Quantitative Diffusive Limits for Singular Nonlocal Transport

We study the nonlocal continuity equation [ partialtμb =operatornamediv!left( μbnablalogbigl((I-b2Delta)-1μbbigr) right) ] on a closed connected Riemannian manifold. For smooth strictly positive initial data, we prove that as b → 0, its global solution converges to heat flow μ(t) at the sharp, uniform-in-time rate [ supt≥0|μb(t)-μ(t)|L1≤ Cb2. ] 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 [ mathbb E!left[ supt≥0W1(μbN(t),μb(t)) right] ≤ C(Nb)-1/2 ] for iid initialization. Consequently, the choice basymp N-1/5 approximates heat flow uniformly in time at rate N-2/5.