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Casimir-Lifshitz force out of thermal equilibrium and heat transfer between arbitrary bodies

We study the Casimir-Lifshitz force and the radiative heat transfer occurring between two arbitrary bodies, each one held at a given temperature, surrounded by environmental radiation at a third temperature. The system, in stationary configuration out of thermal equilibrium, is characterized by a force and a heat transfer depending on the three temperatures, and explicitly expressed in terms of the scattering operators of each body. We find a closed-form analytic expression valid for bodies of any geometry and dielectric properties. As an example, the force between two parallel slabs of finite thickness is calculated, showing the importance of the environmental temperature as well as the occurrence of a repulsive interaction. An analytic expression is also provided for the force acting on an atom in front of a slab. Our predictions can be relevant for experimental and technological purposes.