  A general near-field thermal electromagnetic transport formalism that is
independent of the size, shape and number of heat sources is derived. The
formalism is based on fluctuational electrodynamics, where fluctuating currents
due to thermal agitation are added to Maxwell's curl equations, and is thus
valid for heat sources in local thermodynamic equilibrium. Using a volume
integral formulation, it is shown that the proposed formalism is a
generalization of the classical electromagnetic scattering framework in which
thermal emission is implicitly assumed to be negligible. The near-field thermal
electromagnetic transport formalism is afterwards applied to a problem
involving three spheres with size comparable to the wavelength, where all
multipolar interactions are taken into account. Using the thermal discrete
dipole approximation, it is shown that depending on the dielectric function,
the presence of a third sphere slightly affects the spatial distribution of
power absorbed compared to the two-sphere case. A transient analysis shows that
despite a non-uniform spatial distribution of power absorbed, the sphere
temperature remains spatially uniform at any instant due to the fact that the
thermal resistance by conduction is much smaller than the resistance by
radiation. The formalism proposed in this paper is general, and could be used
as a starting point for adapting solution methods employed in traditional
electromagnetic scattering problems to near-field thermal electromagnetic
transport.
