Quantitative Diffusive Limits for Singular Nonlocal Transport

We study the nonlocal continuity equation d tμ b = div! (μ b grad ((I-b 2Δ)-1μ b)) on a closed connected Riemannian manifold. For smooth strictly positive initial data, we prove that as b to 0, its global solution converges to heat flow μ(t) at the sharp, uniform-in-time rate t 0μ b(t)-μ(t) L 1 Cb squared. The key estimate is the uniform dissipation of a b-weighted higher-order resolvent energy, which yields exponential relaxation despite the absence of a Wasserstein gradient-flow structure. On the circle, we also analyze the corresponding deterministic N-particle dynamics. A weak--strong modulated energy argument gives E! [t 0W 1(μ b N(t),μ b(t))] C(Nb)-1/2 for iid initialization. Consequently, the choice b N-1/5 approximates heat flow uniformly in time at rate N-2/5.