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

We consider overdamped Langevin dynamics for the attractive log gas on the torus T d {T}^d , for d ≥ 1 d≥ 1 . In dimension d = 2 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 β c = 2 d β c={2d} 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 β s β ₛ sharply corresponding to the nonlinear stability of the uniform distribution. We show that for β > β s β >β ₛ , the uniform distribution does not minimize the free energy and moreover is nonlinearly unstable, while for β > β s β >β ₛ , it is stable. We also show that there exists β u β ᵤ for which uniqueness of equilibria holds for β >