2023/11/24 by Antonin Chodron de Courcel, de Courcel, Antonin Chodron, Matthew Rosenzweig +3 · 3 citations
Mathematics · #Mathematical Biology Tumor Growth #Markov Chains and Monte Carlo Methods #Stochastic processes and statistical mechanics
paper · pdf · doi:10.48550/arxiv.2311.14560
We consider overdamped Langevin dynamics for the attractive log gas on the torus \mathbbTd, for d≥ 1. In dimension 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 βc=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 βs sharply corresponding to the nonlinear stability of the uniform distribution. We show that for β>βs, the uniform distribution does not minimize the free energy and moreover is nonlinearly unstable, while for β<βs, it is stable. We also show that there exists βu for which uniqueness of equilibria holds for β<βu. We establish a uniform-in-time rate for entropic propagation of chaos for a range of β<βs. 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 β and provide a counterexample to the mLHLS inequality when β>βs. As a byproduct, we show that it is impossible to have a uniform-in-time rate of propagation of chaos if β>βs.