Thermodynamic definition of mean temperature

• Main Authors: Allahverdyan, A. E, Gevorkian, S. G, Dyakov, Yu. A, Wang, Pao-Kuan
• Format: Journal Article
• Language: English
• Published: 05-07-2022
• Subjects: Physics - Statistical Mechanics
• DOI: 10.48550/arxiv.2207.02343

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.