标题： 吸引性对数气体：稳定性、唯一性和混沌传播 显示英文标题

标题： The attractive log gas: stability, uniqueness, and propagation of chaos

摘要： 我们考虑在环面$\mathbb{T}^\mathsf{d}$上吸引性对数气体的过阻尼朗之万动力学，对于$\mathsf{d}\geq 1$。 在维度$\mathsf{d}=2$中，该模型与趋化作用的抛物-椭圆型 Patlak-Keller-Segel 模型的周期版本一致。 吸引性对数气体（在我们的单位选择下）众所周知具有临界逆温度$\beta_{\mathrm{c}}={2\mathsf{d}}$，对应于自由能从下方有界的时刻。 此外，众所周知，均匀分布无论温度如何总是稳态。 我们明确识别出另一个温度阈值$\beta_{\mathrm{s}}$，它严格对应于均匀分布的非线性稳定性。 我们证明当$\beta>\beta_{\mathrm{s}}$时，均匀分布不使自由能最小化，并且是非线性不稳定的，而当$\beta<\beta_{\mathrm{s}}$时，它是稳定的。 我们还证明了存在$\beta_{\mathrm{u}}$，对于该$\beta<\beta_{\mathrm{u}}$，平衡点的唯一性成立。 我们为一系列$\beta<\beta_{\mathrm{s}}$建立了一个随时间一致的熵传播混沌速率。 据我们所知，这是针对奇异吸引相互作用的第一个此类结果，并正面回答了 Bresch 等人提出的问题。 arXiv:2011.08022。 收敛的证明是通过调制自由能方法进行的，依赖于调制对数 Hardy-Littlewood-Sobolev（mLHLS）不等式。 与 Bresch 等人不同，我们证明了这种不等式在不截断势函数的情况下仍然成立——避免截断对于得到随时间一致的结果至关重要——对于足够小的$\beta$成立，并在$\beta>\beta_{\mathrm{s}}$时提供了 mLHLS 不等式的反例。 作为副产品，我们证明如果$\beta>\beta_{\mathrm{s}}$，则不可能有传播混沌的随时间一致速率。 显示英文摘要

摘要： We consider overdamped Langevin dynamics for the attractive log gas on the torus $\mathbb{T}^\mathsf{d}$, for $\mathsf{d}\geq 1$. In dimension $\mathsf{d}=2$, this model coincides with a periodic version of the parabolic-elliptic Patlak-Keller-Segel model of chemotaxis. The attractive log gas (for our choice of units) is well-known to have a critical inverse temperature $\beta_{\mathrm{c}}={2\mathsf{d}}$ corresponding to when the free energy is bounded from below. Moreover, it is well-known that the uniform distribution is always a stationary state regardless of the temperature. We identify another temperature threshold $\beta_{\mathrm{s}}$ sharply corresponding to the nonlinear stability of the uniform distribution. We show that for $\beta>\beta_{\mathrm{s}}$, the uniform distribution does not minimize the free energy and moreover is nonlinearly unstable, while for $\beta<\beta_{\mathrm{s}}$, it is stable. We also show that there exists $\beta_{\mathrm{u}}$ for which uniqueness of equilibria holds for $\beta<\beta_{\mathrm{u}}$. We establish a uniform-in-time rate for entropic propagation of chaos for a range of $\beta<\beta_{\mathrm{s}}$. To our knowledge, this is the first such result for singular attractive interactions and affirmatively answers a question of Bresch et al. arXiv:2011.08022. The proof of the convergence is through the modulated free energy method, relying on a modulated logarithmic Hardy-Littlewood-Sobolev (mLHLS) inequality. Unlike Bresch et al., we show that such an inequality holds without truncation of the potential -- the avoidance of the truncation being essential to a uniform-in-time result -- for sufficiently small $\beta$ and provide a counterexample to the mLHLS inequality when $\beta>\beta_{\mathrm{s}}$. As a byproduct, we show that it is impossible to have a uniform-in-time rate of propagation of chaos if $\beta>\beta_{\mathrm{s}}$.