Free energy Wasserstein gradient flow and their particle counterparts: toy model, (degenerate) PL inequalities and exit times

TL;DR

The paper investigates Wasserstein gradient flows for mean-field free energies with entropic penalties, focusing on long-time and metastable behavior in both the mean-field PDE and the associated $N$-particle Langevin system. Through carefully designed toy models, it establishes sharp equivalences between PL inequalities for the mean-field free energy and uniform LSIs for the $N$-particle Gibbs measures, and analyzes degenerate minimizers via non-linear Łojasiewicz-type inequalities. A central contribution is the application to the continuous Curie–Weiss model at critical temperature, where degeneracy yields polynomial-in-$N$ convergence with uniform-in-time propagation of chaos and sharp exponents, complemented by Eyring–Kramers-type formulas for transition times. The work also develops local inequality frameworks and practical estimates to handle metastability in finite but large systems, providing benchmarks for higher-dimensional mean-field problems and potential guidance for optimization and sampling in high dimensions. The results offer both rigorous probabilistic-analytic tools and physically meaningful benchmarks to understand how finite-particle approximations inherit long-time dynamical properties from their mean-field limits.

Abstract

In finite dimension, the long-time and metastable behavior of a gradient flow perturbated by a small Brownian noise is well understood. A similar situation arises when a Wasserstein gradient flow over a space of probability measure is approximated by a system of mean-field interacting particles, but classical results do not apply in these infinite-dimensional settings. This work is concerned with the situation where the objective function of the optimization problem contains an entropic penalization, so that the particle system is a Langevin diffusion process. We consider a very simple class of models, for which the infinite-dimensional behavior is fully characterized by a finite-dimensional process. The goal is to have a flexible class of benchmarks to fix some objectives, conjectures and (counter-)examples for the general situation. Inspired by the systematic study of these toy models, one application is presented on the continuous Curie-Weiss model in a symmetric double-well potential. We show that, at the critical temperature, although the $N$-particle Gibbs measure does not satisfy a uniform-in-$N$ standard log-Sobolev inequality (the optimal constant growing like $\sqrt{N}$), it does satisfy a more general Lojasiewicz inequality uniformly in $N$, inducing uniform polynomial long-time convergence rates, propagation of chaos at stationarity and uniformly in time, and creation of chaos.

Figures (2)

• Figure 1: Potential $V_\kappa$ with $V$ from the running example \ref{['eq:VmuEncoder']} plotted for $(m_0,m_1) \in \{(x v,y v),\ x,y\in\mathbb R\}$ where $v$ is a unit eigenvector of $M$ associated to the eigenvalue $\lambda=1$, for $\kappa \in \{0.5,0.9,1,2\}$. The red dots are the minimizers $\pm \sqrt{1-\kappa/\lambda}(v,v)$ (for $\kappa<\lambda$), the black dot is $0$.
• Figure 2: Graph of $f$ given in \ref{['eq:fCW']} at three temperatures $\sigma=0.1$ (blue) $\sigma=0.68 \simeq \sigma_c$ (green) and $\sigma=1$ (magenta). At $\sigma=\sigma_c$ (resp. $\sigma<\sigma_c$, resp. $\sigma>\sigma_c$), $f'(0)=1$ (resp. $f'(0)>1$, resp. $f'(0)<1$).

Theorems & Definitions (79)

• Remark 1
• Example 1
• Lemma 1
• proof
• Remark 2
• Example 2
• Lemma 2
• proof
• Remark 3
• Lemma 3