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We present a detailed derivation of heat radiation, heat transfer and(Casimir) interactions for N arbitrary objects in the framework offluctuational electrodynamics in thermal non-equilibrium. The results can beexpressed as basis-independent trace formulae in terms of the scatteringoperators of the individual objects. We prove that heat radiation of a singleobject is positive, and that heat transfer (for two arbitrary passive objects)is from the hotter to a colder body. The heat transferred is also symmetric,exactly reversed if the two temperatures are exchanged. Introducing partialwave-expansions, we transform the results for radiation, transfer and forcesinto traces of matrices that can be evaluated in any basis, analogous to theequilibrium Casimir force. The method is illustrated by (re)deriving the heatradiation of a plate, a sphere and a cylinder. We analyze the radiation of asphere for different materials, emphasizing that a simplification oftenemployed for metallic nano-spheres is typically invalid. We derive asymptoticformulae for heat transfer and non-equilibrium interactions for the cases of asphere in front a plate and for two spheres, extending previous results. As anexample, we show that a hot nano-sphere can levitate above a plate with therepulsive non-equilibrium force overcoming gravity -- an effect that is not dueto radiation pressure.

Trace formulae for non-equilibrium Casimir interactions, heat radiation and heat transfer for arbitrary objects