# Entropic Gradient Flows on the Wasserstein Space via Large Deviations from Thermodynamic Limits

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## Abstract

In a seminal work, Jordan, Kinderlehrer and Otto proved that the Fokker- Planck equation can be described as a gradient flow of the free energy functional in the Wasserstein space, bringing this way the statistical mechanics point of view on the diffusion phenomenon to the foreground. The aim of this thesis is to show that it is possible to retrieve this natural coupling of functional and metric, by studying the large deviations of particle models. More specically, for the case where the ambient space is the real line, it is proved that the free energy functional can be retrieved as an asymptotic Gamma-limit ( ! 0) of the rate function of a large deviation principle, minus the square of the Wasserstein distance (normalized by time). Furthermore, for a special case where both measures in the denition of the rate function are Gaussians, its value and the rate of convergence are being calculated explicitly.

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