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

Abstract

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.

Jump processes as generalized gradient flows

“…Some generalized gradient-flow structures of evolution equations are generated by the large deviations of an underlying, more microscopic stochastic process [1,2,22,46,59,60]. This explains the origin and interpretation of such structures, and it can be used to identify hitherto unknown gradient-flow structures [36,71].…”

GENERIC framework for reactive fluid flows

“…We point out that evolution equations () and () are rate equations, where the Onsager operator $${\skew{1.5}\bar{{\mathbb{K}}}}$$, or more generally, the non‐linear operator $$\mathrm{D}_{{\bar{\bm{\xi }}}}\bar{\Psi }^*({\bar{\bm{q}}};\cdot )$$ maps the thermodynamic driving force $$\mathrm{D}\bar{{\mathcal {S}}}({\bar{\bm{q}}})\in {\mathcal {Q}}^{*}$$ to a rate $$\partial _t{\bar{\bm{q}}}\in \bar{{\mathcal {Q}}}$$. Other examples for gradient systems can be found for example in [12] for the porous medium equation, in [9, 10] for complex fluids, in [39, 58–60] for (slow & fast) reaction‐diffusion systems with detailed balance, in [61] for the Fokker–Planck equation, and in [62, 63] in the context of large deviations. Mathematical properties of gradient flows are discussed in [11].…”

Importance Sampling for the Empirical Measure of Weakly Interacting Diffusions

“…On the other hand, in recent years the HJB equation on Wasserstein space of [73] (along with the related equations in [21,67]) has received an immense amount of attention both in terms of numerical applications [20,22,47,54,55,60,66] and theoretical results [5,18,19,26,27,56,65,73,80]. The Dawson-Gärtner rate function has previously been related to the theory of mean field games and control through the observation that it can be viewed in terms of derivatives of the free energy associated to the limiting McKean-Vlasov equation viewed as a gradient flow on Wasserstein space in some settings [1,2,4,[41][42][43][44]52]. However, to our knowledge, this paper is the first time that the connection between the LDP rate function (10) and the HJB equation on Wasserstein space (23) has been made explicit in the literature (see Remark 2.2).…”