布朗运动的内外测度
Intrinsic and extrinsic measure for Brownian motion
摘要：
基于曲流形上的Smoluchowski方程，考虑了布朗位移的三个物理观测量：测地线位移$s$、欧氏位移$δ{\bf R}$和投影位移$δ{\bf R}_{\perp}$ . Weingarten-Gauss方程被用来计算在短时间政权的均方欧几里得位移。我们的研究结果表明，从外在的角度来看，空间的几何形状影响的布朗运动在这样一种方式，粒子的扩散是减速，与内在的角度来看，动力学是由高斯曲率的符号[J. Stat. P08006（2010）]。此外，还可以给出球面和极小曲面上的$\left<δ{\bf R}\right>$和$\left<δ{\bf R}^{2}\right>$的精确公式，这些公式对所有时间值都有效。在后一种情况下，令人惊讶的是，布朗运动对应于平坦几何中的通常扩散，尽管极小曲面具有非零高斯曲率。最后，二维的情况下，强调由于其密切关系的流体膜表面自扩散。
英文摘要：
Based upon the Smoluchowski equation on curved manifolds three physical observables are considered for the Brownian displacement, namely, geodesic displacement, $s$, Euclidean displacement, $δ{\bf R}$, and projected displacement $δ{\bf R}_{\perp}$. The Weingarten-Gauss equations are used to calculate the mean-square Euclidean displacements in the short-time regime. Our findings show that from an extrinsic point of view the geometry of the space affects the Brownian motion in such a way that the particle's diffusion is decelerated, contrasting with the intrinsic point of view where dynamics is controlled by the sign of the Gaussian curvature [J. Stat. Mech. P08006 (2010)]. Furthermore, it is possible to give exact formulae for $\left<δ{\bf R}\right>$ and $\left<δ{\bf R}^{2}\right>$ on spheres and minimal surfaces, which are valid for all values of time. In the latter case, surprisingly, Brownian motion corresponds to the usual diffusion in flat geometries, albeit minimal surfaces have non-zero Gaussian curvature. Finally, the two-dimensional case is emphasized due to its close relation to surface self-diffusion in fluid membranes.
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