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Bodies in a steady state but not in thermodynamic equilibrium

While for bodies in their own thermodynamic equilibrium states, the notion of temperature requires that all empirical thermometers must agree as to which of two bodies is the hotter or that they are at the same temperature, this requirement is not safe for bodies that are in steady states though not in thermodynamic equilibrium. It can then well be that different empirical thermometers disagree about which is hotter, and if this is so, then at least one of the bodies does not have a well-defined absolute thermodynamic temperature. Nevertheless, any one given body and any one suitable empirical thermometer can still support notions of empirical, non-absolute, hotness, and temperature, for a suitable range of processes. This is a matter for study in non-equilibrium thermodynamics.