Homogeneity: From Caratheodory's approach to Gibbs thermodynamics

摘要

We consider thermodynamics of homogeneous systems in the frame-work of Caratheodory's approach. We work in a configuration space where indepen-dent variables are all extensive, and we show that the requirement for homogeneity symmetry of the integrable 1-form δQ_(rev), which represents the infinitesimal heat exchanged reversibly by a thermodynamic system, allows to find immediately and explicitly an integrating factor by means of elementary tools of differential geometry. The foliation of the thermodynamic manifold is associated in a natural way with an extensive potential, which is then identified with the metrical entropy S. Both S and the absolute temperature T are derived quantities in this framework. The entropy S corresponds to the fundamental relation in the entropy representation ac-cording to Gibbs' approach to thermodynamics, thus an explicit connection between Caratheodory's approach and Gibbs' approach to thermodynamics is obtained. The relation with statistical mechanical formalism is also pointed out.