Thermodynamics with an Action Principle ( 2 nd edition ) . Heat and gravitation

Some features of hydro-and thermodynamics, as applied to atmospheres and to stellar structures, are puzzling: 1. The suggestion, first made by Laplace, that our atmosphere has an adiabatic temperature distribution, is confirmed for the lower layers, but the reason why it should be so is understood only qualitatively. 2. Arguments in which a concept of energy plays a role, in the context of hydro-thermodynamical systems and gravitation, are often flawed, and some familiar results concerning the stability of model stellar structures, first advanced at the end of the 19th century and repeated in the most modern textbooks, are less than completely convincing. 3. The standard treatment of relativistic thermodynamics does not allow for a systematic treatment of mixtures, such as the mixture of a perfect gas with radiation. 4. The concept of mass in applications of general relativity to stellar structure is unsatisfactory. It is proposed that a formulation of thermodynamics as an action principle may be a suitable approach to adopt for a new investigation of these matters. We formulate thermodynamics of ideal gases in terms of an action principle and study the interaction between an ideal gas and the photon gas, or heat. The action principle provides a hamiltonian functional, not available in traditional approaches where familiar expressions for the energy have no operative meaning. The usual polytropic atmosphere in an external gravitational field is examined, in order to determine to what extent it is shaped by radiation. It is easy to understand that radiation sustains the atmosphere (prevents cooling), but the temperature profile is largely determined by intrinsic properties of the gas and it is difficult to interpret it as an effect of radiation. This has led some people to question whether the rule of uniform temperature as an absolute condition for equilibrium is valid in the presence of a gravitational field. An experiment that involves a centrifuge and that has wider implications in view of the equivalence principle, is proposed, to ascertain the influence of gravitation on the equilibrium distribution with a very high degree of precision. A new formulation of the concept of radiative equilibrium is proposed. The choice of boundary conditions for radial, stellar stability calculations is clarified with the help of a properly defined, conserved mass distribution.