Advective-diffusive motion on large scales from small-scale dynamics with an internal symmetry

TL;DR

This paper analyzes how small-scale coupled diffusions in high-dimensional spaces and on manifolds lead to large-scale advective and diffusive behaviors, using multi-scale analysis and symmetry considerations.

Contribution

It extends previous work by deriving effective large-scale motion for coupled diffusions on complex manifolds like SO(3) and SU(2), highlighting the role of global solvability conditions.

Findings

Effective drift and diffusion are derived via multi-scale analysis.

Global properties of manifolds influence the resulting diffusion.

Diffusions on SU(2) are more isotropic than on SO(3).

Abstract

We consider coupled diffusions in $n$-dimensional space and on a compact manifold and the resulting effective advective-diffusive motion on large scales in space. The effective drift (advection) and effective diffusion are determined as a solvability conditions in a multi-scale analysis. As an example we consider coupled diffusions in $3$-dimensional space and on the group manifold $SO(3)$ of proper rotations, generalizing results obtained by H. Brenner (1981). We show in detail how the analysis can be conveniently be carried out using local charts and invariance arguments. As a further example we consider coupled diffusions in $2$-dimensional complex space and on the group manifold $SU(2)$. We show that although the local operators may be the same as for $SO(3)$, due to the global nature of the solvability conditions the resulting diffusion will be different, and generally more isotropic.