Thermodynamics of Fluid Elements in the Context of Turbulent Isothermal Self-Gravitating Molecular Clouds

In the present work, we suggest a new approach for studying the equilibrium states of an hydrodynamic isothermal turbulent self-gravitating system as a statistical model for a molecular cloud. The main hypothesis is that the local turbulent motion of the fluid elements is purely chaotic and can be regarded as a perfect gas. Then, the turbulent kinetic energy per fluid element can be substituted for the temperature of the chaotic motion of the fluid elements. Using this, we write down effective formulae for the internal and total the energy and for the first principal of thermodynamics. Then, we obtain expressions for the entropy, the free energy, and the Gibbs potential. Searching for equilibrium states, we explore two possible systems: the canonical ensemble and the grand canonical ensemble. Studying the former, we conclude that there is no extrema for the free energy. Through the latter system, we obtain a minimum of the Gibbs potential when the macro-temperature and pressure of the cloud are equal to those of the surrounding medium. This minimum corresponds to a possible stable local equilibrium state of our system.