TY - JOUR
T1 - From a Large-Deviations Principle to the Wasserstein Gradient Flow
T2 - A New Micro-Macro Passage
AU - Adams, Stefan
AU - Dirr, Nicolas
AU - Peletier, Mark A.
AU - Zimmer, Johannes
PY - 2011/11
Y1 - 2011/11
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 Jh 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 Kh. We establish a new connection between these systems by proving that Jh and Kh 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 Jh 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 Kh. We establish a new connection between these systems by proving that Jh and Kh 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
U2 - 10.1007/s00220-011-1328-4
DO - 10.1007/s00220-011-1328-4
M3 - Article
AN - SCOPUS:80053912873
SN - 0010-3616
VL - 307
SP - 791
EP - 815
JO - Communications in Mathematical Physics
JF - Communications in Mathematical Physics
IS - 3
ER -