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Thermodynamic definition of mean temperature

The notion of mean temperature is crucial for a number of fields including climate science, fluid dynamics and biophysics. However, so far its correct thermodynamic foundation is lacking or even believed to be impossible. A physically correct definition should not be based on mathematical notions of the means (e.g. the mean geometric or mean arithmetic), because they ignore the peculiarities of the notion of temperature, and because they are not unique. We offer a thermodynamic definition of the mean temperature that is based upon the following two assumptions. First, as the correct definition should necessarily involve equilibration processes in the initially non-equilibrium system, the mean temperature is bounded from below and above via looking at (respectively) the reversible versus fully irreversible extremes of equilibration. Second, within the thermodynamic approach we assume that the mean temperature is determined mostly by energy and entropy. Together with the dimensional analysis, the two assumptions lead to a unique definition of the mean temperature. The mean temperature for ideal and (van der Waals) non-ideal gases with temperature-independent heat capacity is given by a general and compact formula that (besides the initial temperatures) only depends on the heat-capacities and concentration of gases.