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

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 multiscale analysis. As an example, we consider coupled diffusions in three-dimensional space and on the group manifold SO(3) of proper rotations, generalizing results obtained by H. Brenner [J. Colloid Interface Sci. 80, 548 (1981)JCISA50021-979710.1016/0021-9797(81)90214-9]. We show in detail how the analysis can be conveniently carried out using local charts and invariance arguments. As a further example, we consider coupled diffusions in two-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 differ and generally be more isotropic.

Steady diffusion in a drift field: A comparison of large-deviation techniques and multiple-scale analysis

“…In two additional Appendices C and D we extend for completeness the large deviation analysis to the case of a continuum of internal states, of which one example is a Brownian particle undergoing translation and rotation. We show that an auxiliary equation which appeared in an earlier multiple-scale analysis [20,25], can also be derived by second-order perturbation theory of generating function of the time spent in different orientations. We further show that the large deviation analysis can also be pushed to the third centered moment which we show to generically increase linearly in time, see Appendix D.…”

“…In this Appendix we make the connection to our earlier work on coarse-graining of diffusion in space and over orientations [20,25]. The starting point is the same as in the preceding discussion, except that the internal variable α representing the internal state now lives in a compact manifold M with volume element √ g. We also assume that the spatial motion of the particle takes place in Ddimensional space.…”

“…Rotations in D = 2 can hence be represented by an internal coordinate on a circle (Lie group SO (2)), rotations in D = 3 by an internal coordinate in the solid upper hemisphere (Lie group SO(3)), and so on. In [25] the case of motion in two-dimensional complex space (D = 4) and the group SU (2) was considered as well. Whatever the internal state the process is described by…”

“…For the term B (2) we introduce yet another auxiliary quantity When m 0 is the Haar measure the averages in (D10) (D12) can be computed by invariance arguments as in [20] and [25].…”

“…Tr[µ]1 jl and the quantity in the parenthesis on the right-hand side of (C20) is the traceless part of the mobility matrix. In[20] and[25] this quantity was denotedγ −1 . Eqs.…”