Abstract: | There is a vast literature on how small objects undergo
diffusion when subjected to random forcing, but much less
has been written about how an object rotates due to a random
torque.
There is a dimensionless parameter characterising
this problem: the persistence angle \beta is the typical
angle of rotation during the correlation time of the
angular velocity. When \beta is small, the problem
is simply diffusion on a sphere. But little is known
about models with finite \beta, describing smooth
random motion on a sphere.
I will discuss the formulation and solution of the simplest
model, which is a spherical Ornstein-Uhlenbech process.
In two dimensions (circular motion) this is exactly solvable.
When \beta is large, the solution has a surprising property,
which is analogous to the phenomenon of 'superoscillations'.
In three dimensions we obtain asymptotic solutions for large \beta
which involve a solving a radial Shroedinger equation where the
angular momentum quantum number j takes non-integer values.
The case where j=(\sqrt{17}-1)/2 turns out to be of particular
significance.
As well as discussing random tumbling of a single
body, I will also mention some results on the singularities
of orientation vector fields of small bodies advected in random
flows.
This talk reports joint work with Alain Pumir (ENS, Lyon)
and Vlad Bezuglyy (Open University). |