布朗运动满足黎曼曲率
Brownian motion meets Riemann curvature
摘要：
利用扩散方程的广义协方差来研究d维曲流形上布朗运动的曲率效应。我们使用由所谓的Riemann法向坐标定义的局部框架来推导短时区域均方测地线距离（MSD）的一般公式。这个公式是用依赖于黎曼曲率张量的$O(d)$不变量写成的。我们研究了n维球的情况下，以验证这些结果。我们还表明，正常曲率的扩散是慢于在平面空间中的扩散，而负常曲率的扩散结果是更快。最后，二维的情况下，强调，因为它是相关的单颗粒生物膜上的扩散。
英文摘要：
The general covariance of the diffusion equation is exploited in order to explore the curvature effects appearing on brownian motion over a d-dimensional curved manifold. We use the local frame defined by the so called Riemann normal coordinates to derive a general formula for the mean-square geodesic distance (MSD) at the short-time regime. This formula is written in terms of $O(d)$ invariants that depend on the Riemann curvature tensor. We study the n-dimensional sphere case to validate these results. We also show that the diffusion for positive constant curvature is slower than the diffusion in a plane space, while the diffusion for negative constant curvature turns out to be faster. Finally the two-dimensional case is emphasized, as it is relevant for the single particle diffusion on biomembranes.
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