Gradient flow approach to local mean-field spin systems

TL;DR

This paper develops a gradient flow framework for local mean-field spin systems, enabling the derivation of large deviation principles and hydrodynamic limits through Wasserstein-based techniques.

Contribution

It introduces a novel gradient flow representation for evolution equations with a non-evolving parameter, linked to local mean-field spin systems, extending Wasserstein methods.

Findings

Established a gradient flow representation for the system.

Proved a large deviation principle for the empirical process.

Demonstrated the hydrodynamic limit using established approaches.

Abstract

It is well-known that many diffusion equations can be recast as Wasserstein gradient flows. Moreover, in recent years, by modifying the Wasserstein distance appropriately, this technique has been transferred to further evolution equations and systems. In this paper we establish such a gradient flow representation for evolution equations that depend on a non-evolving parameter. These equations are connected to a local mean-field interacting spin system. We then use this gradient flow representation to prove a large deviation principle for the empirical process associated to this system. This is done by using a criterion that was established by Max Fathi in 2016. Finally, the corresponding hydrodynamic limit is shown by using an approach that was initiated by Sandier and Serfaty in 2004.