Equivalence between N-particle log-Sobolev inequalities and non-linear Łojasiewicz inequalities for a class of mean field systems
Abstract
The convergence rate of a free energy Wasserstein gradient flow is quantified by its so-called Polyak-Lojasiewicz (PL) constant λ, which relates the objective function to its dissipation along the flow. Such a flow is the mean-field limit as N goes to infinity of a system of N interacting particles, whose convergence rate in relative entropy towards its Gibbs measure is quantified by its log-Sobolev constant λN. Different behaviours of λN as N goes to infinity thus describe drastically different phenomena, such as fast relaxation or metastability, with many models undergoing phase transitions between these regimes, depending typically on temperature. Under fairly general conditions, a uniform-in-N log-Sobolev constant (i.e. fast exponential convergence for the particle system) is known to induce a positive PL constant (i.e. exponential convergence for the mean-field flow). A conjecture was stated by Delgadino, Gvalani, Pavliotis and Smith according to which the converse implication was true (λ>0 implies liminf λN >0), even with lim λN =λ. First, we will prove this converse implication, although without the equality lim λN = λ, for a general class of mean-field models. Second, we also notice that this implication fails if the free energy minimiser is not unique, and provide an explicit counter-example. Third, we also consider the same question of relating N-particle and mean-field inequalities in the context of more general Lojasiewicz inequalities, which correspond to polynomial (instead of exponential) convergence rates, and can describe the situation exactly at a phase transition.
Related
- Incremental Stability of Gradient Flows under Generalized Polyak-Łojasiewicz Inequalities and Its Implications for General Systems
- From one particle to N particles: uniform transport-information and log-Sobolev inequalities for Coulomb gases on the torus
- The ballistic limit of the log-Sobolev constant equals the Polyak–Łojasiewicz constant
- Large-time asymptotics for general cross-diffusion systems and novel convex Sobolev inequalities
- Generalized Logarithmic Sobolev inequalities on Bounded Domains and exponential decay to Boltzmann states for Nernst--Planck--Navier--Stokes systems
BibTeX
@misc{monmarche2026equivalence,
title = {{Equivalence between $N$-particle log-Sobolev inequalities and non-linear Łojasiewicz inequalities for a class of mean field systems}},
author = {Pierre Monmarché},
howpublished = {arXiv},
year = {2026},
month = sep,
doi = {10.48550/arxiv.2609.06574},
eprint = {2609.06574},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2609.06574},
}
RIS
TY - UNPB
TI - Equivalence between N-particle log-Sobolev inequalities and non-linear Łojasiewicz inequalities for a class of mean field systems
AU - Pierre Monmarché
PY - 2026
DA - 2026/09/10
PB - arXiv
DO - 10.48550/arxiv.2609.06574
UR - https://arxiv.org/abs/2609.06574
AB - The convergence rate of a free energy Wasserstein gradient flow is quantified by its so-called Polyak-Lojasiewicz (PL) constant λ, which relates the objective function to its dissipation along the flow. Such a flow is the mean-field limit as N goes to infinity of a system of N interacting particles, whose convergence rate in relative entropy towards its Gibbs measure is quantified by its log-Sobolev constant λ_N. Different behaviours of λ_N as N goes to infinity thus describe drastically different phenomena, such as fast relaxation or metastability, with many models undergoing phase transitions between these regimes, depending typically on temperature. Under fairly general conditions, a uniform-in-N log-Sobolev constant (i.e. fast exponential convergence for the particle system) is known to induce a positive PL constant (i.e. exponential convergence for the mean-field flow). A conjecture was stated by Delgadino, Gvalani, Pavliotis and Smith according to which the converse implication was true (λ>0 implies liminf λ_N >0), even with lim λ_N =λ. First, we will prove this converse implication, although without the equality lim λ_N = λ, for a general class of mean-field models. Second, we also notice that this implication fails if the free energy minimiser is not unique, and provide an explicit counter-example. Third, we also consider the same question of relating N-particle and mean-field inequalities in the context of more general Lojasiewicz inequalities, which correspond to polynomial (instead of exponential) convergence rates, and can describe the situation exactly at a phase transition.
ER -