AI 中文总结
本文证明了一类平均场系统中，平均场流的正PL常数蕴含粒子系统对数Sobolev常数的一致下界，并给出自由能最小化器不唯一时该蕴含失效的反例，同时将结果推广至更一般的Łojasiewicz不等式。
AI 中文摘要
自由能Wasserstein梯度流的收敛速率由其所谓的Polyak-Lojasiewicz (PL)常数$\lambda$量化，该常数将目标函数与其沿流的耗散联系起来。这样的流是当$N$趋于无穷大时，由$N$个相互作用粒子组成的系统的平均场极限，该系统相对于其吉布斯测度的相对熵收敛速率由其对数Sobolev常数$\lambda_N$量化。因此，$\lambda_N$随$N$趋于无穷大的不同行为描述了截然不同的现象，如快速弛豫或亚稳定性，许多模型在这些状态之间经历相变，通常取决于温度。在相当一般的条件下，已知一致于$N$的对数Sobolev常数（即粒子系统的快速指数收敛）会诱导正的PL常数（即平均场流的指数收敛）。Delgadino, Gvalani, Pavliotis和Smith提出了一个猜想，认为逆蕴含成立（$\lambda>0$蕴含$\liminf \lambda_N >0$），甚至$\lim \lambda_N =\lambda$。首先，我们将证明这一逆蕴含，尽管不包含等式$\lim \lambda_N = \lambda$，对于一类一般的平均场模型。其次，我们还注意到，如果自由能最小化器不唯一，则该蕴含不成立，并提供一个显式反例。第三，我们还考虑了在更一般的Łojasiewicz不等式背景下，将$N$粒子不等式与平均场不等式联系起来的相同问题，这些不等式对应于多项式（而非指数）收敛速率，并且可以精确描述相变处的情况。
英文摘要
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.
