From a large-deviations principle to the Wasserstein gradient flow : a new micro-macro passage

S. Adams, N. Dirr, M.A. Peletier, J. Zimmer

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Abstract

We study the connection between a system of many independent Brownian particles on one hand and the deterministic diffusion equation on the other. For a fixed time step h > 0, a large-deviations rate functional J h characterizes the behaviour of the particle system at t = h in terms of the initial distribution at t = 0. For the diffusion equation, a single step in the time-discretized entropy-Wasserstein gradient flow is characterized by the minimization of a functional K h . We establish a new connection between these systems by proving that J h and K h are equal up to second order in h as h ¿ 0. This result gives a microscopic explanation of the origin of the entropy-Wasserstein gradient flow formulation of the diffusion equation. Simultaneously, the limit passage presented here gives a physically natural description of the underlying particle system by describing it as an entropic gradient flow.
Original languageEnglish
Pages (from-to)791-815
JournalCommunications in Mathematical Physics
Volume307
Issue number3
DOIs
Publication statusPublished - 2011

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Gradient Flow
Large Deviation Principle
Diffusion equation
Particle System
deviation
gradients
Entropy
K-functional
entropy
Large Deviations
formulations
optimization
Formulation

Cite this

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abstract = "We study the connection between a system of many independent Brownian particles on one hand and the deterministic diffusion equation on the other. For a fixed time step h > 0, a large-deviations rate functional J h characterizes the behaviour of the particle system at t = h in terms of the initial distribution at t = 0. For the diffusion equation, a single step in the time-discretized entropy-Wasserstein gradient flow is characterized by the minimization of a functional K h . We establish a new connection between these systems by proving that J h and K h are equal up to second order in h as h ¿ 0. This result gives a microscopic explanation of the origin of the entropy-Wasserstein gradient flow formulation of the diffusion equation. Simultaneously, the limit passage presented here gives a physically natural description of the underlying particle system by describing it as an entropic gradient flow.",
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From a large-deviations principle to the Wasserstein gradient flow : a new micro-macro passage. / Adams, S.; Dirr, N.; Peletier, M.A.; Zimmer, J.

In: Communications in Mathematical Physics, Vol. 307, No. 3, 2011, p. 791-815.

Research output: Contribution to journalArticleAcademicpeer-review

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T1 - From a large-deviations principle to the Wasserstein gradient flow : a new micro-macro passage

AU - Adams, S.

AU - Dirr, N.

AU - Peletier, M.A.

AU - Zimmer, J.

PY - 2011

Y1 - 2011

N2 - We study the connection between a system of many independent Brownian particles on one hand and the deterministic diffusion equation on the other. For a fixed time step h > 0, a large-deviations rate functional J h characterizes the behaviour of the particle system at t = h in terms of the initial distribution at t = 0. For the diffusion equation, a single step in the time-discretized entropy-Wasserstein gradient flow is characterized by the minimization of a functional K h . We establish a new connection between these systems by proving that J h and K h are equal up to second order in h as h ¿ 0. This result gives a microscopic explanation of the origin of the entropy-Wasserstein gradient flow formulation of the diffusion equation. Simultaneously, the limit passage presented here gives a physically natural description of the underlying particle system by describing it as an entropic gradient flow.

AB - We study the connection between a system of many independent Brownian particles on one hand and the deterministic diffusion equation on the other. For a fixed time step h > 0, a large-deviations rate functional J h characterizes the behaviour of the particle system at t = h in terms of the initial distribution at t = 0. For the diffusion equation, a single step in the time-discretized entropy-Wasserstein gradient flow is characterized by the minimization of a functional K h . We establish a new connection between these systems by proving that J h and K h are equal up to second order in h as h ¿ 0. This result gives a microscopic explanation of the origin of the entropy-Wasserstein gradient flow formulation of the diffusion equation. Simultaneously, the limit passage presented here gives a physically natural description of the underlying particle system by describing it as an entropic gradient flow.

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