TY - JOUR

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

AU - Adams, S

AU - Dirr, Nicolas

AU - Peletier, M

AU - Zimmer, Johannes

PY - 2011/11/1

Y1 - 2011/11/1

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.

UR - http://www.scopus.com/inward/record.url?scp=80053912873&partnerID=8YFLogxK

UR - http://dx.doi.org/10.1007/s00220-011-1328-4

U2 - 10.1007/s00220-011-1328-4

DO - 10.1007/s00220-011-1328-4

M3 - Article

SN - 0010-3616

VL - 307

SP - 791

EP - 815

JO - Communications in Mathematical Physics

JF - Communications in Mathematical Physics

IS - 3

ER -