Brownian motion meets Riemann curvature

Abstract

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.

16 pages and 3 figures

References in corpus (4)

• Lateral diffusion of a protein on a fluctuating membrane
• Hybrid Elastic/Discrete-Particle Approach to Biomembrane Dynamics with Application to the Mobility of Curved Integral Membrane Proteins
• Diffusion on Ruffled Membrane Surfaces
• Diffusing proteins on a fluctuating membrane: Analytical theory and simulations

Cited by in corpus (9)

• Active motion on curved surfaces
• Diffusion in nonuniform temperature and its geometric analog
• Generally covariant state-dependent diffusion
• Intrinsic and extrinsic measure for Brownian motion
• Single-file dynamics of colloids in circular channels: time scales, scaling laws and their universality
• Pinning of Diffusional Patterns by Non-Uniform Curvature
• Tailoring diffusion in analogue spacetimes
• The eta invariant on two-step nilmanifolds
• Physics in precision-dependent normal neighborhoods