Number of pages51
JournalPotential Analysis
Issue number1
Publication statusPublished - Jan 2023

Bibliographical note

Funding Information:
The authors would like to thank Jim Portegies, Oliver Tse, Jasper Hoeksema, Georg Prokert, and Frank Redig for several interesting discussions and insightful remarks. This work was partially supported by NWO grant 613.009.101.

Publisher Copyright:
© 2021, The Author(s).


  • Brownian motion
  • Continuum limit
  • Hard-rod
  • Hard-sphere
  • Large deviations
  • Steric interaction
  • Volume exclusion


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