Quantitative Diffusive Limits for Singular Nonlocal Transport

We study the nonlocal continuity equation $${\mathrm{\partial }}_t{\mu }_b=\mathrm{div}\text{\hspace{0.17em}⁣}({\mu }_b\mathrm{\nabla }\log ((I-b^2\mathrm{\Delta }{)}^{-1}{\mu }_b))$$ 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 $\mu (t)$ at the sharp, uniform-in-time rate $$\underset{t\ge 0}{\sup }\mathrm{\parallel }{\mu }_b(t)-\mu (t){\mathrm{\parallel }}_{L^1}\le C{b}^2\mathrm{.}$$ 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 $$\mathbb{E}\text{\hspace{0.17em}⁣}[\underset{t\ge 0}{\sup }{W}_1({\mu }_b^N(t),{\mu }_b(t))]\le C(Nb{)}^{-1\mathrm{/}2}$$ for iid initialization. Consequently, the choice $b\asymp N^{-1\mathrm{/}5}$ approximates heat flow uniformly in time at rate $N^{-2\mathrm{/}5}$.