AI 中文总结
研究斯托克斯流中刚性粒子静态随机悬浮液沉降，构建无限体积平均沉降速度，通过红外重整化建立稀薄区域重整化团簇展开并计算到两粒子项以证明巴彻勒公式，揭示其与容器形状无关及关键抵消机制。
AI 中文摘要
我们研究了斯托克斯流中刚性粒子的静态随机悬浮液的沉降。巴彻勒公式预测了由于悬浮粒子间流体动力学相互作用对无限体积平均沉降速度的首次稀薄修正。长期以来，由于斯托克斯流的长程性质导致大体积极限下的红外发散，严格推导受阻。在\(d>2\)维中，对于满足定量去相关假设的静态悬浮液，我们构建了无限体积平均沉降速度，并表明它决定了大容器中粒子的相对沉降速度，与容器形状无关。然后我们在稀薄区域建立了该平均沉降速度的重整化团簇展开，并计算到两粒子项，从而证明了巴彻勒公式。证明基于流体动力学相互作用的红外重整化。无限体积可观测量被分解为一个明确的奇异部分，承载不可积的大尺度贡献，以及一个由椭圆估计控制的正则余项。奇异部分通过编码悬浮液产生的发散平均回流的抵消项进行重整化。在稀薄团簇展开层面，重整化逐个团簇实施，奇异 - 正则分解通过受反射方法启发的流体动力学相互作用的有限图解展开实现，该展开分离出主导发散子结构并揭示关键抵消。
英文摘要
We study the sedimentation of stationary random suspensions of rigid particles in Stokes flow. Batchelor's formula predicts the first dilute correction to the infinite-volume mean settling speed due to hydrodynamic interactions between suspended particles. A rigorous derivation has long been obstructed by the long-range nature of the Stokes flow, which gives rise to infrared divergences in the large-volume limit. In dimension $d>2$, for stationary suspensions satisfying quantitative decorrelation assumptions, we construct the infinite-volume mean settling speed and show that it governs the relative settling speed of particles in large containers, independently of the container shape. We then establish a renormalized cluster expansion of this mean settling speed in the dilute regime and compute it up to the two-particle term, thereby justifying Batchelor's formula. The proof is based on the infrared renormalization of hydrodynamic interactions. Infinite-volume observables are decomposed into an explicit singular part, carrying the non-integrable large-scale contribution, and a regular remainder controlled by elliptic estimates. The singular part is renormalized through counterterms that encode the diverging mean backflow generated by the suspension. At the level of the dilute cluster expansion, the renormalization is implemented cluster by cluster and the singular-regular decomposition is achieved through a finitary diagrammatic expansion of hydrodynamic interactions, inspired by the method of reflections, which isolates the leading divergent substructures and exposes the key cancellations.
Comments91 pages
