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

Source abstract

We consider overdamped Langevin dynamics for the attractive log gas on the torus T d {\mathbb {T}}^\mathsf {d} , for d ≥ 1 \mathsf {d}\geq 1 . In dimension d = 2 \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 β c = 2 d \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 β s \beta _{\mathrm {s}} sharply corresponding to the nonlinear stability of the uniform distribution. We show that for β > β s \beta >\beta _{\mathrm {s}} , the uniform distribution does not minimize the free energy and moreover is nonlinearly unstable, while for β > β s \beta >\beta _{\mathrm {s}} , it is stable. We also show that there exists β u \beta _{\mathrm {u}} for which uniqueness of equilibria holds for β > β u \beta >\beta _{\mathrm {u}} . Related to the above findings, we establish a uniform-in-time rate for entropic propagation of chaos for a range of β > β s \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. [Duke Math. J. 172 (2023), pp. 2591–2641]. The proof of the convergence is through the modulated free energy method, in particular relying on a modulated logarithmic Hardy-Littlewood-Sobolev (mLHLS) inequality . Unlike Bresch et al. [Duke Math. J. 172 (2023), pp. 2591–2641], 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—at sufficiently high temperature and provide a counterexample to the mLHLS inequality when β > β s \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 β > β s \beta >\beta _{\mathrm {s}} .