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

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 $\beta>\beta_{\mathrm{s}}$.