On the definition of temperature using time-averages

This paper is a natural continuation of a previous one by the author, which was concerned with the foundations of statistical thermodynamics far from equilibrium. One of the problems left open in that paper was the correct definition of temperature. In the literature, temperature is in general defined through the mean kinetic energy of the particles of a given system. In this paper, instead, temperature is defined à la Carathéodory, the system being coupled to a heat bath, and temperature being singled out as the “right” integrating factor of the exchanged heat. As a byproduct, the “right” expression for the entropy is also obtained. In particular, in the case of a q-distribution the entropy turns out to be that of Tsallis.