Effective Long-Time Diffusivity of Particles of Arbitrary Shape in an External Orienting Field

Abstract Pertaining to the motion of a rigid particle in a flow, several distinct “centers” of the rigid particle can be identified, including the geometric center (centroid), center of mass, hydrodynamic center, and center of diffusion. In this work, we elucidate the relevance of these centers in Brownian motion and diffusion. Starting from the microscopic stochastic equations of motions, we systematically derive the coarse-grained Fokker–Planck equations that govern the evolution of the probability distribution function (PDF) in phase space and in configurational space. For consistency with the equilibrium statistical mechanics, we determine the unknown Brownian forces and torques. Next, we analyze the Fokker–Planck equation for the PDF in the position and orientation space. Through a multiscale analysis, we find the unit cell problem for defining the effective long-time translational diffusivity of a particle of arbitrary shape in an external orienting field. We also show some fundamental properties of the effective long-time translational diffusivity, including rigorous variational bounds for effective long-time diffusivity and invariance of effective diffusivity with respect to change of reference or tracking points. Exact results are obtained in the absence of an orienting field and in the presence of a strong orienting field. These fundamental results hold significant potential for applications in biophysics, colloidal science, and micro-swimmers design.