任意弹性散射积分的水动力学精确重求和
Exact resummation of hydrodynamics for arbitrary elastic scattering integral
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中文总结 AI 辅助
本文针对三维空间中任意旋转不变弹性散射核，推导出频率和波数相关扩散系数的精确表达式，通过无限连分数重求和梯度展开至所有阶，揭示偏离水动力学的指数衰减携带散射微观信息。
中文摘要 AI 辅助
我们考虑在三维空间中传播的单能粒子，并经历外部介质的弹性散射。对于任意旋转不变的散射核，我们推导了频率和波数相关的扩散系数的精确表达式，从而将水动力学梯度展开重求和到所有阶。结果是一个无限连分数，每个连续层级由碰撞算子的更高角谐波的弛豫率决定。因此，虽然普通扩散仅探测第一角谐波，但重求和的本构关系保留了完整散射核的信息。我们展示了这种微观信息如何体现在扩散系数的解析结构以及对局部外部源的响应中。特别地，偏离菲克定律的偏差随距离呈指数衰减，其速率由逆扩散系数最近复波数奇点决定。因此，指数级小的偏离水动力学行为保留了关于底层散射过程的详细信息。
英文摘要
We consider monoenergetic particles propagating in three spatial dimensions and undergoing elastic scattering off an external medium. For an arbitrary rotationally invariant scattering kernel, we derive an exact expression for the frequency- and wave-number-dependent diffusivity, thereby resumming the hydrodynamic gradient expansion to all orders. The result is an infinite continued fraction, with each successive level determined by the relaxation rate of a higher angular harmonic of the collision operator. Thus, while ordinary diffusion probes only the first angular harmonic, the resummed constitutive relation retains information about the full scattering kernel. We show how this microscopic information manifests itself in the analytic structure of the diffusivity and in the response to a localized external source. In particular, deviations from Fick's law decay exponentially with distance, at a rate determined by the nearest complex-wavenumber singularity of the inverse diffusivity. Thus, exponentially small departures from hydrodynamics retain detailed information about the underlying scattering process.
发表机构
- Department of Applied Mathematics and Theoretical Physics, University of Cambridge（剑桥大学应用数学与理论物理系）
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