  We describe a fluctuating volume--current formulation of electromagnetic
fluctuations that extends our recent work on heat exchange and Casimir
interactions between arbitrarily shaped homogeneous bodies [Phys. Rev. B. 88,
054305] to situations involving incandescence and luminescence problems,
including thermal radiation, heat transfer, Casimir forces, spontaneous
emission, fluorescence, and Raman scattering, in inhomogeneous media. Unlike
previous scattering formulations based on field and/or surface unknowns, our
work exploits powerful techniques from the volume--integral equation (VIE)
method, in which electromagnetic scattering is described in terms of
volumetric, current unknowns throughout the bodies. The resulting trace
formulas (boxed equations) involve products of well-studied VIE matrices and
describe power and momentum transfer between objects with spatially varying
material properties and fluctuation characteristics. We demonstrate that thanks
to the low-rank properties of the associatedmatrices, these formulas are
susceptible to fast-trace computations based on iterative methods, making
practical calculations tractable. We apply our techniques to study thermal
radiation, heat transfer, and fluorescence in complicated geometries, checking
our method against established techniques best suited for homogeneous bodies as
well as applying it to obtain predictions of radiation from complex bodies with
spatially varying permittivities and/or temperature profiles.
