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 are not unique, and they ignore the fact that temperature is an ordinal level variable. We offer a thermodynamic definition of the mean temperature that is based upon the following two assumptions. First, the correct definition should necessarily involve equilibration processes in the initially nonequilibrium system. Among such processes, reversible equilibration and fully irreversible equilibration are the two extreme cases. 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 definition of the mean temperature that is determined up to a weight factor that can be fixed to 1/2 due to the maximum ignorance principle. The mean temperature for ideal and (van der Waals) nonideal 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. Our method works for any nonequilibrium initial state, not only two-temperature states.