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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. 

Piezoelectricity in biological soft tissues is a controversial issue with differing opinions. 

Predictive modeling in physical science and engineering is mostly based on solving certain partial differential equations where the complexity of solutions is dictated by the geometry of the domain. Motivated by the broad applications of explicit solutions for spherical and ellipsoidal domains, in particular, the Eshelby’s solution in elasticity, we propose a generalization of ellipsoidal shapes called polynomial inclusions. A polynomial inclusion (or -inclusion for brevity) of degree is defined as a smooth, connected and bounded body whose Newtonian potential is a polynomial of degree inside the body. From this viewpoint, ellipsoids are identified as the only -inclusions of degree two; many fundamental problems in various physical settings admit simple closed-form solutions for general -inclusions as for ellipsoids. Therefore, we anticipate that -inclusions will be useful for applications including predictive materials models, optimal designs, and inverse problems. However, the existence of p-inclusions beyond degree two is not obvious, not to mention their explicit algebraic parameterizations. In this work, we explore alternative definitions and properties of p-inclusions in the context of potential theory. Based on the theory of variational inequalities, we show that -inclusions do exist for certain polynomials, though a complete characterization remains open. We reformulate the determination of surfaces of -inclusions as nonlocal geometric flows which are convenient for numerical simulations and studying geometric properties of -inclusions. In two dimensions, by the method of conformal mapping we find an explicit algebraic parameterization of p-inclusions. We also propose a few open problems whose solution will deepen our understanding of relations between domain geometry, Newtonian potentials, and solutions to general partial differential equations. We conclude by presenting examples of applications of -inclusions in the context of Eshelby inclusion problems and magnet designs.