Brownian motion meets Riemann curvature

Details Brownian motion meets Riemann curvature Brownian motion meets Riemann curvature

2010-05-04

arxiv.org/abs/1005.0650v1

Physics

Condensed Matter

Statistical Mechanics

16 pages and 3 figures

Scientific paper

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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