  Microstructured particles are widely used in industries and state-of-the-art
research and development. Diffusion of particles, particularly, controlled
diffusion by a remotely applied field, has inspired novel applications ranging
from targeted drug deliveries, novel procedures for quantifying physical
properties of nanoparticles and ambient fluids, to fabrication of composites
with enhanced properties. In this work, we report a systematic analysis on
field-controlled diffusion of microstructured particles. In account of shape
anisotropy and structural heterogeneity of a particle, we study coupled
Brownian motions of the particle in $\mathbb{R}^3\times$SO(3). Starting from
the microscopic stochastic differential equations of motions, we achieve the
coarse-grained Fokker-Planck equation that governs the evolution of the
probability distribution function with respect to the position and orientation
of the particle. Under some mild conditions, we identify the long-time
diffusivity for microstructured particles in an external field. The formulation
is applicable to microstructured particles of arbitrary shapes and
heterogeneities. As examples of applications, we analyze the diffusion of a
heterogeneous spheroidal particle and a pair of spheroidal particles bonded by
an elastic ligament. For heterogeneous spheroidal particles, we obtain explicit
generalized Stokes-Einstein's relations for diffusivity that accounts for the
effects of shape anisotropy, heterogeneity, and an external alignment field.
For pairs of spheroidal particles, we consider the superimposed relaxation
process from an initial non-equilibrium state to the final equilibrium state.
The anomalous scaling of Mean Square Displacement (MSD) with respect to the
time of such processes may provide important insight for understanding
anomalous diffusions observed in migration of macromolecules and cells in
complex viscoelastic media.
