  Ionic crystals play a central role in functional applications. Mesoscale
descriptions of these crystals are based on the continuum polarization density
field to represent the effective physics of charge distribution at the scale of
the atomic lattice. However, a long-standing difficulty is that the classical
electrostatic definition of the macroscopic polarization -- as the dipole or
first moment of the charge density in a unit cell -- is not unique. This
unphysical non-uniqueness has been shown to arise from starting directly with
an infinite system rather than starting with a finite body and taking
appropriate limits. This limit process shows that the electrostatic description
requires not only the bulk polarization density, but also the surface charge
density, as the effective macroscopic descriptors; that is, a nonlocal
effective description. Other approaches to resolve this difficulty include
relating the change in polarization to the transport of charge; or, to define
the polarization as the energy-conjugate to the electric field.
  This work examines the relation between the classical electrostatic
definition of polarization, and the transport and energy-conjugate definitions
of polarization. We show the following: (1) The transport of charge does not
correspond to the change in polarization in general; instead, one requires
additional simplifying assumptions on the electrostatic definition of
polarization for these approaches to give rise to the same macroscopic electric
fields. Thus, the electrostatic definition encompasses the transport definition
as a special case. (2) The energy-conjugate definition has both bulk and
surface contributions; while traditional approaches neglect the surface
contribution, we find that accounting for the nonlocal surface contributions is
essential to obtain the correct macroscopic electric fields.
