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Large deviations and gradient flows for the Brownian one-dimensional hard-rod system
Technion.
KTH, School of Engineering Sciences (SCI), Mathematics (Dept.), Mathematical Statistics.ORCID iD: 0000-0001-8702-2293
Eindhoven University of Technology.
(English)Manuscript (preprint) (Other academic)
Abstract [en]

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 shorter 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 onedimensional 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.

Keywords [en]
Steric interaction, volume exclusion, hard-sphere, hard-rod, large deviations, continuum limit, Brownian motion
National Category
Probability Theory and Statistics
Research subject
Mathematics; Physics
Identifiers
URN: urn:nbn:se:kth:diva-266124OAI: oai:DiVA.org:kth-266124DiVA, id: diva2:1381594
Available from: 2019-12-23 Created: 2019-12-23 Last updated: 2020-01-02Bibliographically approved

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https://arxiv.org/abs/1909.02054

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Nyquist, Pierre

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